Refined damped equivalent fluid models for acoustics

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Refined damped equivalent fluid

models for acoustics

Clément Sambuc

Supervisor: Prof. Jean-Louis Migeot

January 2015

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Doctoral committee:

Prof. Gérard Degrez (President)

Free University of Brussels, Fluid’s Mechanics Department, Avenue F.D. Roo-sevelt 50, 1050 Brussels, Belgium.

Prof. Arnaud Deraemaeker (Secretary)

Free University of Brussels, Structural and Material Computational Mechanics Department, Avenue F.D. Roosevelt 50, 1050 Brussels, Belgium.

Prof. Jean-Louis Migeot (Supervisor)

Free University of Brussels, Brussels School of Engineering, Avenue F.D. Roo-sevelt 50, 1050 Brussels, Belgium.

Prof. Jean-Pierre Coyette

Université Catholique de Louvain, École Polytechnique de Louvain, Place du Levant 1, 1348 Louvain-la-Neuve, Belgium.

Dr. Grégory Lielens

Free Field Technologies, Axis Park Louvain-la-Neuve, 9 rue Emile Francqui, 1435 Mont-Saint-Guibert, Belgium.

Dr. Ysbrand H. Wijnant

University of Twente, Structural Dynamics and Acoustics Department, Drienerlolaan 5, 7522 NB Enschede, Netherlands.

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Remerciements

La réalisation de cette thèse n’aurait pas été possible sans le soutien et la coopéra-tion de nombreuses personnes. Avant tout, j’adresse mes plus vifs remerciements aux personnes qui ont supervisé ce travail : Messieurs Jean-Pierre Coyette, Jean-Louis Migeot et Grégory Lielens.

Merci de m’avoir donné l’opportunité de mener à bien ce travail de recherche initié dans le cadre du projet européen Marie-Curie ITN ATCoMe (Advanced Techniques for Computational Mechanics). Merci de m’avoir permis d’intégrer l’équipe de déve-loppement de Free Field Technologies afin d’explorer le monde de la programmation industrielle conjointement à la recherche théorique.

Je tiens particulièrement à exprimer ma gratitude à Grégory Lielens qui ne sera ja-mais qualifié à tort de sage et de savant. Il sait, invente et transmet ja-mais aussi écoute, comprend et tempère. Merci d’avoir su me laisser la liberté nécessaire à l’accomplis-sement de ces travaux, tout en y gardant un oeil critique et avisé.

J’adresse mes sincères remerciements aux professeurs affiliés à l’Université Libre de Bruxelles, Messieurs Gérard Degrez et Arnaud Deraemaker, respectivement pré-sident et secrétaire du jury, pour leur relecture et suggestions sur ces travaux. Je souhaite aussi exprimer ma reconnaissance envers Monsieur Ysbrand Wijnant, chercheur à l’Université de Twente et spécialiste de l’acoustique viscothermique, qui a accepté de faire partie du jury d’examinateurs.

Le cadre particulier de ce travail (à la fois académique et industriel) à constitué un atout important dans le développement de ma carrière.

Ainsi, je souhaite remercier les professeurs Antonio Huerta et Sonia Fernández-Méndez, référents principaux du commité d’accompagnement du projet ATCoMe. Merci à Vladimir, Sudhar, Augusto, Miquel, Sudhakar, César, Aleksandar et Omid, jeunes chercheurs du projet avec qui j’ai pu échanger favorablement sur des sujets scientifiques mais aussi partager de très bons moments lors des réunions et sémi-naires du projet ATCoMe.

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Au cours de ces quatres dernières années, le développement au sein du code in-dustriel Actran à constitué une part importante de mon activité. J’adresse ma sincère reconnaissance à Benoît Van den Nieuwenhof, responsable de l’équipe de dévelop-pement, pour sa pédagogie efficace et sa patience.

De plus, je tiens à remercier l’ensemble de mes collègues de Free Field Technologies pour leur aide qu’ils ont su m’apporter respectivement à leur domain d’expertise. Un grand merci pour l’ambiance chaleureuse et conviviale qu’ils ont su transmettre au quotidient.

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Résumé

Ce travail porte sur l’étude de certains phénomènes d’amortissements intervenant dans l’acoustique des petites cavités.

En physique, un phénomène d’amortissement se traduit par une atténuation des mou-vements élémentaires d’un système mécanique à travers la dissipation d’une partie de l’énergie qui les engendre. Le plus souvent, l’énergie est dissipée sous forme de chaleur. Ces transferts thermiques peuvent provenir de différences locales de tempé-rature ou bien de forces de frottements liées de manière plus ou moins directe à la vitesse de la particule en mouvement. Lorsqu’une petite perturbation se propage au sein d’un fluide newtonien et caloporteur borné par un mur rigide et isotherme, ces mécanismes dissipatifs particuliers se localisent aux abords des parois et jouent un rôle significatif dans certaines situations.

Parmi les exemples d’applications pratiques, il est possible de citer les appareils d’aide auditive, les systèmes microélectromécaniques (transducteurs, microphones et haut-parleurs), les matériaux absorbants constitués de fins réseaux capillaires ou de pores aux dimensions réduites, les systèmes de silencieux, d’échangeurs de cha-leur thermo-acoustiques ou tout autre appareil mettant en jeu des cavités résonantes aux dimensions réduites (micro-acoustique).

Afin de prédire correctement la propagation acoustique au sein de tels micro-systèmes, il est nécessaire de résoudre l’ensemble des équations représentant le com-portement dynamique du fluide considéré. pour résoudre Il est ainsi possible d’ap-procher numériquement les solutions du problème initial, exprimé dans sa forme va-riationnelle(ou forme faible).

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solutions, la convergence ou encore la stabilité du schéma numérique ne seront pas abordées ici.

Cette thèse est scindée en deux parties. Nous nous intéressons tout d’abord à la modélisation de la propagation acoustique au sein de fluides visqueux et conducteurs de chaleur. Les équations linéarisées de Navier-Stokes-Fourier sont rappelées dans un premier temps. Cette introduction est complétée par une interprétation approfon-die des paramètres acoustiques, visqueux et thermiques apparaissant naturellement dans les équations. Seuls les fluides assimilables à des gaz parfaits, pour lesquels les longueurs caractéristiques des effets visqueux et thermiques restent largement infé-rieures aux longueurs d’ondes acoustiques seront considérés.

