PSEUDO-SPECTRAL METHODS APPLIED TO HYDRODYNAMIC AND MAGNETOHYDRODYNAMIC TURBULENCE.

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Universit´ e Libre de Bruxelles Facult´ e des Sciences

PSEUDO-SPECTRAL METHODS APPLIED TO HYDRODYNAMIC AND MAGNETOHYDRODYNAMIC

TURBULENCE.

Olivier Debliquy

Th` ese pr´ esent´ ee en vue de l’obtention du grade de docteur en sciences ` a la facult´ e des sciences de l’universit´ e libre de bruxelles sous la direction du Pr. Carati.

Bruxelles, d´ ecembre 2004.

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“ When I meet God, I am going to ask him two questions: Why relativity ? And why turbulence ? I really believe he will have an answer for the first ”.

W. K. Heisenberg

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Remerciements

Je pensais d’abord écrire mes remerciements en anglais, mais il me paraˆıt plus sincère de les exprimer dans ma langue maternelle, celle de Molière.

Bien que ce travail soit personnel, il se réalise aussi grâce à la présence de plusieurs personnes. Parmi celles-ci, je pense bien sˆ ur tout d’abord à mon directeur de thèse, Daniele Carati. Je lui dois d’avoir acquis la méthode de travail et la rigueur que doivent être celles du chercheur dans sa quête. Durant ces quatre années, le dynamisme et la bonne humeur dont il a fait preuve, ont toujours eu une influence très positive sur moi. De plus, il a pu trouver les mots adéquats dans les moments difficiles. Enfin, malgré un emploi du temps très chargé, il s’est toujours montré très disponible quand cela s’est avéré nécessaire. Pour tout cela, je lui serai éternellement reconnaissant.

Ensuite, je tiens à remercier très amicalement Bernard Knaepen. Malgré nos différents en matière d’ordinateur, cela a été un véritable plaisir de travailler avec lui. A mon arrivée dans le service de Physique Statistique et Plasmas, il m’a pris sous son aile et a témoigné la patience nécessaire pour répondre à mes questions tout au long des années. De même, il a pu bien souvent me conseiller sagement quand j’en avais besoin et je reconnais en lui un véritable ami.

Au cours de l’année 2003, j’ai eu l’immense plaisir de rencontrer le Dr. Mahendra Verma de l’Indian Institute of Technology. Très vite, un contact aisé s’est établi entre nous et a donné naissance à une active collaboration. L’entièreté des résultats du Chapitre VII ont d’ailleurs été obtenus durant sa dernière visite en juin 2004.

Je tiens également à remercier le Pr. Jean Wallenborn sans qui, en réalité, toute cette histoire n’aurait jamais commencé. En effet, c’est grâce à lui que je fus introduit auprès de Daniele alors que je finissais mes études d’ingénieur.

Il va sans dire que je pense aussi `a tous les autres membres du service de Physique

v

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vi Remerciements

Théorique et Mathématique, tant scientifiques qu’administratifs, sans qui le bon fonction- nement et la bonne humeur au sein du groupe ne seraient maintenus. Je n’oublie pas non plus nos chers confrères du Service de Mécanique des Fluides.

Il en va de même pour mes parents à qui j’espère ne pas avoir fait attraper trop de cheveux gris durant ces quatre dernières années.

Le petit (et meilleur) mot de la fin s’adresse à celle à qui je dois plus que de simples remerciements puisqu’elle partage ma vie de tous les jours... J’ai toujours eu beaucoup de chance et celle de t’avoir à mes côtés, mon Amour, en est une encore plus grande.

Ce travail a été supporté financièrement par:

• Le Fonds pour la formation `a la Recherche dans l’Industrie et dans l’Agriculture (FRIA) ;

• La Communaut´e Fran¸caise de Belgique (ARC 02/07-283) ;

• L’association Euratom - Etat Belge. Le contenu de ce travail est la seule responsabilit´e

de l’auteur et ne repr´esente pas n´ecessairement le point de vue de la Commission ou de

ses services.

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Preface

In our everyday life, turbulence is an omnipresent phenomenon and yet remains poorly un- derstood. Air or water flowing around our vehicles, blood moving through the vessels of our bodies, smoke streaming upward from a burning cigarette, motions in the earth’s core are, among many, some examples of this natural process. The random and chaotic nature of turbulence makes it a subject almost impossible to treat from the mathematical point of view. Richard Feynman himself called it “the most important unsolved problem of classical physics” and today there is still no real prospect of a simple analytic theory. Scientists have therefore regarded the numerical simulation as an alternative to compute the relevant prop- erties of turbulent flows. Of course, the continuously growth of available computer power has encouraged this approach and allowed engineers to gain some insights into turbulence. In this context, our thesis aims at developing and using accurate computational methods, namely pseudo-spectral methods, for studying hydrodynamic (1st part) and magnetohydrodynamic (2nd part) turbulence.

In the hydrodynamic part, Chapter I introduces the governing equations of fluid mechan- ics as well as the main issues related to the numerical study of turbulent flows. In particular, the Direct Numerical Simulations (DNS) of turbulence, in which accurate numerical solutions of the Navier-Stokes equations are obtained, are shown to be limited to moderately turbu- lent flows. Chapter II introduces the Large Eddy Simulation (LES) technique which aims at simulating highly turbulent flows and which is based on a separation of scales. In practice, it consists of simulating the large – resolved – scales of the flow explicitly while modelling the small – unresolved – scales. Two different approaches for modelling the kinetic energy of the unresolved scales are proposed and their respective advantages and drawbacks are discussed.

Chapter III is devoted the study of the mixing layer using both DNS and LES. It consists of an inhomogeneous turbulent flow which has been studied experimentally and for which well-documented measurements are available. A highly accurate DNS mimicking the same experiment has been produced. It allows to study the inhomogeneity and anisotropy prop- erties of this flow. Also, LES of the same flow, using different models, have been evaluated.

vii

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viii Preface

In Chapter IV, we explore a pseudo-spectral method to investigate turbulence in a pipe. In this case, the method has to take into account two additional difficulties: i) the presence of the boundary and ii) the axis singularity. We detail how to circumvent these issues.

The second part of the thesis is devoted to magnetohydrodynamic (MHD) turbulence. It

concerns phenomena where electrically conducting flows interact with electromagnetism and

for which governing equations are derived in Chapter V. In Chapter VI, a detailed analysis

of the energy transfers between the magnetic and velocity fields is performed thanks to a

high resolution database of homogeneous MHD turbulence. It provides some insights to

understand the physics of the nonlinear interactions and is also a valuable diagnostic in the

framework of LES modelling. Finally, the inhomogeneous configuration studied in Chapter III

has been extended to MHD. Several statistics related to the kinetic and magnetic energies

are measured and LES of this flow are performed and presented in Chapter VII.

