Yves Detandt

N     

  M   

Yves Detandt

T`  ´ ´    

 ’     ´  

D   S    ’I   ´

 ’U´  L  B

    P  . G. D 

Bruxelles, Aoˆut 2007.

Acknowledgments

Je voudrais tout d’abord remercier le Professeur G´erard Degrez. Ses capacit´es didactiques et ses conseils judicieux ont jalonn´e mon parcours tout au long de cette th`ese. A la fin de quatre ann´ees de th`ese, je conserve un bon nombre de formules griffonn´ees le plus souvent sur une serviette `a l’heure du repas. Ce sont ces quelques formules et les tr`es nombreuses discussions que nous avons eues qui m’ont guid´e dans mes recherches. Je le remercie aussi pour la confiance dont il m’a fait part tout au long de ces 4 ann´ees, me confiant tour `a tour des projets situ´es bien loin de l’a´eroacoustique, mais qui m’ont assur´e une th`ese aussi diverse que passionante.

Je tiens aussi `a remercier tout particuli`erement le Professeur Daniele Carati pour ses conseils avis´es. Au d´ebut de cette th`ese, mes connaissances sur le logiciel libre

´etaient proches du z´ero absolu, mais ses encouragements accompagn´es de ceux de G´erard m’ont amen´e `a prendre une part de plus en plus grande dans la gestion et la conception du parc informatique commun de nos deux services. Le nom d’Anic n’aura certainement plus la mˆeme signifcation pour moi apr`es ces 4 ans, mˆeme si, loi de Murphy oblige, les op´erations se d´eroulaient toujours au moment o`u j’avais le moins de temps `a y consacrer. Je crois avoir beaucoup appris sur le plan personnel et tiens `a remercier Daniele et G´erard pour leur confiance dans ce domaine.

Je voudrais aussi exprimer ma plus profonde gratitude envers David Vanden Abeele. Ton exp´erience en programmation scientifique et ton goˆut pour de ”belles solutions” num´eriques ont ´et´e un exemple tout au long de ces 4 ans. Je garde aussi des souvenirs d’interminables corrections d’articles, mais toujours impressionn´e par la cons´equente am´elioration du papier.

Je remercie ´egalement le Professeur Jean-Louis Migeot. Sa passion et l’int´erˆet pour l’acoustique qu’il a fait naˆıtre lors de son cours ne sont sans doute pas ´etrangers

`a mon choix de sujet de th`ese. Je le remercie aussi, ainsi que toute l’´equipe de Free Field Technologies, avec un attention sp´eciale pour St´ephane Caro et Gr´egory Lie- lens, pour leur disponibilit´e, leur ´ecoute et leur gentillesse lors des nombreux contacts que j’ai eu l’occasion d’entretenir pendant ces 4 ans.

i

J’exprime aussi toute ma reconnaissance vis `a vis de Christophe Schram. Ton manuscrit de th`ese a ´et´e `a la fois une introduction et un fil conducteur me guidant

`a travers les raisonnements parfois compliqu´es des analogies a´eroacoustiques. Tes connaissances et les discussions sur le bruit de jet m’ont aussi beaucoup aid´e pour obtenir une solution num´erique similaire aux exp´eriences.

Je remercie le Professeur G. Winckelmans et le Professeur H. Deconinck pour avoir accept´e de participer au jury de cette th`ese.

Une th`ese est aussi une formidable exp´erience humaine et professionnelle. Je re- mercie Patrick, Pietro, Michel, Franc¸ois, Christophe, Dries, Frank, Didier, Olivier et Pierre-Alexis pour ces temps de midi et moments que nous avons partag´es ensemble.

Ces discussions anodines m’ont souvent permis d’aborder les questions sous un autre angle, me rapprochant ainsi sans aucun doute de la solution.

Dans le cadre de cette th`ese j’ai aussi eu le plaisir d’accompagner Axelle Vir´e, David Vangeenberghe, Isma¨el Yabda, Zakarias Djoudi et Pierre Gousseau. Leur m´emoire a ´et´e l’occasion d’un ´echange de points de vue tr`es int´eressant et qui a conduit `a pas mal d’am´eliorations du code.

Je remercie Shirley Wayne pour sa disponibilit´e et son efficacit´e `a r´esoudre les nombreux soucis administratifs.

Je n’oublie pas non plus l’´equipe technique ; Dany, Guy, Claude, Rapha¨el, Pascal et Yves qui ont toujours ´et´e de la partie depuis la construction du ”poulailler” pour ordinateurs jusqu’aux ”casseroles `a moules” sur lesquelles les ´etudiants de Thermo

´etudieront bientˆot les propri´et´es des cycles frigo.

Enfin, et ce ne sont pas des moindres, j’adresse mes plus vifs remerciements `a ma famille et mes parents en particulier pour leur soutien. A la fin de cette th`ese et de mes

´etudes, je mesure l’importance de leurs encouragements. Je garde le mot de la fin `a l’attention de mon ´epouse, Marie-Astrid. Je n’oublierai jamais ta compr´ehension pour ces longs week-ends de travail, mais aussi pour ta pr´esence `a mes cˆot´es depuis plus de 7 ans. Tu as ´et´e de toutes les d´ecisions et ton sourire est certainement le meilleur des rem`edes pour oublier les tracas et probl`emes dont j’ai l’art de m’entourer.

Cette th`ese a ´et´e support´ee financi`erement par le Fonds pour la formation `a la

Recherche dans l’Industrie et dans l’Agriculture (FRIA). Les simulations ont ´et´e

r´ealis´ees sur le cluster Anic4 financ´e par plusieurs fonds et contrats de recherche

(http ://anic4.ulb.ac.be).