Le chapitre suivant s’intéresse à la modélisation de la propagation acoustique au sein de géométries simples particulièrement utilisées dans l’industrie : les guides d’onde. En particulier, l’effet d’un écoulement de nature hydrodynamique sur la pro-pagation et l’atténuation d’une onde sonore est étudié. La formulation complète de Navier-Stokes-Fourier est assimilée à un système aux valeurs propres, puis résolue à l’aide d’une méthode numérique classique.

Cette solution de référence est ensuite comparée à une formulation simplifiée, basée sur l’extension d’un modèle existant bien connu : le modèle LRF (Low Reduced Fre-quency). Dans ce nouveau modèle, l’hypothèse du fluide au repos est écartée et l’on considère la présence d’un écoulement constant laminaire au sein du guide d’onde. Les solutions correspondant à cette formulation XLRF (eXtended Low Reduced Fre-quency) sont comparées à d’autres solutions analytiques provenant de la littérature. L’effet de cet écoulement sur l’atténuation acoustique est mis en évidence et la vali-dité de certaines approximations proposées est argumentée.

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’grossiers’ construits à partir des critères de l’acoustique classique.

Afin d’illustrer les différentes méthodes proposées dans les chapitres 2 et 3, des modèles d’application simplifiés sont présentés, tels qu’un convertisseur catalytique, ainsi qu’un système absordant large bande consitué de résonateurs quart d’onde.

La seconde partie a pour but l’étude des formulations simplifiées permettant de décrire le comportement acoustique au sein de matériaux biphasiques, plus particuliè-rement des matériaux poro-élastiques (composés d’une partie solide élastique saturée par un fluide). Ce type de composant est très utilisé dans l’industrie en raison de ses caractéristiques absorbantes dans le domaine des moyennes et hautes fréquences. Ce projet s’est orienté vers l’étude d’une particularité importante bien souvent négligée jusqu’alors : l’anisotropie (certaines propriétés intrinsèques du matériau varient en fonction de la direction).

Une étude bibliographique préliminaire nous a permis d’exprimer l’ensemble des équations aux dérivées partielles modélisant à la fois les interactions fluide/structure et l’anisotropie générale des matériaux. La formulation originale de Biot a été modi-fiée afin d’expimer l’ensemble des coefficients de couplage fluide/structure à l’aide de paramètres fluides équivalents (densité et module de compressibilté). Ainsi, la plu-part des micro-modèles fluides disponibles dans la littérature peuvent être utilisés. De plus, l’influence d’un écoulement interne circulant dans le matériau a été étudiée. Cet effet peut directement se traduire par une modification du tenseur de densité ani-sotrope équivalente.

Finalement, cette réflexion nous a permis d’aboutir à un modèle de matériau poro-élastique isotrope transverse intéressant. En effet, en combinant le modèle fluide XLRF présenté dans le chapitre 2 avec la formulation proposée, la modélisation de structures capillaires orientées (comme les matériaux utilisés dans les catalyseurs au-tomobiles) s’en trouve grandement simplifiée. Ce modèle de matériau poreux rigide a été comparé avec l’implémentation directe de la XLRF en utilisant un maillage plus raffiné discrétisant les capillaires.

Le dernier chapitre s’intéresse à la modélisation d’interfaces biphasiques sépa-rant deux domaines fluides. En pratique, ces interfaces peuvent être assimilable à des plaques perforées ou micro-perforées mais aussi des plaques fines de matériaux poro-élastiques. Ce travail clarifie les différents mécanismes décrivant la propagation acoustique à travers ce type d’interfaces tout en utilisant le formalisme bien connu des impédances de transfert. Différents modèles analytiques de plaques perforées rigides ont été confrontées. De plus, l’analogie entre ces modèles et un modèle générique de fluide équivalent a été mise en évidence.

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Summary

This thesis deals with the acoustics of small cavities, where particular damping mech-anisms occur. In physics, the damping of a considered mechanical system corre-sponds to the attenuation of the particle’s movements. A part of the energy which generates these movements is dissipated through different types of phenomena. In fluid dynamics, when a small perturbation is propagating within a Newtonian and heat-conducting fluid bounded by a rigid and isothermal surface, viscous and thermal dissipative mechanisms are generated near the walls and can have a significant im-pact on the acoustic behaviour of the system in some circumstances. Heat transfers representing the dissipation arise from both local fluctuating temperature gradients and viscous forces proportional to the velocity of fluid particle.

Several types of practical applications are of interest, among which: hearing aids, micro-electro-mechanical systems (transducers, microphones and loud-speakers), ab-sorbing materials made of thin capillary nets or small pores, dissipative silencers, thermo-acoustic heat exchangers, or any kind of device bringing into play small res-onant cavities filled with a dissipative fluid (micro-acoustics).

In order to accurately predict the acoustic wave propagation within such micro-systems, it is necessary to express the set of equations that represents the dynamical behaviour of the studied fluid. For industrial purposes, the finite element method is often used. Thus, it becomes possible to numerically evaluate the solutions of the initial system, written in its variational form (or weak form).

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addressed here.

This thesis is divided into two parts. The physical modelling of the acoustic wave propagation within viscous and heat conducting fluids is addressed first. The linearized Navier-Stokes-Fourier equations are introduced and completed by a thor-ough analysis of the different acoustic and visco-thermal parameters that naturally appear in the dimensionless equations. Only perfect gases, for which the charac-teristic lengths of the thermo-viscous effects remain much smaller than the acoustic wavelengths, are studied.

The next chapter deals with acoustic propagation within specific geometries, par-ticularly used in various sectors of the industry: waveguides. In particular, the effect of a hydrodynamic flow on the propagation and attenuation of a sound wave is stud-ied. The Navier-Stokes-Fourier formulation is rewritten into an equivalent eigenvalue problem, and solved by means of a numerical method.

This reference solution is then compared with a new simplified formulation based on the extension of an existing method: the LRF model (Low Reduced Frequency). In this new model, the hypothesis of the fluid at rest is relaxed and we consider a con-stant laminar mean flow circulating inside the waveguide. The XLRF (eXtended Low Reduced Frequency) solutions are also compared with analytical solutions derived for simplified cases. The mean flow effect on the sound attenuation is highlighted and the validity of the approximations involved is discussed.