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Part I

Hydrodynamic Turbulence

1

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2

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Chapter 1

Numerical analysis of turbulent flows

1.1 Governing equations of fluids mechanics

1.1.1 Continuum hypothesis

In most situations, fluids are considered as continuous media, though they are composed of small discrete molecules in large number. This is known as the continuum hypothesis.

The idea to separate the macroscopic motion from the molecular motion is meaningful since they are characterized by length and time scales extremely different. Let’s take air as an example (Pope, 2000, Chap.2): the average spacing between molecules is of the order of 10 ⁻⁹ m, the mean free path, λ, is typically 10 ⁻⁸ m, and the mean time between successive collisions of a molecule is 10 ⁻¹⁰ s. In comparison, the smallest geometric length scales in a flow, `, is seldom less than 10 ⁻⁴ m, so that even a velocity up to 100 m/s would yield a timescale not smaller than 10 ⁻⁶ s. Thus, the flow scales exceed the molecular scales by several orders of magnitude.

Because of this separation of scales, the continuum flow properties can be thought to as the molecular properties averaged over a spherical volume V . This sphere is centered on the point x and has a radius ` ^∗ chosen such that: λ ¿ ` ^∗ ¿ `. Then at time t, the fluid’s density ρ(x, t) is defined as the mass of molecules in the sphere divided by V .

Identically, the fluid’s velocity u(x, t) is the average velocity of the molecules within V . Let’s remark that, due to the separation of scales, the choice of ` ^∗ affects weakly the continuum properties.

Once the continuum hypothesis is invoked to define continuous fields, we can leave behind all notions related to the molecular nature of the fluid so that molecular scales are not relevant anymore.

3

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4 1.1. Governing equations of fluids mechanics

1.1.2 Lagrangian and Eulerian derivatives

The description of fluid motion may be founded on two different approaches. In the La- grangian description, one considers the dynamic of fluid particles - a continuum concept.

By definition, a fluid particle is a point that moves with the local fluid velocity. In this representation, one describes the fluid particle trajectories as a function of space and time.

The equations of motion are thus written for a specific discrete fluid particle (Candel, 1995, Chap.2).

On the contrary, the Eulerian representation adopts a different point of view. One con- siders indeed the time variations of continuous fluid properties at fixed positions, through which a succession of fluid particles will move.

Therefore, the rate of change will be different as they are expressed in the Lagrangian or Eulerian description. To illustrate that matters, let’s consider a scalar function of position and time, f(x, t). The rate of change of f associated with a given element of fluid will be written here as D(f )/Dt (Lagrangian derivative). It represents the rate of change of f of a fluid lump as it moves around. This should not be confused with ∂f /∂t, which is, of course, the rate of change of f at a fixed point in space (Eulerian derivative) (Davidson, 2001, Chap.3).

An expression for D(f )/Dt may be obtained as follows. Indeed, assuming a Cartesian system for simplicity, we have for the function f (x, t):

δf = ∂f

∂t δt + ∂f

∂x δx + ∂f

∂y δy + ∂f

∂z δz . (1.1)

If we are interested in the change in f following a fluid particle, then δx i = u i δt (i = x, y, z), so that:

D f

Dt = ∂f

∂t + u _x ∂f

∂x + u _y ∂f

∂y + u _z ∂f

∂z (1.2)

= ∂f

∂t + (u · ∇ ) f (1.3)

or, in Einstein’s notations (i.e., summation over repeated indices is assumed), D f

Dt = ∂f

∂t + u _i ∂f

∂x _i . (1.4)

The derivative D( · )/Dt is also called the material derivative or substantial derivative.

Due to its ease of use to solve practical problems, the Eulerian representation is the most

common in fluids mechanics and will be adopted in the following chapters.

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1.1. Governing equations of fluids mechanics 5

1.1.3 Mass conservation

The mass conservation or continuity equation can be derived using control volumes or material volumes ¹ and is given by:

∂ρ

∂t + ∇ · (ρ u) = 0 (1.5)

Using the vector identity:

∇ · (ρ u) = ρ ∇ · u + u · ∇ ρ (1.6)

and substituting this relation in (1.5), one obtains:

∂ρ

∂t + ρ ∇ · u + u · ∇ ρ = 0 (1.7)

If we consider constant-density flows (i.e., flows in which ρ is independent both of x and t so that eq. 1.7 vanishes), the evolution equation eq. (1.5) degenerates to the kinematic condition that the velocity field be solenoidal or divergence-free:

∇ · u = 0 or ∂u _i

∂x _i = 0 (1.8)

Sometimes, the last expression is referred to as the incompressibility condition. This condition is also valid for non constant density flow as long as δρ/ρ ¿ δu i /u i .

1.1.4 Momentum conservation

The momentum equation, based on Newton’s second law, relates the fluid particule acceler- ation D(u i )/Dt to all forces experienced by the fluid. In most (classical) situations, these forces are separated in two kinds:

• the volume forces or body forces. The body force of interest are gravity (whose component is noted g i ) and other external forces represented here by f i . Denoting ψ as the gravitational potential (i.e, the potential per unit mass associated with gravity), the body force per unit mass is

g _i = − ∂ψ

∂x _i (1.9)

• the surface forces acting through the surface of a fluid particle. The surface forces are described by the stress tensor T _ij - which is symmetric, i.e., T _ij = T _{j i} . It is convenient to decompose this tensor into two contributions: the pressure, P , and the viscous stress tensor, σ _ij :

T _ij = − P δ _ij + σ _ij (1.10)

1 These approaches should be familiar to the reader and will not be developed here.

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6 1.1. Governing equations of fluids mechanics

The pressure acts isotropically and depends only of the thermodynamic condition of the fluid while the viscous stress tensor is related to its distorsion (Candel, 1995, Chap.5).

These forces cause the fluid to accelerate according to the momentum equation ρ D(u _i )/Dt = P F _i and using the material derivative defined in (1.4), we obtain:

ρ µ ∂u i

∂t + u j ∂u i

∂x _j

¶

= ρ (g i + f i ) + ∂ T ij

∂x _j . (1.11)

Substituting (1.9) and (1.10), the expression becomes ρ

µ ∂u _i

∂t + u _j ∂u _i

∂x _j

¶

= − ρ ∂ψ

∂x _i − ∂P

∂x _i + ρ f _i + ∂ σ _ij

∂x _j . (1.12)

Further, defining the modified pressure, P , by

P = P + ρ ψ , (1.13)

this equation simplifies to ρ

µ ∂u _i

∂t + u _j ∂u _i

∂x _j

¶

= − ∂ P

∂x _i + ρ f _i + ∂ σ _ij

∂x _j . (1.14)

We now specialize the momentum equation to the case of Newtonian fluids. In that particular case, the viscous stress tensor, σ _ij , is a linear and isotropic function of the shear stress tensor, S _ij . Using simple arguments, one can show that this relation is given by (Landau

& Lifshitz, 1987, Chap.2):