Contents

1 Introduction 1

1.1 Context . . . . 1

1.2 Aeroacoustic simulations . . . . 2

1.2.1 Direct approach . . . . 2

1.2.2 Hybrid approach . . . . 2

1.3 Flow simulations . . . . 3

1.4 Turbulence . . . . 4

1.5 Acoustic simulations . . . . 4

2 3D Flow solver 7 2.1 Incompressible Navier-Stokes equations . . . . 8

2.1.1 Conservation laws . . . . 10

2.1.2 Weak forms . . . . 11

2.2 Discretized equations . . . . 13

2.2.1 Finite element method . . . . 13

2.2.2 Spectral expansion . . . . 14

2.2.3 Convective Terms . . . . 16

2.2.4 Matrix form . . . . 21

2.2.5 Time discretization . . . . 24

2.2.6 Conclusion . . . . 29

2.3 Boundary conditions . . . . 30

2.3.1 Inlet boundary . . . . 31

2.3.2 Outlet boundary . . . . 32

2.3.3 Wall . . . . 33

2.3.4 Axis boundary . . . . 34

2.4 Stabilization . . . . 38

2.4.1 Pressure oscillations . . . . 38

2.4.2 Convective oscillations . . . . 41

2.5 Conclusion . . . . 44

iii

3 Flow test cases 47

3.1 Computed quantities . . . . 47

3.1.1 Force - Torque . . . . 48

3.1.2 Kinetic energy . . . . 50

3.1.3 Enstrophy . . . . 50

3.1.4 Angular momentum . . . . 51

3.2 Flow past a sphere at moderate Reynolds numbers . . . . 51

3.2.1 Stokes flow . . . . 52

3.2.2 Steady axisymmetric flow . . . . 53

3.2.3 Unsteady non-axisymmetric flows . . . . 57

3.3 Taylor-Couette flow . . . . 64

3.3.1 Axisymmetric Taylor-Couette flow . . . . 66

3.3.2 Turbulent Taylor-Couette flow . . . . 68

3.4 Conclusion . . . . 75

4 Turbulence modeling 77 4.1 Turbulence concepts . . . . 77

4.2 LES modeling . . . . 79

4.2.1 Eddy viscosity models . . . . 85

4.2.2 Dynamic procedure . . . . 90

4.2.3 Variational multiscale approach . . . . 91

4.3 Conclusion . . . . 94

5 Aeroacoustics 95 5.1 Acoustic equations . . . . 99

5.2 Analogies . . . . 102

5.2.1 Lighthill analogy . . . . 103

5.2.2 Vortex sound . . . . 105

5.3 Acoustic solutions . . . . 106

5.3.1 Integral methods . . . . 107

5.3.2 Variational solution . . . . 111

5.4 Conclusion . . . . 116

6 Aeroacoustic simulations 119 6.1 Introduction . . . . 119

6.2 Cylinder . . . . 119

6.2.1 Hydrodynamic simulation . . . . 120

6.2.2 Noise extraction . . . . 123

6.2.3 Acoustic solution . . . . 126

6.2.4 Conclusion . . . . 132

6.3 Jet noise . . . . 132

6.3.1 Re=5000 . . . . 133

CONTENTS v 6.3.2 Re = 14000 . . . . 150 6.4 Conclusion . . . . 173

7 Conclusions 177

A Finite element integrals 191

B Derivation of regularity conditions on the axis 193

C Stokes solution past a sphere 197

D Computation of force and torque 201

E Dispersive character of di ff erent schemes 205

F Damping of noise sources 211

Chapter 1

Introduction

1.1 Context

The present work focuses on the computation of the noise generated by unsteady tur- bulent flows. The conversion from flow energy into acoustics has been neglected in the past due to the relative ine ffi ciency of the transfer. Even if generally flow and sound propagate in the same medium, the acoustic and fluid research communities have been interested by completely different topics. Some fluid dynamicists however used the acoustic signature of vortex shedding to identify the frequency associated to this pure flow phenomenon and acoustic excitation has been extensively used in flow experiments to trigger or stabilize some periodic mechanisms. These are probably precursors of aeroacoustics which is a branch between acoustics and fluid dynamics analyzing the sound generated by unsteady flows.

The development of jet engines in the 1950’s triggered the interest of engineers and scientists who attempted to estimate the noise generated by di ff erent designs of this noisy propulsion technique. Lighthill, a fluid dynamics specialist, proposed a theoretical explanation for jet noise which was subsequently extended to more com- plicated flows. At this time, theory was restricted by flow analysis tools and scaling laws derived from theory were the basic tools for the design of quiet flows.

In the 1990’s, the development of efficient computers at an affordable cost led to a deeper analysis of flow phenomena and opened new trends for aeroacoustic methods.

Economical development of civil air transport tends to increase the number of flight operations at different airports and mainly during night. People living around the airports claim for silent aircraft and the international rules for aircraft noise are more and more drastic. The last Silencer program funded by the European Union shows significant noise reduction, but also illustrates that these reductions are probably the best compromise for the actual design. New technologies have to be evaluated to

1

meet the future imposition of noise reduction. The development of new numerical tools, which compare and point out the best designs are therefore necessary. This tendency is not limited to the civil aircraft industry. People are now used to higher and higher comfort level, and require reduction of noise generated for instance by their cars air supply system. These methods combining tools developed for flow and acoustic simulations are the central interest of this thesis and will be briefly discussed to outline the di ff erent chapters which subdivide the present work.

1.2 Aeroacoustic simulations

Aeroacoustics is based on the common research e ff ort of two distinct branches of physics. Fluid dynamics solutions are generally classified by a set of relevant non- dimensional numbers. For aeroacoustic problems, the Reynolds number, the Strouhal number and the Mach number fully define the problem. For a given geometry, the Reynolds number compares convective and viscous effects. The Strouhal number relates the frequency of unsteady flow mechanisms to the geometrical and velocity scales of the problem. The Mach number compares the flow velocity to the sound speed. This sound speed is defined by measuring the ratio between pressure to den- sity fluctuations. Two methods are presented in the following to solve aeroacoustic problems.

1.2.1 Direct approach

Acoustic signals are low amplitude pressure fluctuations in a compressible medium like air. These pressure fluctuations satisfy the compressible Navier-Stokes equations which are the fundamental fluid dynamics equations. The first method computes flow and acoustic quantities together during the same simulation. Flow and acous- tics are however characterised by length and velocity scales which are completely di ff erent. The acoustics propagates over large domains which eventually extend to infinity. Flows are generally confined in a smaller region of space. For low Mach numbers, this solution leads to maximal computational requirements as illustrated in chapter 5. The direct approach of the problem is mainly suited for flows at Mach number close to or higher than unity where the acoustic requirements are closer to flow requirements.

1.2.2 Hybrid approach

At low Mach numbers, acoustics is weakly coupled to fluid dynamics and we can

compute the flow field separately from the acoustic solution. This phenomenon is

mainly due to the disparity between di ff erent scales characterizing flow and acous-

tics. In this hybrid approach we compute first the flow, extract the equivalent source

distribution and propagate these sources up to the listener position. For low Mach

1.3 Flow simulations 3 number flows, density fluctuations have a negligible impact on the flow. The Incom- pressible Navier-Stokes equations can be used and the solver provides a flow field which is post-processed in order to extract the equivalent sound sources. This ex- traction is based on analogies derived by Lighthill, Powell, M¨ohring and many other scientists who devoted a large attention to flow noise.

1.3 Flow simulations

In the present work, we compute the incompressible flow field around axisymmetric geometries. A cylindrical coordinate system is the natural choice since it facilitates the imposition of the boundary conditions on the axisymmetric boundaries. For low Mach numbers, density fluctuations do not affect the flow and density can be consid- ered as constant. We therefore solve the incompressible Navier-Stokes equations for pressure and three flow velocity components. Acoustics, which is the compressible part, is not present in this solution, but the equivalent acoustic sources are extracted using different means which allow to recover an equivalent compressible information.