In chapter 3, no action of the mean flow on the acoustic part is considered, the fluid is assumed at rest. No geometrical restriction is applied and we address the visco-thermal acoustic modelling of 3D arbitrary geometries. First, a particular in-terest is given to the equivalent fluid formulations due to their ease of implementa-tion. Afterwards, a new method for the estimation of the acoustic field within small arbitrary cavities is presented. This model is based on different considerations com-ing from existcom-ing models (Boundary Layer Impedance model and Sequential Lin-earized Navier-Stokesmodel) as well as the estimation of a wall-distance field. This DBLNSF (Distance-Based Linearized Navier-Stokes-Fourier) model shows practical benefits in comparison with others methods, especially in a finite element context. Indeed, it becomes possible to diminish the error caused by the coarsening of the mesh, providing a filtering technique of the visco-thermal solutions. Consequently, coarses meshes based on classical isentropic acoustic criteria can be used to solve 3D visco-thermal acoustics keeping good accuracy and efficiency.

In order to illustrate the proposed methods of the chapters 2 et 3 some simplified application cases are also presented, such as a catalytic exhaust converter system and a broadband noise absorber made of quarter-wave resonators.

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fluid). This type of acoustic component is widely used in industry because of their good absorbing properties in the medium- and high-frequency range. This project focuses on a physical property that was frequently neglected until now: anisotropy (some intrinsic properties of the material vary depending on the direction).

A preliminary bibliographic study deals with the derivation of the set of partial order differential equations that account for both fluid/structure interactions and the anisotropy of a given poro-elastic material. It has been shown that the original Biot’s formulation can be slightly modified in order to express all the coefficients of the coupled system in function of some equivalent fluid parameters (density and bulk modulus). Thus, most micro-models for equivalent fluids available in the literature can be used to express the full elasto-acoustic problem.

In addition, the effect of an internal mean flow circulating inside the material has been studied. This mechanism results in a modification of the density into an equivalent anisotropic tensor.

Finally, this consideration induced the formulation of an original model of transverse isotropic poro-elastic material. Actually, by combining the XLRF model presented in chapter 2 with the proposed anisotropic formulation, the modelling of transversely orientated capillary materials (for instance catalyst substrates) is greatly simplified. This rigid-porous formulation has been compared with the direct implementation of the XLRF involving a fully discretized capillary net.

The last chapter addresses the modelling of the acoustic transmission between two acoustic domains separated by a biphasic interface. In practice, such interfaces correspond to either perforated or micro-perforated plates or thin plates of poro-elastic materials. This work clarifies the physical mechanisms involved in these types of acoustic interfaces by using the so-called transfer impedances formalism.

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Introduction to visco-thermal

wave propagation

1.1 Visco-thermal wave propagation

1.1.1 Background

Acoustic science refers to a special case of fluid dynamics, describing the behavior of a material submitted to harmonic excitations in terms of pressure. The fluid’s pressure repeatedly fluctuates but so do its velocity, density and temperature. A conversion of energy occurs periodically between kinetic energy (linked to velocity perturbation) and potential energy (linked to the increased pressure and temperature).

In isentropic acoustics (also called ‘classical’ or ‘standard’ acoustics), all the fluctuating variables can be efficiently expressed in terms of pressure fluctuations. A description of the wave propagation in function of compressibility and inertia effects condensed within a single equation is therefore possible. However, in that case, some intrinsic properties of the fluid material are not taken into account, such as its vis-cosity and its thermal conductivity. This corresponds to approximating the fluid as inviscid and adiabatic and considering the acoustic propagation as governed by an isentropic process (there are no energy losses over time and space).

Visco-thermal acousticscan be regarded as a generalization of standard acous-tics. Shear forces or heat flows due to thermal conduction in the fluid are no longer neglected and might play an important role. Figure 1.1 should help the reader in un-derstanding the mechanisms of visco-thermal acoustics.

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stated previously in the case of isentropic acoustics can become inaccurate. If the boundary surface is fixed and rigid, the local fluid velocity at the boundary is zero. Conceptually, there is no difference between the surface velocity and the fluid veloc-ity at this location. Such a statement is called the no-slip velocveloc-ity condition. At the same time, the acoustic waves propagate imposing a fluctuating movement to the fluid velocity and the velocity does not equal zero at some points of the bounded medium. Thus, important velocity gradients appear in a region close to the fixed boundaries, locally creating a shearing motion in the fluid. Because of the fluid’s viscosity, some forces (called viscous forces) are created and oppose the shear forces generated by the acoustic waves. This reaction depends on the dynamic viscosity of the fluid and involves the dissipation of some part of the acoustic wave’s energy as heat (noted v in figure 1.1). This results in slowing down and damping out locally the acoustic

wave. Beyond a certain distance from the surface, called the viscous boundary layer thickness(noted viscous BLT), the shear forces become negligible. Finally, this im-plies that the viscous dissipations are confined inside this boundary layer (here the viscous boundary layer). Even if the viscosity is very small, the velocity gradients can be very large and the viscous effects are still significant.

Similarly, the fluid experiences temperature fluctuations inside the medium due to the passing pressure wave. The fluctuating temperature is also called acoustic temperatureor perturbed temperature. Throughout this thesis, the term ‘temperature’ alone always refers to as the fluctuation of temperature around a quiescent state, or mean value. If the thermal conductivity of the boundary surface is much higher than the thermal conductivity of the fluid, the acoustic temperature locally goes to zero. Such a condition is called the isothermal boundary condition and also induces large gradients of the fluctuating temperature, confined within another boundary layer (called the thermal boundary layer). The heat flux created in the normal direction to the boundary vanish at a certain distance from the wall (called the thermal boundary layer thickness, and noted thermal BLT), where energy is no longer dissipated. This characteristic length depends on the thermal conductivity property of the medium and, for most gases and fluids, the viscous BLT and thermal BLT have the same order of magnitude.

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1.1 Visco-thermal wave propagation 3 Acoustic fluctuations Boundary layers Bulk domain δv Viscous BLT δ_t Thermal BLT Energy dissipations ǫv,t δ_t ǫ_t ǫ_v v′wall = 0 ∇v′ bulk= 0 T_wall′ = 0 ∇T′ bulk= 0

Figure 1.1: Visco-thermal boundary layers close to a rigid no-slip/isothermal surface.

1.1.2 Applications

It is worth noting that viscous forces and heat flux generated by viscid and heat-conducting fluids are irreversible processes, dissipating part of the acoustic energy. Those damping phenomena depend on the shear viscosity and the thermal conduc-tivity of the fluid. As a matter of fact, they are usually neglected in a lot of acoustic applications. This is related to the very small order of magnitude of the viscous and thermal boundary layers compared to the dimensional scale of the system.

Nevertheless, once the total volume enclosed within the boundary layers becomes significant compared to the total volume of the acoustic domain, the impact of visco-thermal dissipations on the acoustic response cannot be neglected anymore. Due to the development of miniaturization in many fields of industry, these conditions are met in an increasing number of applications. Below is a non-exhaustive list of prac-tical cases where visco-thermal acoustics turn out pertinent:

I Dynamic microphones consisting of a membrane backed with a thin fluid layer

[17, 116, 30, 81].