σ _ij = 2µ S _ij + µ

ζ − 2µ 3

¶

S _kk δ _ij , (1.15)

where µ and ζ are respectively the dynamic and bulk viscosity. The shear stress tensor is given by the following linear combination of the velocity derivatives:

S _ij = 1 2

µ ∂u _i

∂x j

+ ∂u _j

∂x i

¶

, (1.16)

so that replacing (1.15) and (1.16) in the momentum equation leads to the Navier-Stokes equation :

ρ µ ∂u i

∂t + u _j ∂u i

∂x _j

¶

= − ∂ P

∂x _i + ρ f _i + µ ∂ ² u i

∂x ² _i + ³ ζ + µ

3 ´ ∂ ² u j

∂x _i ∂x _j , (1.17) or, in vector notation,

ρ µ ∂u

∂t + (u · ∇ ) u

¶

= −∇P + ρ f + µ ∇ ² u + ³ ζ + µ

3 ´

∇ ( ∇ · u) . (1.18)

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1.2. Turbulence 7

1.1.5 The limit of incompressible flow

In this thesis, we will assume that the fluid velocity is much smaller than the speed of sound so that compressibility effects can be neglected. This corresponds to the situation already discussed in Section 1.1.3 where the density ρ is constant so that the continuity equation reduces to eq. (1.8):

∇ · u = 0 (1.19)

This simplifies strongly the description of the flow and the Navier-Stokes equation (1.18) reduces to:

∂u

∂t + (u · ∇ ) u = −∇ p + f + ν ∇ ² u , (1.20) where p denotes the pressure divided by the mass density and ν = µ/ρ is the kinematic viscosity. The value of p can be determined by taking the divergence of the last equation which leads to the following Poisson equation:

∇ ² p = −∇ · £

(u · ∇ ) u ¤

. (1.21)

taking into account the condition (1.19). Thus, for incompressible flows, pressure is not related with density and temperature by an equation of state as we may be accustomed to thinking of pressure as a thermodynamic variable. Indeed, the pressure is uniquely determined by the velocity field through eq. (1.21). Therefore, satisfying eq. (1.21) is a necessary and sufficient condition for a solenoidal velocity field to remain solenoidal (Pope, 2000, Chap.2).

Moreover, one could add that the pressure appears in the Navier-Stokes equations through its gradient only. Hence, any constant pressure does not modify the right-hand side of eq. (1.20).

1.2 Turbulence

1.2.1 Reynolds number

In nature, fluids may experience two very different regimes: laminar and turbulent. The flow is said to be laminar when the fluid motion is steady, smooth, and presents certain symmetries. On the contrary, in a turbulent regime, the flow is unsteady, irregular, seemingly random and chaotic: the velocity is a rapid fluctuating function of time.

These two regimes may be observed while looking at the trickle of water coming out of a

tap. When the velocity is slow, the fluid is regular and laminar. When increasing the flow

rate, trickles begin to oscillate around a mean position and the fluid becomes irregular and

turbulent.

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8 1.2. Turbulence

Figure 1.1: Instability of smoke from a cigarette (Perry & Lim, 1978). Picture from Van Dyke (2002).

Another well known situation is the natural convection of the smoke from a cigarette.

The smoke streaks are first laminar. Then an instability appears and precedes the transition to turbulence. Finally, the flow is totally unordered and chaotic which is characteristic of turbulence (see Fig. 1.1).

In each phenomena described above, three important parameters appear: i) the charac- teristic length of the flow L ; ii) the characteristic velocity of the flow U and iii) the viscosity of the fluid ν. These quantities can be combined into a dimensionless parameter, the Reynolds number :

Re = U L

ν . (1.22)

This number depends on both the external conditions of the flow through U and L and the physical nature of the flow through ν. One observes that for small values of Re, the fluid remains laminar. The experimental data seem to indicate that, when the Reynolds number increases, it eventually reaches a critical value Re _cr (the critical Reynolds number) beyond which the flow is unstable with respect to small disturbances. This value is of course not universal but changes for each type of flow since it depends strongly of the shape and dimensions of the body, the inflow condition, ... Although, for systems in everyday life, one can estimate its value to be about 10 ³ . For Re much bigger than this critical value, one observes fully developed turbulence.

This can be understood by comparing the non-linear term and the viscous term from the

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1.2. Turbulence 9

Navier-Stokes equation (1.20). The order of magnitude for the non-linear term should be of U ² /L. This term is responsible of the convection and has a tendency to destabilize the flow. Indeed, fluctuations are convected by this term and contaminate a fraction of the fluid proportional to the large scale velocity. Similarly, the amplitude of the viscous term is of the order of νU/L ² . Its effect is to eliminate the instabilities and it is thus a stabilizing term.

Therefore the Reynolds number (which is simply the ratio of these two terms) indicates how important are the non-linearities compared to the viscous term.

Finally, one has to mention that the characteristic velocity U and length scale L that enter formula (1.22) are not, contrary to the viscosity, intrinsic quantities. They may be chosen differently according to the context. It is thus important when reporting results on turbulent flows, to specify carefully which definitions of these parameters is adopted.

1.2.2 Energy cascade and Kolmogorov’s theory

[Due to its clarity, the book ’Turbulent Flows’ by Pope has served as a guide to this section].

In examining previous examples, we have observed that the turbulent motions range in size from the width of the flow to much smaller scales so that turbulence can be considered to be composed of eddies of different sizes. In 1922, Richardson has introduced the idea that the kinetic energy entering the turbulence at large scales (through the production mechanism) is transferred to smaller and smaller scales until, at the smallest scales, the energy is dissipated by viscous action. This phenomena is called the energy cascade. This picture is important notably because it places the dissipation at the end of a sequence of processes. The rate of dissipation ε is determined therefore by the first process in the sequence which is the transfer of energy from the largest eddies.

The largest eddies are characterized by a lengthscale ` ₀ (comparable to the flow scale L) and a velocity u ₀ (comparable to U ). The Reynolds number of these eddies Re ₀ = u ₀ ` ₀ /ν is therefore large (i.e. comparable to Re) so the direct effects of viscosity at this level are negligibly small. Their energy is of order of u ² ₀ and have a timescale τ ₀ = ` ₀ /u ₀ so that the rate of transfer of energy can be supposed to scale as u ² ₀ /τ ₀ = u ³ ₀ /` ₀ . Consequently, this picture of the cascade indicates that ε scales as u ³ ₀ /` ₀ , independent of ν.

Although this description furnished by Richardson was the basis of the energy cascade theory, Kolmogorov (1941b) refined some ideas and, in particular, he identified the smallest scales of turbulence that now bear his name. The theory advanced by Kolmogorov is stated in the form of three hypotheses.

The first hypothesis concerns the isotropy of the small-scale motions. In general, large

eddies are anisotropic. According to Kolomogorov, this anisotropy of the large scales is lost in

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10 1.2. Turbulence

the chaotic scale-reduction process so that the small scale motions are statistically isotropic.