Except for some particular configurations, the analytical solution of the Navier- Stokes equations is not available, and different simulations provide a numerical so- lution which approximates the exact one. The accuracy of these approximations de- pends mainly on the number of parameters which define the numerical solution and on the order of interpolation used for the approximate solution. CFD codes are based on two di ff erent classes of discretization techniques:

• Finite volumes of finite differences methods relate nodal values (which ap- proximate the sampled values of the exact solution) using Taylor’s expansions or balances over small control volumes surrounding each of these points.

• Finite elements or spectral methods use a continuous description of the numer- ical solution. A number of parameters define the numerical solution and are the unknowns of the simulation (solution at nodal positions for finite element method and mode amplitudes for spectral methods). Unknowns are related us- ing weighted residual techniques and a number of submethods differ by the weight applied on the residual during integration.

In the present work, we solve the incompressible Navier-Stokes equations in a cylin-

drical coordinate system. The azimuthal direction is periodic by definition and a

spectral method in this direction is used for all unknowns (p, v _z , v _r , v _θ ). For radial

and axial directions which define meridional planes, we use a Galerkin finite ele-

ment method on unstructured meshes. The flow solver is therefore able to handle

flow around axisymmetric geometries which can be very complex in a meridional

plane. This combination between these two discretization techniques has a number

of advantages which are presented in chapter 2.

1.4 Turbulence

The Navier-Stokes equations involve linear and non-linear terms. In general, en- gineering applications are convection dominated and the solutions are completely di ff erent for di ff erent values of Reynolds number, which compares convective and viscous effects. Turbulent flow solutions are made out of a large number of vortices which are characterized by their size. The smallest size of vortices decreases as the Reynolds number increases. The numerical cost associated to a simulation in which all turbulent structures are resolved, including the smallest ones, corresponds to the product of the number of points used for spatial resolution by the number of time steps for one characteristic period, and scales like Re ³ . This cost indicates that most of engineering flows are untractable with the present computational power. Different modeling techniques reduce these requirements to an a ff ordable cost for presently available computational power.

• Reynolds Averaged Navier Stokes (RANS) where an ensemble average is applied on the equations. As a result, high frequency (temporal) fluctuations and high wavenumber (spatial fluctuations) are discarded.

• Large Eddy Simulation (LES) approach is based on spatial filtering. Mesh spacing cannot resolve the smallest turbulent structures and subgrid scale mod- els replace their influence on the large resolved structures.

LES models and their practical implementations are presented in chapter 4.

1.5 Acoustic simulations

Acoustics corresponds to small compressible fluctuations in the medium. In the present work, we demonstrate the practical ability of an hybrid method for aeroa- coustics problems. The equivalent noise sources are extracted from the computed flow field. The sound is propagated to the listener position which is usually located at a large number of acoustic wavelengths from the flow itself. The acoustic solver propagates sound over these large distances and evaluates the sound level at listener position. The same discretization technique as the one proposed for flow simulations can be used. The numerical scheme directly influences the way numerical acoustic waves propagate:

• The amplitude of the acoustic waves can be a ff ected along the propagation.

This is related to the dissipative characteristic of the scheme used for the spatial discretization.

• For different frequencies, the numerical acoustic waves do not propagate at the

speed of sound. This affects the solution and creates interferences which affect

1.5 Acoustic simulations 5 the sound evaluated at listener position. This corresponds to the dispersive characteristic of the discretization.

These two points are discussed in chapter 5 where we derive the acoustic equa- tions, present different analogies to extract the equivalent noise sources and different methods to compute the acoustic propagation of this noise. The last chapter 6 presents some cases where the noise is generated by periodic processes in a laminar solution.

These tests show the accuracy of the method and are compared with literature results.

Chapter 2

3D Flow solver

The resolution of an aeroacoustic problem involves flow simulations in order to com- pute the equivalent acoustic source terms. The temperature e ff ects will not be con- sidered in the present work and the flow variables reduce to pressure, density and velocity components which are related trough 4 di ff erential equations and an alge- braic equation:

∂ρ

∂t + ∂ρv i

∂x i = 0 (2.1)

ρ ∂v i

∂t + ρv j

∂v i

∂x j = − ∂p

∂ x i

− ∂τ i j

∂x j

(2.2)

p = F (ρ) (2.3)

The first one corresponds to mass conservation or continuity equation. The second, which is in fact a vectorial relation, corresponds to the Newton’s law applied to flu- ids and involves a non-linear convective term. The last equation relates pressure to density (generally called a state equation) and is fluid specific. The non-linear char- acter introduced by the convective term in the momentum equations is the origin of the turbulent mechanisms which makes Fluid Dynamics a so interesting but complex science. Turbulence concepts and modelling aspects are discussed in chapter 4. Dif- ferent parameters are derived from these equations. These non-dimensional numbers fully define the problem and are used to scale observation from experimental mock up to real size applications:

• Strouhal number is defined as the ratio between the convective time _U ^L and the period associated to flow mechanisms T : S t = _UT ^L = ^{f L} _U

• Mach number relates the flow velocity scale U to sound speed c ₀ = q _∂p

∂ρ

s : M = _c ^U

₀

. The sound speed definition also shows that this non-dimensional number is a measure of the importance of compressible effects.

7

• Reynolds number compares convective time _U ^L to viscous time ^L _ν

²

: Re = ^{U L} _ν In the present work, we focus on low Mach number flows. The sound speed is very high compared to flow speed and density is almost insensitive to pressure vari- ations. Density is considered as constant in the flow simulations and therefore drops out of the unknowns. The incompressible flow equations reduce only to momentum and continuity equations:

∂v i

∂x i = 0 (2.4)

ρ ∂v i

∂t + ρv j

∂v i

∂ x _j = − ∂ p

∂x i

− ∂τ i j

∂x j

(2.5) In practice, the incompressible limit is valid up to a Mach number of around 0.3. The incompressible flow solution does not di ff er from the real flow solution as long as the Mach number is in this range. This solution however does not involve any acoustic part, which is related to compressible effects. Acoustic informations will therefore need to be recovered in a source term propagated to a listener position as explained in chapter 5.

2.1 Incompressible Navier-Stokes equations

In the present thesis, we develop a solver for axisymmetric geometries. A cylindrical coordinate system is a natural choice for this type of geometry. Axial and radial co- ordinates refer the position in a meridional plane referenced by the azimuthal angle θ from a reference direction. Di ff erent techniques are used in the Computational Fluid Dynamics (CFD) community to transform the set of differential equations into a dis- crete system relating numerical values at some dedicated points. Except for special cases, these numerical values are not a sampling of the exact solution : the numer- ical solution approximates the exact solution and the convergence property ensures that increasing this number of sampling positions will lead to a better approximation, closer to exact solution.