I In-ear hearing aids [77]

I Vocal tract, trachea and lungs [45]

I Inkjet print heads and propagation in narrow ink channels [139, 15]

I Plates and double wall panels for reduction of sound transmission [12, 109, 82]

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I Charge air coolers used on turbocharged internal combustion engines [84]

There are many other types of applications of interest. In some cases where the boun-dary layer thicknesses are small compared to the dimension of the domain, visco-thermal effects can still become significant near resonance. At resonance, those ef-fects prevent the amplitude of the acoustic waves to rise up to infinity if the fluid is considered perfectly inviscid and adiabatic. Visco-thermal dissipations damp out the acoustic waves but also cause a small shift of the resonance frequency. Finally, the handling of such mechanisms can provide a much better accuracy than classi-cal acoustic models in order to estimate the resonance frequencies or the pressure amplitude within some systems where the design of the resonance is very important.

1.1.3 History

This section gives a short overview of the scientific developments accomplished in the field of visco-thermal wave propagation, starting from the first considerations about adiabatic processes within a fluid, to the last numerical methods that have been proposed to deal with visco-thermal acoustic dissipations.

The first effects of heat and viscosity on the propagation of waves were intro-duced by Laplace, who recognized that for gases with a constant heat capacity, a polytropic law agrees with an adiabatic process [115]. Then Stokes [130], following the works of Navier, derived the momentum equation applied to fluid dynamics and introduced a frictional resistance related to the coefficient of viscosity.

The major reference that laid down the basis of visco-thermal theory has been done by Kirchhoff [83] who showed that the losses due to heat conductivity and the ones due to viscosity are of the same order of magnitude. Consequently, the internal en-ergy equation had to be included to the set of Navier-Stokes equations (continuity and momentum conservation equations) to account for thermal effects.

Kirchhoff derived a fourth-order dispersion equation from the resulting set of equa-tions and proposed afterwards a solution corresponding to acoustic propagation within wide tubes (when the boundary layers are small compared to the cross-section of the tube). Then, Rayleigh [120] derived the solutions for narrow tubes using Kirchhoff’s equations.

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1.1 Visco-thermal wave propagation 5

Later on, Tijdeman [135] validated Zwikker and Kosten’s approximations by using a numerical solution of Kirchhoff’s dispersion equation. He introduced the term Low Reduced Frequency(LRF). Two decades later, Beltman [15, 16, 17] introduced a use-ful dimensionless notation and formulated the LRF model for thin layers and other types of waveguides with particular geometries.

Concurrently, with the technological advances of computer science, numerical methods have been increasingly used for scientific research in physics. A pioneering work was presented by Shields et al. [126] in 1965, involving an iterative numerical procedure in order to solve Kirchhoff’s dispersion equation. Several authors, such as Bruneau et al. [31], or Liang and Scarton [91], successively reported numerical techniques to find the root of Kirchhoff’s dispersion equation. Thanks to that, the existence of vortical (or vorticity) modes linked to viscous effects and entropic (or thermal) modes linked to heat conductivity has been highlighted.

More recently, an hybrid analytical and numerical method to model visco-thermal effects has been proposed by Bossart et al. [26]. The dissipations occurring within the boundary layers are condensed within an equivalent boundary condition. This reduction is combined with a classical isentropic acoustic equation and solved by means of Finite Element (FE) or Boundary Element (BE) methods. This formulation remains valid only for ‘slightly’ visco-thermal conditions (which means that the vis-cous and thermal boundary layers have to remain small compared to the dimensions of the domain).

In addition, some studies focusing on the resolution of the fully coupled set of equations describing visco-thermal acoustic wave propagation have been succes-sively completed . Among them, one can cite the works of Malinen et al. [97] Joly [75], Nijhof et al. [108] and Kampinga et al. [79]. After discretization of the full set of equations, one obtains a matrix system quite complex to solve. The stabil-ity, conditioning, convergence and robustness of the finite element formulation has been subjected to careful analyses in order to provide efficient and accurate resolution methods.

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1.1.4 Modelling techniques

Before going onto a mathematical description of the physics involved in visco-thermal wave propagation (which starts with the set of so-called Full Linearized Navier-Stokes-Fourierequations), an overview of the different types of models available in the literature is given here. For detailed explanations about the different strategies presented, the reader can refer to Nijhof [107], who realised an extensive review of the solutions available in the literature.

The resolution techniques can be separated according to several criteria. For all of those techniques, the starting point is the full set of Linearized Navier-Stokes-Fourier equations that will be presented later on. On the one hand, some strategies bring into play the full set of equations without any reduction but on the other hand, some ap-proximations can be done in order to reduce the complexity of the problem. Similarly, the models vary from being purely analytical to purely numerical. Those aspects will have a strong influence on the accuracy, the efficiency and the versatility (or appli-cability) of the models. They each have their own advantages and disadvantages depending on the configuration of the physical system to model. All the physical models can be described following the classification below.

Reduced and non-reduced models

Reduced models are coming from reduction techniques of the full set of LNSF equa-tions. They usually have the advantage of being more efficient compared to non-reduced models (or fully coupled visco-thermal models). This is not always the case but this gain of efficiency is generally made at the expense of a lower accuracy. The use of a given reduced model is thus driven by a tradeoff between the expected ac-curacy and the computational costs involved to reach the solution. For instance, the LRF approximate models are only available for particular geometries.

Although such models only remain valid in certain conditions, they have the advan-tage of providing extremely fast and (in most cases) accurate solutions. The use of analytical expressions is particularly well suited to those types of strategies.

Finite Element and analytical methods

With the technological advancements and increases in memory, cpu-power and solver efficiency, the use of Finite Element (FE) formulation has gradually increased over the last few years. Usually, one of the main advantages of the FE models is that solutions remain valid for any types of arbitrary geometries, which is certainly not the case with analytical solutions. Ensuring stability of the numerical scheme and a suitable mesh refinement lead to solutions of good accuracy.

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com-1.2 Derivation of the basic set of equations 7

pute, however they have some limitations dealing with the geometry or the coordinate system, for instance.

Reduced strategies can involve either analytical or numerical (using FE) methods. Similarly, FE methods can be used to solve the full set of coupled equations as well as a reduced formulation (involving a given set of approximated equations or a unique wave equation). The present thesis mainly focuses on both aspects of the modelling strategies presented above. Existing reduced models for simulating visco-thermal wave propagation are presented and an emphasis is put on new types of extensions or reduction strategies. Moreover, because of the particular framework of this research project, a particular interest is given to FE methods.