Just as the directional information of the large scales is lost as the energy cascades down to the small scales, Kolmogorov argued that all information about the geometry of the large- eddies - determined by the mean flow field and boundary conditions - is also lost. As a consequence, the statistics of the small-scale motions are in a sense universal - similar for every high Reynolds number flow. In the energy cascade (for ` ¿ ` 0 ), the two dominant processes are the transfer of energy to successively smaller scales, and viscous dissipation.

A plausible hypothesis, then, is that important parameters of “this universal state” are the rate at which the small scales receive energy from the large scales, i.e. ε, and the kinematic viscosity ν. This is known as the Kolmogorov’s second hypothesis. Given the two parameters ε and ν, there are (to within multiplicative constants) unique length, velocity and time scales that can be formed. These are the Kolmogorov scales:

` _η = (ν ³ /ε) ^1/4 , (1.23)

u _η = (εν) ^1/4 , (1.24)

τ _η = (ν/ε) ^1/2 . (1.25)

Two identities stemming from these definitions clearly indicate that the Komogorov scales characterize the very smallest dissipative eddies. First the Reynolds number based on the Kolmogorov scales is unity, i.e., u _η ` _η /ν = 1, which is consistent with the notion that the cascade proceeds to smaller and smaller scales until the Reynolds number is small enough for dissipation to be effective. Second, the dissipation rate is given by

ε = ν(u _η /` _η ) ² = ν/τ _η ² . (1.26) As it will be shown in Chapter 2 and 3, (u _η /` _η ) = 1/τ _η provides a consistent characterization of the velocity gradients of the dissipative eddies.

Now that we have an estimate for the characteristic scales of the largest and smallest eddies, let’s analyze their ratios:

` ₀ /` _η ∼ Re ^3/4 , (1.27)

u ₀ /u _η ∼ Re ^1/4 , (1.28)

τ ₀ /τ _η ∼ Re ^1/2 , (1.29)

taking into account that the rate of transfer of energy is given by u ³ ₀ /` 0 . Evidently, the ratio

` 0 /` η increases with increasing Re. As a consequence, at sufficiently high Reynolds number,

there is a range of scales ` that are very small compared to ` 0 and yet very large compared

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1.2. Turbulence 11

PSfrag replacements

` _η ` ` ₀

Transfer of energy to successively smaller scales

Dissipation Dissipation

Inertial Energy-containing

range range

range

Production

Figure 1.2: Schematic description of the energy cascade.

to ` _η , i.e. ` _η ¿ ` ¿ ` ₀ . Since eddies in this range are much bigger than the dissipative eddies, it may be supposed that their Reynolds number u`/ν is large, and consequently that their motion is little affected by viscosity. That’s why this interval of scales is also qualified as the inertial range (their motions are dominated by inertial effects - viscous effects being negligible). Hence, following from this and from the previous assumption, the third hypothesis states: “ at high Reynolds number, the statistics of the motions of scale ` in the range

` _η ¿ ` ¿ ` ₀ have a universal form that is uniquely determined by ε and is independent of ν ”. However, lengthscales, velocity scales and timescales cannot be formed from ε alone but require a parameter that has the dimension of a length. Thus, the most reasonable choice for the lengthscale is thus to take the eddy size `, so that the two others characteristic scales are those formed from ε and `:

u _` = (ε`) ^1/3 , (1.30)

τ _` = (` ² /ε) ^1/3 . (1.31)

This is consistent with the idea that the velocity scales and timescales decrease as ` decreases.

A schematic diagram of the energy cascade is given on Fig 1.2.

1.2.3 The energy spectrum

It remains to determine how the kinetic energy is distributed among eddies of different sizes.

This is easily done for homogeneous turbulence so that the kinetic energy can be written in terms of its spectral density of energy E(k):

E = Z

dk E (k) . (1.32)

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12 1.3. Simulations of turbulent flows

Introducing this definition is convenient since it allows to associate an energy to the structures with a characteristic length scale ` ∼ k ⁻¹ . It follows from the third Kolmogorov’s hypothesis that, in the inertial range, the energy should scale as u ² _` . Taking into account that E(k) has the dimension of an energy per unit length and that u _` is given by eq. (1.30), it is easy to show that:

E(k) ∼ ε ^2/3 k ^−5/3 . (1.33)

In the “turbulence” community, this famous formula is known as the Kolmogorov -5/3 (or k ^−5/3 ) spectrum. In order to illustrate this universal law, we have plotted on Fig. 1.3 a compilation of measurements of one dimensional longitudinal velocity spectra, E _xx (k _x ).

Indeed, the same arguments as in the case with E(k) could be used to show that the quantity E _xx (k _x ) is a universal function of k _x , at sufficiently high Reynolds number. All data are taken from Saddoughi & Veeravalli (1994) and have been non-dimensionalized using the viscosity and the dissipation. It is remarkable how well all the measurements lie on a single curve for k _x η > 0.1 and validate in that sens the second Kolmogorov hypothesis. The high Reynolds number data exhibit power-law behaviour for k _x η < 0.1, the extent of the power-law region generally increasing with Re. This is consistent with the picture given by the third hypothesis and the formula (1.33).

However, as we will see in the next section, the wide range of scales present in high Reynolds number flows makes them difficult to approach by numerical tools and very often imposes modelling efforts.

1.3 Simulations of turbulent flows

Following the previous section, the random and chaotic nature of turbulence makes it a subject almost impossible to treat from the mathematical point of view. So far, only a few fundamental results have been obtained and there are no prospects of a simple analytic theory. In this context, scientists have regarded the numerical simulation as an alternative to achieve the objective of calculating the relevant properties of turbulent flows.

However, one could argue that a numerical description is not fully adapted for fluids since

they are considered as continuous media (see Section 1.1.1). In that view, describing a fluid

would theoretically require the knowledge of its velocity at each point of the domain, at all

times. This infinite amount of information is of course unmanageable on a computer which

is limited to the the amount of data that can be stored in its memory. Fortunately, this

scenario is a little bit too pessimistic. Indeed, as shown in the previous section, the second

Kolmogorov’s hypothesis assumes there exist scales beyond which variations of velocity and

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1.3. Simulations of turbulent flows 13

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PSfrag replacements

k _x η E xx ( k x ) / ( εν 5 ) 1 / 4

Figure 1.3: Measurements of one dimensional longitudinal velocity spectra. The experimental

data are taken from Saddoughi & Veeravalli (1994) where references to the various exper-

iments are given. For each experiment the final number in the legend is the value of the

Reynolds number based on Taylor micro-scale R _λ (see Chapter 3).

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14 1.3. Simulations of turbulent flows

time can be neglected: they are the Kolmogorov scales. Therefore it seems reasonable to consider the lengthscale (1.23) and the timescale (1.25) as the spatial and temporal limit of discretization for a numerical experiment in fluid turbulence. This choice is traditionally adopted in Direct Numerical Simulations.