Spectral and finite element methods use a continuous description of the numerical

solution based on some nodal (or modal for spectral methods) values. On the other

hand, finite differences and finite volume methods express derivatives and fluxes as

a function of nodal neighbouring values of the solution. Spectral methods have a

high level of accuracy and lead to easily invertible systems. This method is proba-

bly optimal for turbulence physics analysis, but is mainly restricted to fully periodic

problems or designed for particular applications. The other discretization techniques

are able to compute flow in general three-dimensional geometries, but create large

systems of coupled equations which are difficult to solve.

2.1 Incompressible Navier-Stokes equations 9 In the present chapter, we develop an incompressible flow solver based on a mixed spectral/finite element discretization. The combination of spectral and finite element discretization has been used by Snyder [95] for flow computations in a Carte- sian coordinate system. For axisymmetric geometries (jets, sphere, ...), a cylindrical coordinate system is the most natural choice. Positions are referenced in a meridional plane by the axial and radial coordinates and the azimuthal coordinate measures the position of this plane according to a reference direction. In this coordinate system, the azimuthal direction is periodic by definition and leads naturally to a spectral expan- sion. On the other hand, for the sake of generality of problems investigated using this solver, we use a linear finite element interpolation on unstructured triangular meshes.

The solver is able to compute fully three-dimensional flows with the restriction that boundaries have an axisymmetric shape.

Continuity equation - Conservation of mass

L _c (~ u, p) = ∂rv r

∂r + ∂v _θ

∂θ + r ∂v z

∂z = 0, (2.6)

where L _c is the linear operator associated with the continuity equation. For incom- pressible flows, the pressure is not explicitly present in the continuity equation. This causes problems in our discretization as presented in section 2.4.1. This equation has to be solved together with the momentum balance :

z-momentum equation

∂rv z

∂t + L _z (~ u, p) = −C _z , (2.7)

where

• L z = r ∂ p

∂z − ν( ∂

∂r (r ∂v z

∂r ) + 1 r

∂ ² v z

∂θ ² + r ∂ ² v z

∂z ² ) is the linear operator associated with the z-momentum equation,

• C z = rv r

∂v z

∂r + v _θ ∂v z

∂θ + rv z

∂v z

∂z + α c v z

" ∂rv r

∂r + ∂v _θ

∂θ + r ∂v z

∂z

#

is the corresponding convective term.

r-momentum equation :

∂rv r

∂t + L _r (~ u, p) = −C _r , (2.8)

where

• L _r = r ∂ p

∂r − ν(r ∂

∂r ( 1 r

∂rv r

∂r ) + 1 r

∂ ² v _r

∂θ ² − 2 r

∂v θ

∂θ + r ∂ ² v _r

∂z ² ) is the linear operator

associated with the r-momentum equation,

• C r = rv r

∂v r

∂r + v _θ ∂v r

∂θ − v ² _θ + rv z

∂v r

∂z + α c v r

" ∂rv r

∂r + ∂v _θ

∂θ + r ∂v z

∂z

#

is the corre- sponding convective term.

θ-momentum equation :

∂rv θ

∂t + L _θ (~ u, p) = −C _θ , (2.9)

where

• L _θ = ∂ p

∂θ − ν(r ∂

∂r ( 1 r

∂rv _θ

∂r ) + 1 r

∂ ² v _θ

∂θ ² + 2 r

∂v r

∂θ + r ∂ ² v _θ

∂z ² ) is the linear part associated with the θ-momentum equation,

• C _θ = rv r

∂v _θ

∂r + v _θ ∂v _θ

∂θ + v r v _θ + rv z

∂v _θ

∂z + α c v _θ

" ∂rv r

∂r + ∂v _θ

∂θ + r ∂v z

∂z

#

is the corre- sponding convective term.

The parameter α c leads to di ff erent expressions of the convective terms which are equivalent at the continuum level since they differ by a term proportional to the di- vergence of velocity (which is zero for incompressible flows). In practice, only three di ff erent values are of interest, leading to the following formulas:

• Convective form (α c = 0) : C _i = v j ∂v

i

∂x

j

• Skew-symmetric form (α c = 1/2) : C i = ¹ ₂ v j ∂v

i

∂x

j

+ ¹ ₂ ^∂v _∂x

ⁱ

^v

_j^j

• Divergence form (α c = 1) : C _i = ^∂v _∂x

ⁱ

^v

_j^j

2.1.1 Conservation laws

The Navier-Stokes equations are derived from classical mass balance and Newton’s laws applied to an infinitesimal volume (dxdydz in a Cartesian reference frame, drdzrdθ in a cylindrical coordinate system). Global balances can be derived in a same manner by integrating the Navier-Stokes over an open volume Ω delimited by the surface ∂ Ω .

The first equation expresses the conservation of mass in an open system:

I

∂ Ω

rv _r n ^r _seg + rv _z n ^z _seg

dS = 0 (2.10)

For incompressible flow, density is constant and does not allow any mass variation

inside the system and fluxes trough boundaries have to compensate each other.

2.1 Incompressible Navier-Stokes equations 11 If we integrate the Momentum equation, we recover Newton’s law for an open system ( conservation of impulse )

d dt

Z

Ω ρv i dV = − I

∂ Ω (ρv i v j + pδ i j + τ i j )n j dS + Z

Ω ρ f i dV (2.11)

This relation clearly shows the origin of convective terms which are the momen- tum fluxes trough the boundaries. The last conservation principle corresponds to an equation for kinetic energy k e = ^ρv ₂

ⁱ

^v

ⁱ

. This equation is derived by taking the scalar product of the momentum equation with velocity:

d dt

Z

Ω k e dV = − I

∂ Ω (k e δ i j + pδ i j + τ i j )v i n j dS + Z

Ω ρ f j v j dV + Z

Ω τ i j

∂v i

∂x j

dV (2.12)

The surface term corresponds to energy introduced trough the boundaries. The last volumic term R

Ω τ i j ∂v

i

∂x

j

dV represents viscous dissipation and is always negative.

2.1.2 Weak forms

The first part of the discretization process relies on a spatial representation of the solution as a function of sampled values which are the unknowns of the discrete problem. The second step of the procedure derives as many discrete equations as unknowns. This procedure will be based on weighted residual techniques. Residuals are weighted by a function denoted by w and integrated over the volume. In practice the support of this function is limited such that a minimal number of unknowns are involved in each relation.