1.2 Derivation of the basic set of equations

This section details the statements and the underlying assumptions that are made to describe the dynamic behaviour of viscous, compressible and heat-conducting fluids. The equations governing such models are usually named “Navier-Stokes Compress-ible equations” by fluid dynamicists or often “Barotropic equations” by mathemati-cians. Furthermore, the term “Navier-Stokes equations” is often used (and sometimes misused) to refer to different descriptions of fluid dynamics. A lot of works available in the literature suffer from deficiencies in clearly describing the assumptions and statements considered to establish a given set of equations. Sometimes, the “Navier-Stokes equations” can refer to incompressible or compressible Newtonian fluids and may or may not be linearized.

The following derivation intends to give a concise formulation of the “Navier-Stokes Compressible equations” for Newtonian fluids. The term “Navier- “Navier-Stokes-Fourier equations” seems more appropriate to describe such kinds of fluids than the term “Navier-Stokes Compressible equations”. Indeed, since the fluid’s temperature follows Fourier’s law of heat conduction (see [142]), such a designation is considered as clearer. From this point onwards, the name “Navier-Stokes-Fourier” (or NSF) will be retained.

1.2.1 Continuum hypothesis

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Statistical mechanics have to be used when the continuum hypothesis is not re-spected. The choice between conventional fluid dynamics or statistical mechanics is driven by the Knudsen number, which is defined as the ratio of the molecular mean free path to a representative physical length scale. If the Knudsen number exceeds one, reliable solutions must be found using statistical mechanics. Air under standard atmospheric conditions is regarded as a continuum medium once the dimensions of the domain exceed the order of the micrometre. Hence, microfluidics with smaller characteristic lengths or NEMS1 fall out of the framework of this thesis. Acoustics within MEMS2becomes more and more popular due to the trend of miniaturization. Such systems must be studied carefully because the Knudsen number approaches unity in some circumstances. In some cases, the perfect gas can become flawed and the molecular movements should also be taken into account.

1.2.2 Conservation laws

The basic equations of continuum mechanics are the conservation laws for mass, mo-mentum and energy. Those conservation laws describe how a particular measurable property of an isolated physical system changes with respect to time as a control vol-ume deforms. This property is space- and time- dependent and corresponds to a fluid quantity per unit volume. The Reynolds transport theorem, derived in [89] can be used to express the time rate of change of a given quantity:

DΨ Dt = Z V ∂ρψ ∂t dV+ Z S ρψ(v · n)dS, (1.1) noting that:

I Ψ(t) is the total amount of the extensive property I ψ(x, t) is the extensive property per unit mass I ρ is the density of the material

I Vand S are respectively the control volume and the surface bounding the

con-trol volume

I _∂t∂ and _DtD are respectively the partial and material time derivatives I v is the total velocity of the medium

1_{Nano Electro Mechanical Systems (NEMS) are a class of devices integrating electrical and}

me-chanical functionality in the nanometer range.

2_{Micro Electro Mechanical Systems are made up of components between 1 to 100 micrometres in}

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1.2 Derivation of the basic set of equations 9

I n is the normal vector of the surface S

This theorem states that the rate of change of a given fluid property in a given control volume, equals the time rate of change at a point plus the net flow of the property through the control surface.

1.2.2.1 Conservation of mass

The fluid propertyΨ is substituted by the mass m of the fluid medium. Consequently, the extensive property per unit mass ψ becomes unity. The rate of change of mass (material derivative) equals the total amount of generated (or destroyed) mass in the control volume. A volumetric external source of mass qm can also be introduced

within the system. Finally, the transport theorem expresses as: Z V ∂ρ ∂tdV+ Z S ρ(v · n)dS = Z V ρqmdV. (1.2)

This equation states that the time rate of change of mass in the control volume V combined with the flux of mass flowing out of V through the surface S equals the total rate of mass generated (resp. destroyed) by external sources (resp. sinks if qm< 0).

1.2.2.2 Conservation of momentum

If the propertyΨ is chosen to be the material momentum M = m · v, the associ-ated time rate of change corresponds to the net force F from Newton’s second law. Thus, the specific quantity ψ equals the velocity of the fluid and the transport theorem yields: DM Dt = Z V ∂ρv ∂t dV+ Z S ρv(v · n)dS = ΣF = Z S (σ · n)dS + Z V ρqbdV. + Z V ρqmvdV (1.3)

The left-hand side of the equation represents the addition of the time rate of change of the momentum at a given point with the momentum of the mass flowing out of V through S per unit time. The net force (right-hand side) is divided into three parts:

I the surface forces described by the dot product of the stress tensor σ and the

normal direction to the surface S

I the external body forces such as gravity, introduced by the vector qb

I and the momentum production due to the external volumetric source of mass

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1.2.2.3 Conservation of energy

Replacing the extensive property per unit mass by the total specific energy E leads to the derivation of the energy equation. The power is the rate of change of the extensive property and equals the substantive derivative of ρE. The energy equation finally reads: Z V ∂ρE ∂t dV+ Z S ρE(v · n)dS =Z S (σ · v) · ndS − Z S (qh· n)dS + Z V qedV +Z V ρ(qb· v)dV+ Z V ρqm(v · v)dV. (1.4)

The left-hand side represents the time rate of change of E in V combined with the energy flow out of V through S per unit time. The power on the right-hand side is divided into five parts:

I the net work done per unit time by the stress tensor σ acting on S

I the thermal diffusion through S per unit time provided by the heat flux vector

qh

I the volumetric energy supply to V per unit time due to the external rate of

specific energy supply qe

I the work done per unit time by the body forces qbexerted on V

I the additional volumetric energy supply to V per unit time due to source of

mass qm

1.2.3 Differential form

The conservation laws have been given in integral form, which is also referred to as the ‘weak’ representation. The different integrands must be integrable on any arbi-trary volume V and such a restriction justifies the use of the ‘weak’ term. It is possible to express the governing equations in local form by applying Gauss’s theorem to the surface integrals. For an arbitrary control volume V, the partial differential equation (PDE) conservation form can be written as follow:

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These balance equations or continuity equations are the so-called ‘strong’ represen-tationof the conservation laws. The different space and time derivatives must exist. This is a ‘stronger’ restriction compared to the ‘weak’ form in the sense that it limits the use of partial differential equations to continuous solutions. Discontinuous be-haviours require the ‘weak’ representation of the balance equations or some added terms modelling jump shock conditions known as the Rankine-Hugoniot jump con-ditions.