1.3.1 Direct Numerical Simulation

Thee Direct Numerical Simulation (or DNS) is a numerical experiment in which all the macroscopic hydrodynamic lengthscales and timescales are resolved, i.e. they take account of the dynamics of all eddies present in the turbulent motion. Thus, it can be considered as an accurate numerical description of the flow.

Let us evaluate what it represents in terms of computational cost. Since all lengthscales are represented, we have to consider a discretization resolving both the largest scale determined by ` ₀ and the smallest scale ` _η . Hence, for a three dimensional geometry, the number of points N , required for describing the fluid is given in terms of the ratio between these two lengthscales:

N ≈ µ ` ₀

` η

¶ 3

≈ Re ^9/4 , (1.34)

using the relation (1.27).

As for the number of time steps, a similar argument could be used. The total time of the simulation has to be long enough so that the fluid can move across one typical length of the system. It can be estimated by the timescale of the largest eddies: τ ₀ = ` ₀ /u ₀ . On the other side, the time step of the temporal integration has to be small enough to allow the convection of a small structure on a distance ` _η by the large scale velocity u ₀ :

τ _η ^c ≈ ` _η u ₀ ≈ ` _η

` ₀

u ₀ ≈ Re ^−3/4 τ ₀ . (1.35)

Thus, the total number of time steps M for describing an experiment is given by:

M ≈ τ ₀

τ _η ^c ≈ Re ^3/4 . (1.36)

As a consequence, the computational cost of a DNS increases according to N × M ≈

Re ³ . Let’s quantify this value by considering the example of a man walking: the typical

lengthscale L is of the order of 1 m and its velocity U is typically 1 m /s. The kinematic

viscosity of the air is of the order of 10 ⁻⁵ m ² /s so that the Reynolds number for a man walking

is Re ≈ 10 ⁵ . Therefore a numerical simulation would require N ≈ 10 ¹¹ number of points

and, for each point, about 10 real values have to be stored in the computer memory. That

represents a total amount of 10 ¹² real values or 4000 Gbytes of memory ! Moreover, a program

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1.3. Simulations of turbulent flows 15

for simulating a turbulent flow contains about one thousand instructions and each instruction would have to be executed N × M ≈ 10 ¹⁵ times. Hence, in terms of computational time, the total number of floating point operations (or flops) would be of the order of 10 ¹⁸ . If one could use the biggest computer facility on earth, the Earth simulator, capable of doing 10 ¹³ flops per second, simulating one second of the walk of a man would take approximately one day.

Following the same logic, simulating a civil aircraft during a flight with a characteristic speed of 100 m /s and a characteristic length of 100 m would require approximately 10 ¹² Gbytes of memory and 10 ¹² years of computation time ! Through these examples, one understands directly that DNS is restricted to flows with low-to-moderate Reynolds number.

This difficulty has therefore motivated scientists to develop alternate numerical techniques among which the Reynolds Averaged Navier-Stokes (or RANS).

1.3.2 Reynolds Averaged Navier-Stokes

The basic principle of a RANS is to decompose the velocity field u _i into its averaged value U _i = h u _i i and its fluctuating part u ⁰ _i = u _i − U _i and to compute the evolution of the averaged value only. The averaging operator usually refers to an ensemble averaging over a large number of equivalent experiments. It is also sometimes considered as a time averaging when the flow is statistically stationary. The decomposition u _i = U _i + u ⁰ _i seems appropriate since the average velocity U _i contrary to the instantaneous velocity u _i , is expected to be a reproducible and slowly varying quantity. Its numerical description is thus much cheaper than the description reflected by DNS.

The equation for U _i is obtained by applying the averaging operator h . . . i to the Navier- Stokes equation ² (using the same kind of decomposition for the pressure p = P + p ⁰ and the forcing term f _i = F _i + f _i ⁰ ):

∂ _t U _i + ∂ _j U _j U _i = − ∂ _i P + F _i + ν ∇ ² U _i + ∂ _j R _ij , (1.37) where the additional term R _ij , referred to as the Reynolds Stress tensor, is nothing else than:

R _ij = h u _i ih u _j i − h u _i u _j i = h u ⁰ _i u ⁰ _j i . (1.38) In the last expression, we use the property:

hh u _i ih u _j ii = hh u _i i u _j i = h u _i ih u _j i . (1.39) An important consequence coming of the identity (1.38) is that the equation describing U _i is not closed. Indeed, R _ij cannot be evaluated from the knowledge of U _i only and will

2 Here and in the next chapters, we will use the simplified notation: ∂ t = ∂/∂t and ∂ i = ∂/∂x i .

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16 1.3. Simulations of turbulent flows

therefore require a modelling at some point. At this level, we are faced with two options.

The first possibility is to model R ij in terms of the averaged velocity U i . More precisely, models are often expressed in terms of the gradient of the averaged velocity and some other statistical quantities like the turbulent kinetic energy k = h u ⁰ _i u ⁰ _i i /2, and the dissipation rate ε for which model transport equations are solved. This particular choice is better known as the k − ε model. The second strategy is to derive an equation for the Reynolds stress obtained by writing first the equations for the fluctuations u ⁰ _i :

∂ _t u ⁰ _i + ∂ _k U _k u ⁰ _i + ∂ _k u ⁰ _k U _i = − ∂ _i p ⁰ + f _i ⁰ + ν ∇ ² u ⁰ _i − ∂ _k (u ⁰ _i u ⁰ _k − R _ik ) , (1.40) and multiplying them by u ⁰ _j . Then a second equation for u ⁰ _j multiplied by u ⁰ _i is added to the first one and the Reynolds stress equations are just obtained by taking the averaged of the result. Of course, this set of equations are again not closed and must be modelled. It will not be detailed here. Further information are given in Pope (2000) [Chap.11].

One realizes immediately that, compared to DNS which provides a complete description of the flow, RANS deals only with its averaged properties. Therefore, it limits severely the kind of statistics that can be measured. Moreover, RANS rely strongly on the models used to take into account the fluctuations u ⁰ _i of the velocity field. Indeed, the modelling has to correctly describe most of the complexity of turbulence which makes this approach highly dependent on the quality of the models. However, its simplicity makes this numerical tool of common use for industrial applications.

The difficulties encountered in both DNS and RANS have prompted the development of an intermediate approach that has received many attention since a decade: Large-Eddy simulation (or LES).

1.3.3 Large Eddy Simulation

In LES, only the large scales of the flow are simulated while the small scales are modelled.

The idea to separate large and small scales is mainly justified by Kolmogorov’s theory. As we know, larger-scale motions depend strongly of the external conditions of the flow through the geometry, the inflow, the boundary condition, ... ; they are thus expected to be anisotropic and contain most of the energy as confirmed by Fig 1.3. On the contrary, smaller scales are responsible of the dissipation; they are less energetic and, because of their isotropic nature, they present a more universal behaviour. In this prospect, computing large scales explicitly and modelling small scales seem an appropriate strategy.