For the Navier-Stokes equations in a cylindrical coordinate system, we derive the

following weak forms:

Z 2π 0

Z

Ω wL _c drdzdθ = 0, (2.13)

Z 2π 0

Z

Ω w ∂rv z

∂t + C _z − p ∂rw

∂z

!

drdzdθ + I

∂ Ω wr

"

pn ^z _seg − ν ∂v z

∂z n ^z _seg − ν ∂v z

∂r n ^r _seg

# dldθ

+ ν Z _2π

0

Z

Ω

"

r ∂w

∂z

∂v z

∂z + r ∂w

∂r

∂v z

∂r − w r

∂ ² v z

∂θ ²

#

drdzdθ = 0, (2.14)

Z 2π 0

Z

Ω w ∂rv r

∂t + C r − p ∂rw

∂r

!

drdzdθ + I

∂ Ω wr

"

pn ^r _seg − ν ∂v r

∂z n ^z _seg − ν ∂v r

∂r n ^r _seg

# dldθ

+ ν Z 2π 0

Z

Ω

"

r ∂w

∂z

∂v r

∂z + r ∂w

∂r

∂v r

∂r − w r

∂ ² v r

∂θ ² − 2 ∂v _θ

∂θ − v r

!#

drdzdθ = 0, (2.15) Z 2π

0

Z

Ω w ∂rv θ

∂t + C _θ + w ∂p

∂θ

!

drdzdθ + I

∂ Ω wr

"

−ν ∂v θ

∂z n ^z _seg − ν ∂v θ

∂r n ^r _seg

# dldθ

+ ν Z 2π

0

Z

Ω

"

r ∂w

∂z

∂v θ

∂z + r ∂w

∂r

∂v θ

∂r − w r

∂ ² v _θ

∂θ ² + 2 ∂v r

∂θ − v _θ

!#

drdzdθ = 0. (2.16)

These weak forms involve a volume integral and are therefore strongly linked to conservation principles derived in section 2.1.1. In particular the choice w = 1 leads to the conservation of mass and momentum if the conservative expression (α c = 1) is used for the convective terms. Even if all convective expressions are equivalent at the continuum level, the parameter α c ensures conservation principles for aforemen- tioned discrete solutions.

Most operators involved in the Navier-Stokes equations are singular on the axis.

The Navier-Stokes equations presented in Bird [7] are obtained by a transformation from a Cartesian to a cylindrical coordinate system and involve axis singularity for convective, pressure and viscous terms. In the present case, we multiply the equations by a radial coordinate r. Equations are mathematically the same, but the weak forms show that some conservation principles are observed in the numerical solution. This form multiplied by the radial coordinate corresponds to a natural expression with a physical interpretation ¹ . There is only a remaining axis singularity involved in viscous terms which will be removed by appropriate regularity conditions on the axis of revolution 2.3.4.

1

Alternatively, one can keep the standard form of the governing equations and perform a volume

integral using the volume element rdrdθdz. The resulting expressions are identical but the latter point

of view is maybe more natural.

2.2 Discretized equations 13

2.2 Discretized equations

For flow around axisymmetric geometries, the problem (boundary conditions and azimuthal variable’s definition) is perfectly periodic in the azimuthal direction. Az- imuthal dependence is therefore well represented by a spectral expansion. In merid- ional planes, we use a Galerkin linear interpolation over unstructured triangular meshes.

The axisymmetric character is clearly a restriction for the application of the code to general geometries. The finite element representation in meridional planes preserves however the ability of the solver to deal with non-trivial geometries.

q ^h (z, r, θ n = n2π N , t) = 1

N

N

_nodes

X

j = 0

N/2−1

X

k = −N/2+1

Q ^k _j (t)

N j (z, r)e ^Ikθ

ⁿ

(2.17) with I ² = −1 and Q ^k _j (t) stands for the Fourier coe ffi cient of mode k in a node j at a given moment in time t.

2.2.1 Finite element method

In meridional planes defined by θ = constant, we use a P1 finite element repre- sentation for both velocity and pressure (P1 / P1 element). The flow equations are discretized using the pressure stabilized Petrov-Galerkin (PSPG) formulation.

(l c ) i = Z

Ω N i

∂rv ^h _r

∂r + N i

∂v ^h _θ

∂θ + N i r ∂v ^h _z

∂z dS +

Z

Ω τ pspg ( ∂N i

∂z



 

 

 r ∂v ^h _z

∂t + L ^h _z + C ^h _z



 

 

 + ∂N i

∂r r ∂v ^h _r

∂t + L ^h _r + C ^h _r

! )dS

= 0, (2.18)

where τ pspg is the Petrov stabilization parameter which is piecewise constant over elements. This pressure stabilization will be discussed later in section 2.4.1.

Since we adopt a finite element discretization only in the meridional plane, the PSPG stabilization term involves only axial and radial derivatives of the finite element shape functions.

The usual Galerkin technique is used to discretize the momentum equations (2.19 to 2.21) with the viscous and pressure terms integrated by parts (2.22 to 2.24).

Z

Ω N _i r ∂v ^h _z

∂t dS + (l z ) i = −(c z ) i = − Z

Ω N _i C ^h _z dS (2.19) Z

Ω N _i r ∂v ^h _r

∂t dS + (l r ) i = −(c r ) i = − Z

Ω N _i C ^h _r dS (2.20) Z

Ω N i r ∂v ^h _θ

∂t dS + (l θ ) i = − (c θ ) i = − Z

Ω N i C ^h _θ dS (2.21)

(l z ) i = − Z

Ω p ^h ∂rN i

∂z dS + Z

∂ Ω N _i r



 

 

 p ^h n ^z _seg − ν ∂v ^h _z

∂z n ^z _seg − ν ∂v ^h _z

∂r n ^r _seg



 

 

 dl + ν

Z

Ω r ∂N _i

∂z

∂v ^h _z

∂z + r ∂N _i

∂r

∂v ^h _z

∂r − N _i r

∂ ² v ^h _z

∂θ ² dS (2.22)

(l r ) i = − Z

Ω p ^h ∂rN i

∂r dS + Z

∂ Ω N i r

"

p ^h n ^r _seg − ν ∂v ^h _r

∂z n ^z _seg − ν ∂v ^h _r

∂r n ^r _seg

# dl

+ ν Z

Ω r ∂N i

∂z

∂v ^h _r

∂z + r ∂N i

∂r

∂v ^h _r

∂r − N i

r



 

 

 

∂ ² v ^h _r

∂θ ² − 2 ∂v ^h _θ

∂θ − v ^h _r



 

 

  dS (2.23) (l θ ) i =

Z

Ω N _i ∂ p ^h

∂θ dS + Z

∂ Ω N _i r



 

 