1.2.4 Constitutive relations

Solids, fluids and visco-elastic materials deform under the action of external forces, and the deformations are not directly related to the internal forces. In order to ensure the uniqueness of the solution of the set of conservation equations, it is necessary to provide the missing connections that close the system.

The constitutive relations or constitutive laws introduce the expressions of the stress tensor σ, heat flux vector qh and total energy E in function of the fluid dynamic

quantities, such as pressure, temperature and velocity.

The coefficients used in the equations below are chosen in such a way that fluid friction always opposes to shear gradients and dilatation in a flow. Moreover, heat flows are defined by default from hot to cold in order to satisfy the second law of thermodynamics.

1.2.4.1 Fourier’s law

Fourier’s law, also known as law of heat conduction, states that the time rate of heat transfer is proportional to the negative gradient of the temperature. Therefore, the heat flux vector is written as:

qh = −λ∇T. (1.6)

The thermal conductivity, λ, is often treated as a constant. Though this is not always true, the conductivity of a material can be submitted to small variations that depend on the mean temperature. In any case, the second law of thermodynamics requires that λ ≥ 0.

1.2.4.2 Stresses in isotropic Newtonian media

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I The stress is hydrostatic and the pressure is the thermodynamic pressure when

the fluid is at rest.

I The stress tensor is linearly proportional to the deformation tensor.

I The viscous "intrinsic" torque per unit volume on a fluid element

(antisymmet-ric part of the viscous tensor) is negligible.

I The fluid is isotropic; which means that the magnitude of the resistance of the

fluid is the same along each directions.

The above statements lead to an expression of the stress tensor σ as a function of the viscous stress tensor τ and the pressure as:

σ = −pI + τ. (1.7)

I is the identity tensor and τ represents the viscous stress tensor which reads:

τ = ν(∇ · v)I + µ(∇v + (∇v)T

). (1.8)

µ is the first coefficient of viscosity (also called dynamic or shear viscosity) and ν is the second coefficient of viscosity. This last parameter relates to the so-called bulk viscosity or volume viscosity coefficient µ0 through the expression µ0 = ν + 2/3µ. Throughout this thesis, the viscosity coefficients are considered scalar and constant.

The symmetric part of the tensor τ (first term in the right-hand side of equa-tion (1.8)) is the isotropic stress. This stress may be observed when the material is compressed or expanded at the same rate in all the directions and it acts as an extra pressure that appears only while the material is being compressed. However, unlike the true hydrostatic pressure, this is proportional to the rate of change rather than the amount of compression. This term remains very small compared to the second one, as soon as the volume stops changing.

The antisymmetric part of τ (linked to the shear viscosity µ) corresponds to the stress associated to the progressive shearing deformation. This is the viscous stress that resists to the fluid motion occurring within a tube of uniform cross-section (Poiseuille flow) or between two parallel moving plates (Couette flow).

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1.2.4.3 Equations of state

The equations of state are thermodynamic equations describing the state of matter under a given set of physical conditions. The ideal gas law is most accurate for monatomic gases at high temperatures and low pressures since it neglects both molec-ular size and intermolecmolec-ular attractions. However, this equation becomes increasingly inaccurate at higher pressures and lower temperatures. The relative importance of in-termolecular attractions diminishes with increasing thermal kinetic energy, i.e., with increasing temperatures. Therefore, a number of more accurate equations of state have been developed for gases and liquids.

The model of ideal gas follows from a statistical treatment of particles which ex-change kinetic and vibrational energy in perfect elastic collisions. The molecules, or atoms, of the gas are assimilated to points having significant mass but no significant volume. Hence, the potential energy due to intermolecular forces is neglected in the model.

The constitutive ideal gas law, also called thermal equation of state, is given by :

p= ρRT, (1.9)

noting R the specific gas constant.

The caloric equation of state for an ideal gas yields:

de= CVdT (1.10)

where d indicates a differential change and CV the specific heat at constant volume.

Rrelates to the specific heats at constant volume or constant pressure, respectively CV and Cpfollowing:

R0= Cp− CV (1.11)

1.2.5 Navier-Stokes-Fourier conservation equations

1.2.5.1 Mass conservation

The second term in the left-hand side of equation (1.5a) can be rewritten using the gradient decomposition rule.

∂ρ

∂t +(v · ∇)ρ+ ρ(∇ · v) = ρqm. (1.12)

Knowing that the material derivative expresses as: D

Dt{?} = ∂

∂t{?} + (v · ∇){?}, (1.13)

the equation (1.12) becomes: Dρ

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1.2.5.2 Linear momentum conservation

The equation (1.5b) is expanded with the chain rule. Subsequently, the time derivative of the density is substituted using the equation (1.5a), leading to:

ρ "∂v

∂t +(v · ∇)v #

= ∇ · σ + ρqb (1.15)

Note that the volumetric source of mass disappears and only the body forces remains as an external source term.

Using equations (1.7) and (1.8), ∇ · σ can be expressed as:

∇ ·σ = ∇ · (τ − pI),

= ν∇ · (∇ · v)I + µ∇ · ∇v + µ∇ · (∇v)T

− ∇ · pI, = (ν + µ)∇(∇ · v) + µ∇2_{v − ∇p.}

And finally, the linear momentum equation yields:

ρDv

Dt = (ν + µ)∇(∇ · v) + µ∇

2_{v − ∇p}_{+ ρq}

b. (1.16)

1.2.5.3 Energy conservation

The total energy E splits into an internal energy contribution e and a kinetic contri-bution, so that:

E= e + 1

2|v · v|. (1.17)

The partial time derivative in equation (1.5c) is rewritten using the chain rule and the term ∂ρ_∂t is substituted using equation (1.5a). Finally the left-hand side of equation (1.5c) writes: ∂ρE ∂t +∇ ·ρEv = ρ ∂E ∂t +ρv · ∇E + E ∂ρ ∂t +ρ∇ · v + v · ∇ρ ! , = ρ∂E_{∂t +}ρv · ∇E + ρqmE, = ρD Dt e+ 1 2|v · v| ! + ρqm e+ 1 2|v · v| ! . (1.18)

Applying the dot product of the velocity vector to the linear momentum equation (1.15) leads to the following relationship,

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which could be identified as a mechanical (or kinetic) energy equation.