From the mathematical point of view, the scale separation is introduced by a filtering

operator, F ^∆ . This spatial filter is often seen as the convolution between the velocity u i and

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1.3. Simulations of turbulent flows 17

gihjkilj*mLn1oAp qRkj'risisis

0 2 4 6 8 10

-1 0 1 2 3 4

t

PSfrag replacements

∆

x u , u

u − u

Figure 1.4: A sample of the velocity field in one dimension u(x) ( ) and the corresponding filtered field u(x) ( ), using the Gaussian filter with ∆ ≈ 0.35. The residual field u − u is represented by the dotted line.

a kernel K:

u _i (x) = F ^∆ £ u _i (y) ¤

= Z

K (x − y, ∆) u _i (y) dy , (1.41) where ∆ represents the filter width. In this work, we will restrict our studies to homogeneous filtering operators, i.e. the same operator is used in the whole domain. The major advantage of homogeneous filters is that they commute with the spatial derivatives:

£ ∂ _i , F ^∆ ¤

= 0 . (1.42)

The three traditional kernels used in the LES literature define the Gaussian filter, the top-hat filter and the (sharp Fourier) cutoff filter. In one dimension, they are given by the following expressions:

K _Gaussian (x, ∆) = r 2

∆ ² exp µ

− 2πx ²

∆ ²

¶

, (1.43)

K _top-hat (x, ∆) = 1

∆ H µ ∆

2 + x

¶ H

µ ∆ 2 − x

¶

, (1.44)

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18 1.3. Simulations of turbulent flows

K _cutoff (x, ∆) = 1

πx sin ³ πx

∆

´

, (1.45)

where H(x) is the Heaviside function. In order to illustrate the effect of the filter, let’s take a look at Fig. 1.4 where a Gaussian filter has been applied to a random function u(x).

One observes that the filtered field u(x) follows the general trends of the original field u(x).

However, the highly oscillating parts of the field have been removed. This is confirmed by looking at the residual field u − u. The high fluctuations are therefore characteristic of the small scales while the large scales characterize the global behaviour of the field.

Applying this filtering operator F ^∆ to the Navier-Stokes equations leads to the LES equations:

∂ _t u _i + ∂ _j (u _i u _j ) = − ∂ _i p + f _i + ν ∇ ² u _i − ∂ _j τ _ij , (1.46) where τ _ij = u _i u _i − u _i u _i is an unknown term (referred to as subgrid-scale stress tensor ) ³ since it is not expressed in terms of the filtered velocity u _i only and will therefore require some modelling effort. Not surprisingly, the structure of these equations is similar to the RANS equations, though the meaning of the unknown term has an important difference with the Reynolds stress R _ij . In the LES case, it represents the effects of the filtered scales u _i − u _i and not the effect of the complete turbulent fluctuations u ⁰ _i like in RANS. The modelling of τ _ij , though very important in LES, is thus expected to have a smaller impact on the prediction than the modelling of R _ij . The global effect of τ _ij is expected to remove energy from the resolved scales u _i . In the most simple approach, scientists have thus proposed to use models based on the eddy viscosity concept:

τ _ij ≈ − 2 ν _e S _ij (1.47)

where S _ij = (∂ _i u _j +∂ _j u _i )/2 is the filtered rate of strain tensor and ν _e is the eddy viscosity. Due to the phenomenology, it is normal that this model finds similarities with the formula (1.15) for the viscous stress tensor. Concerning the eddy viscosity ν _e , there exist several forms among which the most famous is the Smagorinsky model (1963) described with some others in the next chapters.

Since its position lies at the intermediate level between DNS and RANS, LES are cheaper in terms of computational cost than DNS but remain in many cases too expensive to compute real life flows. It is not long ago that complex flows can be simulated using this numerical tool which is still in development. Its use for industrial applications is one of the major challenges in the numerical study of turbulence. On the other side, modelling only small scales and

3 Although this nomenclature is not accepted unanimously in the LES community.

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1.4. Discretization methods 19

simulating directly large scales set LES at a higher level of accuracy than RANS since only part of the turbulence complexity is modelled. Results are thus usually more robust than those obtained in RANS, though they still depend of the quality of the models.

In this work, we will focus our numerical study on the use of DNS and LES to reproduce turbulent flows characteristics in simple geometries.

1.4 Discretization methods

After selecting the level of accuracy to describe numerically the turbulence, one has to choose a suitable discretization method, i.e. a method for approximating the derivatives present in the Navier-Stokes equations. Indeed, as already mentioned, the computer imposes us to evaluate these equations at a finite set of points in space and time though Navier-Stokes equations describe a continuous phenomenon. In that prospect, one has to give another representation to the derivative, which is a continuous mathematical concept. There are many approaches and the most important are: finite difference, finite volume, finite elements and spectral methods.

1.4.1 Finite differences

This is the oldest method for numerical solution of partial differential equations. It consists to cover the solution domain by a grid forming a set of grid points. At each of the points, the derivative is approximated in terms of the nodal values of the function. The result is one algebraic equation per grid node, which depends on the variable value at a certain number of neighbours nodes. Taylor series expansion or polynomial fitting is used to obtain approximations to the first and second derivatives. Sometimes, it can be also necessary to obtain variable values at locations other than grid points so that interpolation is used.

Although it could be applied to any grid type, the finite difference method has been especially applied to structured grids for which it has shown to be very simple and effective. On the other side, these methods are not very successful to conserve quantities such as energy, ...

unless special care is taken. Moreover, it is restricted to simple geometries which makes it disadvantageous for complex flows.

1.4.2 Finite volumes

Contrary to finite differences, the finite volumes method uses integral forms (i.e. the equations

are expressed in terms of integrals on the domain) of the mass and momentum conservation

equations. To that end, one decomposes the domain into a finite number of contiguous

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20 1.4. Discretization methods

control volumes where the conservation equations are enforced. The centroid of each volumes constitutes the point, or node, at which the variable values are to be calculated. In order to express variable values at the control volume surface, one uses interpolation in terms of these nodal points. As for the surface and volume integrals, they are evaluated by quadrature formulae. This results in an algebraic equation for each control volume, in which a number of neighbour nodal values appear.

The main advantage of this method is its ability to adapt to complex geometries because the control volume boundaries might take any shape and need not to be related to a par- ticular coordinate system. Moreover, the use of integral forms guarantees the method to be conservative as long as surface integrals are the same for the control volumes sharing the boundary.

Its ease of use and understanding makes this method very popular in engineering compu- tations. Nevertheless, implementing schemes of order higher than second can be a hard task for three dimensional problems.

1.4.3 Finite elements

Very similar to the finite volumes method, finite elements breaks the geometry into a set of unstructured pieces; in two dimensions, these pieces are usually triangles or quadrilaterals while, in three dimensions, tetrahedra or hexahedra are of most common use. However, this method differs from finite volumes in the way to express the equations.