  −ν ∂v ^h _θ

∂z n ^z _seg − ν ∂v ^h _θ

∂r n ^r _seg



 

 

  dl

+ ν Z

Ω r ∂N i

∂z

∂v ^h _θ

∂z + r ∂N i

∂r

∂v ^h _θ

∂r − N i

r



 

 

 

∂ ² v ^h _θ

∂θ ² + 2 ∂v ^h _r

∂θ − v ^h _θ



 

 

  dS (2.24) Herein, n ^z _seg and n ^r _seg are the axial and radial components of the unit vector normal to the domain boundary (∂ Ω). The azimuthal component of this vector is always zero due to the axisymmetric nature of the boundaries. Because of the P1 finite element representation, meridional velocity gradients are constant over elements. Therefore, the meridional velocity gradient components involved in the boundary integrals take the value in the adjacent elements. These boundary integrals involve the radial coor- dinate and vanish automatically on the axis due to the multiplication of the Navier- Stokes equations by the radial coordinate (or else by the volume element rdrdθdz).

Some axis boundary conditions are automatically satisfied, which is an additional ad- vantage of this multiplication, but we postpone this discussion to the section devoted to the boundary conditions. In the most general case, these boundary integrals do not require any specific implementation. On a slip wall boundary segment for instance, the tangential stress vanishes. On other boundary types, momentum equations are substituted by the boundary condition.

2.2.2 Spectral expansion

The finite element discretization done in the previous subsection does not take care of the azimuthal dependence of the variable. The cylindrical coordinate system ensures by nature the periodicity in the azimuthal direction (q(z, r, θ, t) = q(z, r, θ + 2π, t)), which leads naturally to a Fourier series expansion which has to be truncated to N values (N represents the number of cross-planes in which the discrete solution is to be determined).

q(z, r, θ, t) = A ⁰ +

N/2−1

X

k = 1

A ^k cos(kθ) + B ^k sin(kθ) + A ^N/2 cos( Nθ

2 ) (2.25)

2.2 Discretized equations 15 The amount of A ^k and B ^k unknowns is strictly identical to the number of azimuthal positions where the unknown is evaluated. The derivatives in the azimuthal direction need to be evaluated for the Navier-Stokes equations. The mode k = N/2 causes a problem since its derivatives cannot be distinguished from the axisymmetric mode k = 0. We therefore drop the term A ^N/2 in the development 2.25. Instead of using the sine and cosine basis functions for the development, we use the e ^Ikθ (I ² = −1) functions which are more appropriate for the computation :

q i (θ n , t) = 1 N

N/2−1

X

k = −N/2+1

Q ^k

i e ^Ikθ

ⁿ

, (2.26)

Q ^−k _i = (Q ^k _i ) ^∗ (2.27)

where Q ^k _i are the Fourier modes associated to node i of the finite element mesh. The Fourier expansion is particularly efficient at low Reynolds numbers, for which flow fields commonly exhibit elongated helicoidal structures.

Relation 2.27 computes the di ff erent Fourier modes according to N equally spaced values q i (θ n , t) in the azimuthal direction. This Fourier transform operation is per- formed for each nodal position at each iteration and relation 2.27 which is O(N ² ) is very ine ff ective. To reduce the cost of Fourier transforms, Cooley and Tukey [19]

proposed in 1965, an algorithm which subdivides the problem into successily smaller ones. Their algorithm performs successive division by two for the size of the vector and can therefore only handle problems where N is a power of two (N = 2 ⁿ ). This de- composition which is O (N.log ₂ N) is referred to as the Fast Fourier Transform (FFT) algorithm.

In the 1990’s, new algorithms were identified to compute rapidly the Fourier Transform of 2,3,5,7,9 and 11 q i (θ n , t). Combining these efficients algorithms [33, 34, 32] to the subdivision techniques of the problem into smaller ones developed by Cooley and Turkey extended the FFT’s to problems where N is not only a power of two but can be any integer of the form N = 2 ^a · 3 ^b · 5 ^c · 7 ^d · 11 ^e .

Each mode k of the physical quantities (v z , v _r , v _θ , p) can be shown to obey the follow- ing set of relations obtained from the discretized Navier-Stokes equations and from Fourier properties:

Z

Ω N _i ∂r(V _r ^k ) ^h

∂r + Ik(V _θ ^k ) ^h + r ∂(V _z ^k ) ^h

∂z dS = 0, (2.28)

Z

Ω N _i r ∂(V _z ^k ) ^h

∂t dS + (L ^k _z ) i = −(C ^k _z ) i , (2.29) Z

Ω N _i r ∂(V _r ^k ) ^h

∂t dS + (L ^k _r ) i = −(C ^k _r ) i , (2.30) Z

Ω N i r ∂(V _θ ^k ) ^h

∂t dS + (L ^k _θ ) i = − (C ^k _θ ) i . (2.31)

(L ^k _z ) i = − Z

Ω (P ^k ) ^h ∂rN i

∂z dS + Z

∂ Ω

N _i r



 

 





 

 

 (P ^k ) ^h − ν ∂(V _z ^k ) ^h

∂z



 

 

 n ^z _seg − ν ∂(V _z ^k ) ^h

∂r n ^r _seg



 

 

 dl + ν

Z

Ω r ∂N i

∂z

∂(V _z ^k ) ^h

∂z + r ∂N i

∂r

∂(V _z ^k ) ^h

∂r + k ² N _i

r (V _z ^k ) ^h dS (2.32)

(C ^k _z ) i =

N/2−1

X

n = −N/2 + 1

(c z ) i (θ n , t)

e ^−Ikθ

ⁿ

(2.33)

(L ^k _r ) i = − Z

Ω (P ^k ) ^h ∂rN i

∂r dS + Z

∂ Ω N _i r

"

−ν ∂(V _r ^k ) ^h

∂z n ^z _seg + (P ^k ) ^h − ν ∂(V _r ^k ) ^h

∂r

! n ^r _seg

# dl

+ ν Z

Ω r ∂N i

∂z

∂(V _r ^k ) ^h

∂z + r ∂N i

∂r

∂(V _r ^k ) ^h

∂r + N i

r

h (1 + k ² )(V _r ^k ) ^h + 2Ik(V _θ ^k ) ^h i dS

(2.34) (C ^k _r ) i =

N/2−1

X

n = −N/2+1

[(c r ) i (θ n , t)] e ^−Ikθ

ⁿ

(2.35)

(L ^k _θ ) i = Ik Z

Ω

N _i (P ^k ) ^h dS + Z

∂ Ω

N _i r



 

 

  −ν ∂((V _θ ^k ) ^h ) ^h

∂z n ^z _seg − ν ∂(V _θ ^k ) ^h

∂r n ^r _seg



 

 

 

dl (2.36) + ν Z

Ω r ∂N i

∂z

∂(V _θ ^k ) ^h

∂z + r ∂N i

∂r

∂(V _θ ^k ) ^h

∂r + N i

r

h (1 + k ² )(V _θ ^k ) ^h − 2Ik(V _r ^k ) ^h i dS

(C ^k _θ ) i =

N/2−1

X

n = −N/2+1

[(c θ ) i (θ n , t)] e ^−Ikθ

ⁿ

(2.37)

This hybrid discretization creates as many equations as unknowns, which are the nodal Fourier components of velocity and pressure. This discretization falls inside the weighted residual methods. For both the finite elements and spectral parts, a Galerkin approach is used, i.e. w = N i e ^−Ikθ .