Substituting the previous expression (1.19) into the energy balance equation (1.5c) allows to express the internal energy equation:

ρDe

Dt + ρqme= σ: (∇v) − ∇ · qh+ ρqe+ 1

2ρqm|v · v|, (1.20) noting that the double dot tensor product is defined by:

σ: (∇v)= ∇ · (σ · v) − (∇ · σ) · v = −pI: ∇v + τ: ∇v = −p∇ · v + Φ. (1.21) The functionΦ = τ: ∇v represents a specific heat source generated by the vortical (shear) waves. The complete expression of this function reads:

Φ = (ν(∇ · v)I + µ(∇v + (∇v)T )) : ∇v, = ν(∇ · v)2_{+ 2µ} 3 X i=1 (∇ivi)2+ µ 3 X i, j=1 j,i (∇ivj+ ∇jvi)2.

The material derivative of the internal energy is deduced from the caloric equation of state (1.10):

De Dt = CV

DT Dt,

and using thermal equations of state (1.9) and (1.11), one can write:

DT Dt = D Dt " 1 R0 p ρ # = 1 Cp− CV 1 ρ Dp Dt + p ρ∇ · v − pqm ρ ! . (1.22)

Finally, De/Dt expresses as: De Dt = CV DT Dt = Cp DT Dt − 1 ρ Dp Dt − 1 ρp∇ · v+ 1 ρqmp (1.23) = CV ρR0 _Dp Dt + p∇ · v − pqm (1.24)

The final expression of the thermal energy equation is obtained by substituting the expression (1.23) into the left-hand side of equation (1.20) and using Fourier’s law to replace the heat flux vector.

ρCp DT Dt + qmCVT −λ∇2T = Dp Dt − qmp+ Φ + 1 2ρqm|v · v|+ ρqe. (1.25) An aternative form of the previous equation is retrieved by substituting the expression (1.24) instead of (1.23) within the equation (1.20).

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1.3 Linearization of Navier-Stokes-Fourier equations

1.3.1 Linear acoustic assumptions

The system of Navier-Stokes-Fourier equations presented above is nonlinear. Actu-ally, the unknown functions of pressure, density, velocity and temperature appear as variables of polynomials of degree higher than one. Even if nonlinear effects can be important in some cases, the modelling of those phenomena requires more general models, which is not the topic of this work.

To circumvent the non-linear effects, the so-called linearization of variables around a mean state is generally required. All the fluid variables are decomposed as a sum of a quiescent contribution (also called average or mean value) and a fluctuating or acousticcontribution.

Acoustic/flow interactions have raised more and more interest over the past few years. The complete system of Linearized Navier-Stokes-Fourier (LNSF) equations, de-rived previously, brings into play a steady mean velocity field of arbitrary magnitude. Among other things, this velocity field accounts for convection of mass, momentum and energy.

Some assumptions have to be made in order to linearize the set of equations. Density, pressure and temperature fluctuations are assumed to be small compared to their respective quiescent values. However, acoustic velocity is considered smaller than the isentropic speed of sound. No specific order of magnitude is imposed for the mean velocity, although the flow will be assumed incompressible in the remainder of this project.

Moreover, the variables are formulated using the harmonic form or phasor notation. This representation is efficient because the partial differential equations do not ex-plicitly depend on time anymore. The system can be solved frequency by frequency, provided that boundary conditions also decompose into a discrete frequency spec-trum. The complete time-dependent solution can be reconstructed as a superposition of harmonic solutions using an inverse discrete Fourier transform. The harmonic for-mulation is done at the expense of the introduction of a complex-valued notation for all the acoustic fluid variables. Finally, the total state of the different fluid variables express as follows: ˇ ρ(x, t) = ρ0(x)+ Rehρ(x, ω)eiωt i , with |ρ| ρ0, ˇp(x, t)= p0(x)+ Re h

p(x, ω)eiωti, with |p| p0,

ˇ

T(x, t)= T0(x)+ Re

h

T(x, ω)eiωti, with |T | T0,

ˇv(x, t)= v0(x)+ Re

h

v(x, ω)eiωti, with |v| c.

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fre-1.3 Linearization of Navier-Stokes-Fourier equations 17

quency and time. The total fluid variables are identified with the symbol ˇ and the quiescent space-dependent variable denoted with subscript0. The external loads qm,

qband qeare all considered as harmonic perturbations too:

qm(x, t)= Re h qm(x, ω)eiωti, qb(x, t)= Re h qb(x, ω)eiωti, qe(x, t)= Re h qe(x, ω)eiωti.

Additional remark can be done about the existence of evanescent waves, local-ized near sharp edges and corners, point sources or any kind of non-smooth boun-dary conditions. At those locations, the linear acoustic assumptions become invalid. However, evanescent waves also occur in isentropic acoustics and are not necessarily related to visco-thermal effects. The linearized Navier-Stokes-Fourier equations are insufficient in any case because the contribution of non-linear terms can be signifi-cant. Nevertheless, these simplifications are often made and widely accepted as the standard description of acoustics in the scientific community.

1.3.2 Governing equations

Provided that all the introduced assumptions are valid, the set of Linearized Navier-Stokes-Fourierequations for an ideal gas can be decomposed into zeroth order and first order expressions. The zeroth order equations are not of interest in this project, because all mean flow fields are supposed to be known a priori. In addition, it is assumed that the mean flow part are not influenced by the acoustic fluctuations. The non-linear effects due to acoustic streaming are not considered and this is a reason-able assumption as long as the amplitude of the perturbations remains small.

The first order acoustic equations are found to be:

iωρ+ v0· ∇ρ + v · ∇ρ0+ ρ0∇ · v+ ρ∇ · v0 = ρ0qm, (1.27a) ρ0[iωv+ v0· ∇v+ v · ∇v0]+ ρv0· ∇v0− (ν+ µ)∇(∇ · v) − µ∆v = −∇p + ρ0qb, (1.27b) ρ0 h iωCpT + Cp(v0· ∇T+ v · ∇T0)+ qmCVT0i + ρCpv0· ∇T0−κ∆T = iωp + v · ∇p0+ v0· ∇p+ Φ − p0qm+ 1 2ρ0|v0· v0|qm+ ρ0qe. (1.27c)

The fluctuating viscous heating function yields:

Φ = 2ν(∇ · v0)(∇ · v)+ µ(∇v + (∇v)T) : ∇v0+ µ(∇v0+ (∇v0)T) : ∇v, (1.28)

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1.3.3 Inviscid and adiabatic wave propagation

As soon as internal viscous forces are assumed to be small with respect to inertial forces, the fluid is considered inviscid. Analogously, if the changes in internal energy caused by the thermal conductivity of the fluid are supposed to be small with re-spect to the changes in internal energy due to expansion, the fluid can be regarded as adiabatic. Such a situation can be seen as an asymptotic case of the system of equa-tions (1.27) and mathematically corresponds to a zero-valued thermal conductivity coefficient and zero-valued viscosity coefficients:

ν = 0, µ = 0, λ = 0.