In each cell, the solution is approximated by a (linear or higher order) shape function that has to be chosen to satisfy continuity across element boundaries and expressed in terms of the values at the corners of the cell. Then, one substitutes this approximation into the weighted integral form of the conservation equations. The equations to be solved are derived by requiring the derivative of the integral with respect to each nodal value to be zero; this ensures the solution to be of minimum residual and leads to a set of non-linear algebraic equations.

The major feature of finite elements is their ability to deal with complex geometries

since the grids can easily be refined where it is necessary. On the other hand, the use of

unstructured grids leads to matrices of the linearized equations that are not as regular as

those for structured meshes: their algebraic manipulation by numerical methods is thus less

efficient. This constitutes the main drawback of this method.

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1.4. Discretization methods 21

1.4.4 Spectral methods

Although previous methods seem more appropriate for computational fluid dynamics, their relative low accuracy on one side and the progress made on computer capabilities on the other side have prompted the revival of spectral methods in the 1970’s. The prototype of these methods is the well-known Fourier method which consists of representating the solution as a truncated series expansion; the unknowns being the expansion coefficients. The Fourier basis is appropriate for periodic problems. For non periodic problems, there exist other bases such as Chebyshev or Legendre polynomials but other functions could be considered according to the problem.

One of the major motivation for using spectral methods has been the development of an efficient algorithm to compute the Fourier transform which is known as the Fast Fourier Transform (or FFT) algorithm. These improvements were fundamental for a fast calculation of the nonlinear terms through the pseudospectral technique. This method consists in doing the differentiation in the spectral space (i.e., the space of the expansion coefficients) while the products are performed in the physical space, the connection between both spaces being done through the FFT algorithm.

The main property of spectral methods is their fast rate of convergence, which is exponen- tial in the case of Fourier or Chebyshev series. In previous methods, the level of accuracy is often limited to second or fourth order. Therefore this constitutes a serious advantage compared to other methods.

However, spectral methods are not adapted for complex geometries. Indeed, any change in geometry or boundary conditions requires a profound modification of the algorithm, making spectral methods difficult to extend.

These last two remarks leads to the conclusion that spectral methods are ideally suited

for simulations of turbulence in geometrically simple domains since the discretization error

will be almost negligible. This is the precise context of our researches and this explains why

we have opted for these methods. Some of the more fundamental aspects about this technic

will be addressed in the next chapters.

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22 1.4. Discretization methods

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Chapter 2

Modelling aspects of k

2.1 Notations and definitions

As developed in Chapter 1, the LES equations for incompressible flows can be written in the following form:

∂ _t u _i + ∂ _j (u _j u _i ) = − ∂ _i p + ν ∇ ² u _i − ∂ _j τ _ij , (2.1) where u _i is the LES field, p is the pressure divided by the density and ν the kinematic viscosity ¹ . Like any second-order symmetric tensor, the unknown subgrid-scale stress tensor τ _ij may be decomposed into an isotropic part and a traceless part:

τ _ij = 2

3 k δ _ij + τ ^∗ _ij , (2.2)

where τ ^∗ _ij = τ _ij − ¹ ₃ τ _ll δ _ij and

k = 1

2 τ _ll = 1 2

¡ u _l u _l − u _l u _l ¢

. (2.3)

The first term in (2.3) represents the resolved part of the total energy while the second term u _l u _l is the resolved part of the energy contained in the resolved velocity field.

Defining globally the resolved and subgrid scale kinetic energies is easy. Introducing the notation h . . . i for the integration over the entire volume of the system, the kinetic energy is defined by,

E ≡ 1 2 h u _l u _l i

1 It must be stressed that every term in the LES equations (2.1) are projected quantities so that they can be characterized by the same amount of data and represented on the same numerical grid. This is not the usual presentation of the LES equation in which the nonlinearities, both the convective term and the subgrid-scale stress tensor, are traditionally not projected yielding a somewhat inconsistent picture:

∂ t u i + ∂ j (u j u i ) = −∂ i p + ν∇ ² u i − ∂ j τ ij .

where τ ij = u i u j − u i u j . Obviously, for purely projective filter, both pictures are mathematically equivalent and τ ij = τ ij . We prefer however the formalism of equation (2.1) since it contains only projected quantities.

23

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24 2.1. Notations and definitions

= 1

2 h u _l u _l i

= 1

2 h u _l u _l i + h k i , (2.4)

where the general property h a i = h a i has been used. Another way of putting things is obtained by splitting the velocity u _i into a resolved part u _i and an unresolved part u ⁰ _i ≡ u _i − u _i :

E = 1

2 h (u _l + u ⁰ _l ) (u _l + u ⁰ _l ) i

= 1

2 h u _l u _l i + 1

2 h u ⁰ _l u ⁰ _l i . (2.5)

The cross term disappears due to the fact that · · · is assumed to be a projective filter.

Comparing (2.4) and (2.5), immediately yields h k i = ¹ ₂ h u ⁰ _l u ⁰ _l i and it is reasonable to interpret the volume average of k as the subgrid scale energy E _sgs = ¹ ₂ h u ⁰ _l u ⁰ _l i . The kinetic energy appears as the sum of E _sgs and of the resolved kinetic energy E _r = ¹ ₂ h u _l u _l i .

Locally the situation is more complex and denoting by lowercase letters all energy densities we have: e = ¹ ₂ u _i u _i = e _r + e _sgs + u _i u ⁰ _i , where e _r = ¹ ₂ u _i u _i and e _sgs = ¹ ₂ u ⁰ _i u ⁰ _i . Because of the

“interference” term u _i u ⁰ _i there is no clear separation of scales in the local energy. However, since the quantity e _sgs represents the energy density of the subgrid scale part of the velocity field, it may authentically be referred to as the subgrid scale kinetic energy.

According to this comment, the terminology traditionally adopted for k (turbulent kinetic energy) can be misleading and one must be careful about the interpretation of k which is given by

k = e _sgs + u _i u ⁰ _i , (2.6)

meaning that k represents, in the energy balance, the correction to the resolved energy. From the previous expression, it is clear that the actual subgrid-scale energy e sgs = ¹ ₂ u ⁰ _i u ⁰ _i , which is positive definite, differs from k, which, on the contrary, is not positive definite because of the correlation between resolved and unresolved velocity fields u _i u ⁰ _i . This aspect will be more detailed in the Section 2.3. So k should be considered as a non-resolved quantity that has to be modelled and which is only on average equal to the global subgrid-scale kinetic energy.

The interpretation in terms of energy can only serve as a guide for elaborating a model.