2.2.3 Convective Terms

As explained in the introduction of the present chapter, the non-linear terms have a dominant contribution for industrial applications since they correspond to mech- anisms generating turbulence. Some flow phenomena are directly related to their presence in the Navier-Stokes equations and to their non-linear character:

• Boundary layers: Close to solid boundaries, a competition between viscous and convective e ff ects creates a zone of intense shear called boundary layer.

This region is relatively small but needs mesh refinement to capture the large

velocity gradients and the small eddies generated near the wall and convected

2.2 Discretized equations 17 in the outer part of the boundary layer where they form the large turbulent structures.

• Instabilities : When increasing the Reynolds number, the laminar shear layers become unstable in the so-called Kelvin-Helmholtz instabilities. These fluctu- ations are characterized by an intrinsic time scale which is directly related to the sound frequency.

These phenomena are relevant for turbulent aeroacoustic predictions and high- light the importance of these terms in the discretized equations. According to Fourier Transform properties, the Fourier transform of a product is a convolution between the Fourier transform vectors of the corresponding terms:

c(θ n ) = a(θ n ) · b(θ n ) (2.38)

C ^k = Σ ^N/2−1 _{n = −N/2 + 1} c(θ n )e ^Ikθ

ⁿ

(2.39)

C ^k = Σ ^N−1 _{n = −N + 1} A ⁿ B ^k−n (2.40) For convective terms, signal a and b represents respectively velocity and its gradi- ent. Convolution formula 2.40 shows that convective terms have a bandwith which is twice larger than the original bandwith of the velocity field (−N < k < N). Boundary conditions impose modal values at the boundaries and activate therefore the corre- sponding azimuthal modes. Convective terms transfer informations from one mode to the others through this convolution product. The active modes will therefore excite di ff erent modes automatically. Except for the inlet, most of the boundary conditions set non-axisymmetric modes to zero. If only axisymmetric flow conditions are im- posed at the inlet, only mode 0 is activated and will not excite any higher azimuthal mode. This blockage is particular for mode 0: an axisymmetric simulation will not trigger by itself three-dimensional non-axisymmetric structures. Perturbations have to be added to trigger the three-dimensional instabilities which after a transitional phase will develop a structured periodic process or generate random chaotic turbu- lent structures.

In practice, this is not an artificial numerical trick: most of three-dimensional flow solvers have already a non-axisymmetry in the mesh and perturbations are al- ways present in experimental setups. These fluctuations can be added even for lam- inar flows; small perturbations analysis confirms the damping effect of the flow on these.

Convective terms are however not the unique non-linear contribution to the dis-

crete Navier-Stokes equations. The pressure and convective stabilization and subgrid

scale models involve square roots, ratios, and more complicated formulas which in

general cannot be computed by a convolution of the different spectra. We use a

k G(k)

k L

Figure 2.1: Fourier transform and discrete Fourier transform (DFT)

pseudo spectral approach [8] in which the non-linear terms are computed in physical space before transformed into Fourier components in order to supply the appropriate terms in the discrete Navier-Stokes equations.

The Fourier transform decomposes a signal into a continuous spectrum G(k). In fluid mechanics, the velocity or pressure spectra are always limited to a wavenumber denoted by k L which is proportional to the inverse of the smallest structure in the flow (refer to chapter 4). The discrete Fourier transform (DFT) operator samples this spectrum into a finite number of values which are called Fourier modes of the repre- sentation. According to the Shannon-Nyquist [92] theorem, DFT does not translate accurately the spectral information if spatial resolution ∆ θ = _N ^2π

_m

is not sufficient:

k L = ^N ₂

^m

. An harmonic signal (e ^I(k

^L

^{+ k}

⁺

^)θ ) with a wavenumber which falls outside this criterion (k ₊ > 0) will not be discarded by the discrete Fourier transform as it should, but introduces a wrong contribution to mode k L − k ₊ . This phenomenon which affects the numerical results is called aliasing.

Except for the non-linear terms, the computation is based on Fourier modes. Sim- ulations restrict the analysis to a limited part of the spectrum (k ≤ N _a ). The Numerical velocity field spectrum will therefore automatically be limited to N a at the end of each iteration. Based on this numerical field, the computation of convective terms induces higher frequencies as illustrated by the convolution formula (−2N a < k < 2N a ).

The spatial resolution ∆ θ fixes the number of terms used by DFT operator N = _∆ ^2π _θ .

The Shannon-Nyquist criterion (2N a < N) would require that no convective term

should be a ff ected by aliasing. This is too restrictive since only half of the convective

spectrum range ( −N _a < k < N a ) is useful for the computation. Convective modes

k > N a are not used and can suffer from aliasing, which leads to the famous two-

2.2 Discretized equations 19

Aliased part (removed) Aliased

Removed part Removed part

part (removed)

! u(! x,t

ⁿ

)

! u · !( u) !

Ideal Fourier Transform Discrete Fourier Transform

Resolution Resolution

N

m

modes 2N

a

modes

N

a

modes N

a

modes

Figure 2.2: Computation of convective terms in SFELES. Left part correspond to an ideal Fourier transform respecting Shannon-Nyquist criterion. Right part is the more economical procedure which satisfies two-thirds rule.

thirds rule [79]:

N a < N − (2N a − N) ⇒ N a < 2

3 N (2.41)

Convective contributions have di ff erent expressions according to the α c param- eter. These formulations differ by a divergence of velocity field which is identi- cally zero at the continuous level for incompressible flows. At the discrete level, the solenoidal character of the numerical velocity field is not strongly imposed and these di ff erent expressions may ensure at the discrete level some conservation principles according to the choice of α c .

The kinetic energy conservation principle corresponds to the integral over a vol- ume V of the scalar product of velocity and momentum equations:

Z

V

v i ρ ∂v i

∂t + ρv i v j

∂v i

∂x j

+ v i

∂p

∂ x _i + v i

∂τ i j

∂x j

dV = 0 (2.42)

For incompressible flows, the kinetic energy conservation simplifies further to:

Z

V

∂

∂t ρv l v _l

2

dV + Z

∂V v _i n _j ρv l v _l

2 + p

δ i j + τ i j

dS =

Z

V

τ i j

∂v i

∂x j

dV (2.43)

This last equation corresponds to conservation of kinetic energy at the continuum

level (i.e satisfying exactly the Navier-Stokes equations). The discrete field however

satisfies a finite number of relations which mimic, up to a certain accuracy, continuity

and momentum equations, in particular the divergence of discrete field is not imposed strongly to zero. The spatial discretization involves a spectral and finite element interpolation (v ^h _i = _N ¹ (V _i ^k ) α N _α e ^Ikθ ).

Using Parseval’s theorem (equation 2.44)

Σ ^N/2−1 _−N/2+1 kq(z, r, θ)k ² = Σ ^N/2−1 _−N/2+1 Q ^k (Q ^k ) ^∗ (2.44) and introducing the spectral expansion in azimuthal direction into equation 2.42 shows that kinetic energy will be preserved if K _e ^k = ^ρV

^kl

^(V ₂

^lk

⁾

^∗

, which represents the kinetic energy associated to a given mode k (no summation is applied on index k in the following relations) satisfies:

Z

Ω

∂K _e ^k

∂t rdzdr + Z

∂ Ω (V _i ^k ) ^∗ n _j

DFT ^k ( ρv l v _l 2 ) + P ^k

δ i j + τ ^k _{i j} rdl =

Z

Ω τ ^k _{i j} ∂V _i ^k

∂ x j

rdzdr (2.45) The correct application of dealiasing techniques presented in the preceding para- graphs ensures the correct kinetic energy preservation during the Fourier transform of convective terms (DFT ^k (v l v _l )). If we take the scalar product between nodal mo- mentum equations and nodal velocity unknowns, we get:

(V _i ^k ) ^∗ _α Z

Ω N _α ∂(V _i ^k ) ^h

∂t rdzdr + (V _i ^k ) ^∗ _α Z

Ω rN _α DFT ^k (v ^h _j ∂v ^h _i

∂x j + α c v ^h _i

∂v ^h _j

∂ x j

)dzdr

−(V _i ^k ) ^∗ _α Z

Ω

∂rN _α

∂x i

(P ^k ) ^h dzdr + Z

∂ Ω (V _i ^k ) ^∗ _α N _α n j

h P ^k δ i j + τ ^k _{i j} i

rdl = Z

Ω (V _i ^k ) ^∗ _α ∂N _α

∂ x j

τ ^k _{i j} rdrdz

Z

Ω

∂

∂t



 

 



ρ(V _i ^k ) ^h ((V _i ^k ) ^h ) ^∗ 2



 

 

 rdzdr + Z

Ω

DFT ^k (v ^h _i v ^h _j ∂v ^h _i

∂x j + α c v ^h _i v ^h _i

∂v ^h _j

∂x j

)rdzdr

− Z

Ω (P ^k ) ^h _α N _α ∂r((V _i ^k ) ^h ) ^∗

∂x i

| {z }

= 0

dzdr + Z

∂ Ω ((V _i ^k ) ^∗ ) ^h n j

h P ^k δ i j + τ ^k _{i j} i

rdl = Z

Ω r ∂((V _i ^k ) ^h ) ^∗

∂x j

τ ^k _{i j} drdz

Z

Ω

∂

∂t K _e ^k h

rdzdr + Z

Ω DFT ^k ( ∂

∂ x j



 

 

 v ^h _j ρv ^h _i v ^h _i 2



 

 

 + (α c − 1 2 )v ^h _i v ^h _i

∂v ^h _j

∂x j

)rdrdz + Z

∂ Ω ((V _i ^k ) ^∗ ) ^h n j

h P ^k δ i j + τ ^k _{i j} i

rdl = Z

Ω r ∂((V _i ^k ) ^h ) ^∗

∂x j

τ ^k _{i j} drdz (2.46)

If we apply Green’s theorem to the convective terms, the first convective term com-

pletes the boundary integral over ∂ Ω and the second vanishes only for α c = ¹ ₂ . Look-

ing back to the kinetic energy conservation principle for azimuthal mode k shows that

the kinetic energy will be preserved if the skew-symmetric form (α c = 0.5) is used

for the convective terms. The kinetic energy of the numerical field follows the same

physical principle, i.e. global kinetic energy content is a balance between the vis-

cous and pressure forces acting on the boundaries, kinetic energy exchange through

2.2 Discretized equations 21 the domain boundaries and losses due to viscous dissipation. In practice, the stabi- lizations and the turbulence models will add artificial dissipation on the numerical solution. This conservation is however used to control the simulation at runtime and stop it if the kinetic energy exceeds a maximal value (parameter MAXKE).

The convective terms are computed in physical space and transformed to Fourier space to obtain the correct right hand side for each Fourier system governing each mode. Since they involve products of quantities, their computation in Fourier space would require a convolution product. The use of the FFT algorithm allows to by-pass this convolution step at a minimal cost (∝ N log N).

The pseudo-spectral approach outlined here above has the important advantage of leading to system matrices which are constant in time. The coupling between Fourier modes occurs only through the non-linear right hand side terms, which are treated explicitly. Modes excited by, for instance, boundary conditions can start exciting other Fourier modes because the products present in the non-linear terms generate Fourier contributions over the entire spectrum.

2.2.4 Matrix form

The discretization of the Navier-Stokes equations is a two step process:

1. We select the positions where we evaluate the solution. These positions are the nodes of the finite element mesh and the number of Fourier modes in the spec- tral expansion in the azimuthal direction. This selection is directly related to the interpolation of the numerical solution (spectral expansion in the azimuthal direction combined to finite element interpolation on unstructured triangular meshes).

2. The problem leads to a solution if the number of unknowns equals the number of independent equations relating these unknowns. This second step corre- sponds to the technique used to derive this set of equations.

For the incompressible Navier-Stokes equations, the nodal velocity components and

pressure ((V _z ^k ) j , (V _r ^k ) j , (V _θ ^k ) j , P ^k _j ) are the unknowns. The spatial discretization leads to

a su ffi cient number of relations between these unknowns (4 complex values for each

node). The weak forms involve classical integrals over the whole domain of products

of residuals by the Galerkin weight functions. These integrals can be computed by

assembling the contributions over each element composing the mesh. This classical

assembling procedure builds the global system by adding the contribution of all pos-

sible relations between the weight function N i and nodal values supported by node

j. Thanks to the compact support of Galerkin weight functions, the contributions are

zero if node i and j do not belong to the same element. For the linear part of the

equations (left hand side of the system of ODEs):