The motivation to estimate the turbulent kinetic energy is many fold. First, k directly enters the definition of the subgrid-scale stress tensor τ _ij since it is proportional to its trace:

τ ll = 2k. In the case of incompressible flows, this is not a serious issue because it is common

practice to merge τ _ll and the filtered pressure p into a pseudo-pressure term. Indeed, as long

as the time advance of LES for incompressible flows is concerned, no model for k is needed

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2.1. Notations and definitions 25

and an equation for the pseudo pressure P = p + ² ₃ k is obtained by enforcing the continuity condition:

∇ ² P = − ∂ _i ∂ _j ¡

u _i u _j + τ ^∗ _ij ¢

. (2.7)

In LES of compressible flows (Yoshizawa, 1986; Speziale et al., 1988; Moin et al., 1991;

Erlebacher et al., 1992; Zang et al., 1992), this way of proceeding is not adequate and one cannot avoid the modelling of k. A second motivation to study models for the turbulent kinetic energy comes from the fact that it may be used as a scaling for the eddy-viscosity in traditional eddy-viscosity models for the deviatoric part of τ _ij (see Sagaut, 2001 for a review).

Finally, let us mention that its knowledge (Winckelmans et al., 2002) is required to predict simple quantities like the total energy,

E = 1

2 h u _i u _i i = 1

2 h u _i u _i i + h k i , (2.8) or the pressure

p = − 2

3 k − ∇ ⁻² ∂ _i ∂ _j u _i u _j − ∇ ⁻² ∂ _i ∂ _j τ ^∗ _ij . (2.9) There are thus several reasons that motivate the exploration of explicit models for k.

Modelling of the turbulent kinetic energy is not new. One proposal was made by Yoshizawa (1986) (later generalized by Moin et al., 1991) in which the author introduced a model based on the LES filter width and the norm of the filtered rate of strain tensor (see eq. 2.18). The validity of this model was discussed in Speziale et al. (1988) and will be further investigated later in this chapter. A more recent model was introduced by Misra & Pullin (1997) in the framework of vortex-based subgrid stress modelling. In their proposal, the turbulent kinetic energy is obtained by introducing a model for the energy spectra and invoking a local balance between the total dissipation and the sum of the resolved-scale dissipation and the produc- tion of the turbulent kinetic energy. A similar model for the subgrid-scale energy was later formulated in physical space by Voelkl et al. (2000).

In this work, two different approaches for modelling k have been investigated. The first one is based on a simple formalism that takes advantage of the dynamic procedure and will include a space averaged version as well as a local version (Knaepen et al., 2002). In the second approach, an equation for the turbulent kinetic energy, k, is carried out explicitly.

Although this approach is not new in LES (Ghosal et al., 1995; Wong, 1992; Fureby et al., 1997), important differences with the existing works come from the emphasis on the non positivity of k, that was known but usually not fully accounted for (Debliquy et al., 2004a ).

The evolution equation for k contains two unknown terms corresponding to a turbulent

flux and a turbulent dissipation. Both these terms require additional modelling efforts which

PSEUDO-SPECTRAL METHODS APPLIED TO HYDRODYNAMIC AND MAGNETOHYDRODYNAMIC TURBULENCE.

Universit´ e Libre de Bruxelles Facult´ e des Sciences

PSEUDO-SPECTRAL METHODS APPLIED TO HYDRODYNAMIC AND MAGNETOHYDRODYNAMIC

TURBULENCE.

Olivier Debliquy

Th` ese pr´ esent´ ee en vue de l’obtention du grade de docteur en sciences ` a la facult´ e des sciences de l’universit´ e libre de bruxelles sous la direction du Pr. Carati.

Bruxelles, d´ ecembre 2004.

“ When I meet God, I am going to ask him two questions: Why relativity ? And why turbulence ? I really believe he will have an answer for the first ”.

W. K. Heisenberg

Table of Contents

Table of Contents i

Remerciements v

Preface vii

I Hydrodynamic Turbulence 1

1 Numerical analysis of turbulent flows 3

1.1 Governing equations of fluids mechanics . . . . 3

1.1.1 Continuum hypothesis . . . . 3

1.1.2 Lagrangian and Eulerian derivatives . . . . 4

1.1.3 Mass conservation . . . . 5

1.1.4 Momentum conservation . . . . 5

1.1.5 The limit of incompressible flow . . . . 7

1.2 Turbulence . . . . 7

1.2.1 Reynolds number . . . . 7

1.2.2 Energy cascade and Kolmogorov’s theory . . . . 9

1.2.3 The energy spectrum . . . . 11

1.3 Simulations of turbulent flows . . . . 12

1.3.1 Direct Numerical Simulation . . . . 14

1.3.2 Reynolds Averaged Navier-Stokes . . . . 15

1.3.3 Large Eddy Simulation . . . . 16

1.4 Discretization methods . . . . 19

1.4.1 Finite differences . . . . 19

1.4.2 Finite volumes . . . . 19

1.4.3 Finite elements . . . . 20

1.4.4 Spectral methods . . . . 21

2 Modelling aspects of k 23 2.1 Notations and definitions . . . . 23

2.2 Algebraic models . . . . 26

2.2.1 Model for the averaged subgrid scale kinetic energy . . . . 26

i

ii CONTENTS

2.2.2 Numerical results . . . . 29

2.2.3 Local model for k . . . . 34

2.3 Models based on the transport equation for k . . . . 37

2.3.1 Modelling of the traceless subgrid-scale stress tensor . . . . 40

2.3.2 Modelling of the turbulent flux . . . . 41

2.3.3 Modelling of the subgrid-scale energy dissipation . . . . 42

2.3.4 Numerical results . . . . 46

2.4 Discussion . . . . 57

3 Inhomogeneous flow: the shear-free mixing layer 61 3.1 Introduction . . . . 61

3.2 Initial condition . . . . 63

3.3 DNS . . . . 65

3.3.1 Parameters of the simulation . . . . 65

3.3.2 Kinetic energy diagnostics . . . . 68

3.3.3 Variance profiles . . . . 75

3.3.4 Intermittency . . . . 78

3.3.5 Anisotropy . . . . 81

3.4 LES with simple model . . . . 82

3.4.1 SGS model . . . . 82

3.4.2 Comparison of the filtered DNS and the LES . . . . 83

3.5 LES with model based on k . . . . 88

3.5.1 SGS model . . . . 88

3.5.2 Turbulent flux model . . . . 89

3.5.3 SGS dissipation . . . . 89

3.5.4 Results . . . . 89

3.6 Example of LES at infinite Reynolds number . . . . 94

3.7 Discussion . . . . 98

4 Wall-bounded flow in pipe 99 4.1 Introduction . . . . 99

4.2 Numerical schemes . . . 101

4.2.1 Temporal integration . . . 102

4.2.2 Axial and azimuthal discretization . . . 104

4.2.3 Radial treatment . . . 106

4.2.4 Pressure correction scheme . . . 109

4.3 Results . . . 110

4.3.1 Analytical cases . . . 110

4.3.2 Turbulent pipe flow . . . 112

4.4 Discussion . . . 118

CONTENTS iii

II Magnetohydrodynamic Turbulence 119

5 MHD formalism 121

5.1 MHD equations . . . 121

5.1.1 Lorentz force . . . 122