Under these assumptions, LNSF equations (1.27) reduce to the so-called Linearized Euler Equations (LEE). This set of equations describes reversible adiabatic pro-cesses (also called isentropic propro-cesses). Such statements remain valid as long as the boundary layer thicknesses are small compared to the dimensions of the bulk do-main. When no external sources of any type (mass, forces or energy) are considered (qm= 0, qb = 0, qe= 0) and combining the thermal equation of state (1.9) and the

energy conservation equation (1.25) leads to: Dp Dt = γRT Dρ Dt = c 2Dρ Dt. (1.30)

This equation introduces the isentropic speed of sound, since acoustic disturbances at a given point remain very small. Under the ideal gas assumption, the isentropic speed of sound c is given by:

c2= γp0 ρ0

or c2= γR0T0 (1.31a,b)

with γ being the ratio of specific heats Cpand CV.

If one considers a uniform, incompressible and subsonic (v0 < c) mean flow, as

well as inviscid and adiabatic acoustic perturbations, then it is possible to reduce the LNSF equations into a scalar wave equation. The mean temperature, mean density and mean pressure are all constants inside an incompressible flow.

The suitable combination of the equation (1.30) with the time derivative of the mass continuity equation (1.27a) (equivalent to multiplying the equation by iω) and the divergence of the momentum continuity equation (1.27b) induces the so-called con-vected Helmholtzequation.

" ∇2+ k0− i v0 c · ∇ 2# p= 0, (1.32)

where k0 = ω/c is the isentropic wave number.

If the mean velocity is equal to zero, this equation reduces to the well known Helmholtz equation that simply yields,

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1.4 Dimensionless forms and parameters 19

1.4 Dimensionless forms and parameters

Dimensional analysis is the study of the dimensions associated to some physical quantities. This analysis facilitates the general understanding of the physical mech-anisms that are part of a complex mechanical system. This enables one to predict the behavior of large systems by studying small-scale models. Appropriate scaling (also called nondimensionalization) makes sense to recover intrinsic properties of a system.

The system defined by equations (1.27) is rewritten in dimensionless form in the fol-lowing. A set of visco-thermal parameters comes from the dimensionless equations and their physical meanings are discussed below.

Subsequently, a suitable analysis of the different orders of magnitude involved in the dimensionless groups is provided. Doing so, the reduction of the LNSF equations becomes possible by removing the negligible terms.

1.4.1 Dimensionless equations

All the fluid variables such as density, pressure, temperature and velocity are rendered dimensionless using relevant reference values (denoted with the subscript∞). For

instance, such reference values can be taken as the hydrodynamic values when the fluid is considered at rest. The dimensionless variables are identified by the overbar notation, as follows: ¯ ρ = _ρρ ∞, ¯p = p p∞, ¯T = T T∞, ¯v = v c∞ , (1.34) ¯ ρ0= ρ0 ρ∞ , ¯p0 = p0 p∞ , ¯T0= T0 T∞ , M0= v0 c∞ , (1.35)

knowing that M0represents the Mach number.

Subsequently, the linearized mass conservation equation (1.27a) is divided by the factor ωρ∞, the linearized momentum conservation equation (1.27b) is divided by

ωρ∞c∞and the linearized energy conservation equation (1.27c) is divided by ωCpρ∞T∞.

The external sources of volumetric mass, body forces or energy are omitted and the full system of dimensionless Linearized Navier-Stokes-Fourier equation reads:

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¯ ρ0 " i ¯T+ 1 k₀(M0· ∇ ¯T + ¯v · ∇ ¯T0) # + 1 k₀ρM¯ 0· ∇ ¯T0− 1 k2_T∆ ¯T = γ − 1 γ " i ¯p +1 k0 (M0· ∇ ¯p+ ¯v · ∇ ¯p0) − ¯p0¯qm # + ¯Φ + 1 2(γ − 1) ¯ρ0|¯v0·¯v0| ¯qm+ ¯ρ0¯qe. (1.36c)

The dimensionless viscous fluctuating function ¯Φ reads: ¯ Φ = _ωρ Φ ∞T∞Cp = γ − 1 ¯k2 v h 2ξ( ¯∇ · M0)( ¯∇ ·¯v)+ ( ¯∇¯v + ¯∇¯vT) : ¯∇M0 +( ¯∇M0+ ¯∇MT₀) : ¯∇¯vi , (1.37)

together with the dimensionless source terms as:

¯qm= qm ω , ¯qb= qb ωc∞ , ¯qe = qe ωc∞T∞ . (1.38)

The perfect gas law can also be written in dimensionless form: ¯p ¯p0 = ¯ ρ ¯ ρ0 + ¯ T ¯ T0 . (1.39)

A total of five parameters γ, ξ, k0, kv and kT have been introduced. The next section

gives a physical interpretation of those parameters. Their impact on the behavior of a viscous and heat conducting ideal gas is discussed.

1.4.2 Visco-thermal parameters

For ideal gases, the following two dimensionless ratios appear in thermodynamics:

I The adiabatic index,

γ = Cp CV

, (1.40)

reflects the ratio between heat changes due to variations in density and heat changes due to variations in pressure when an acoustic wave circulates through a medium. For most gas mixtures like air under standard atmospheric condi-tions, γ ≈ 1.4.

I The viscosity ratio,

ξ = _µν, (1.41)

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1.5 Physical length scales 21

Besides those two ratios, the so-called acousto-visco-thermal wavenumbers k0,

kvand kT (inverse of a distance) are of major importance.

I The isentropic acoustic wavenumber, is defined as:

k0=

ω c∞

. (1.42)

For a propagating acoustic wave, k0represents the number of wavelengths per

2π unit distance - in other words, the amount of phase change of the wave-form per 2π metre. This parameter describes the wave propagation in a non-dissipative medium. Throughout this thesis an important distinction has to be made between the isentropic acoustic wave and the (physical) acoustic wave especially if visco-thermal effects dominate.

I The shear wavenumber, writes:

kv =

rωρ

∞

µ . (1.43)

It is linked to the viscous boundary layer and represents the ratio between in-ertial and viscous forces. The quotient k0/kv = pωµ/ρ∞/c∞ determines the

relative importance of the wavelength, compared to the viscous boundary layer thickness. In most gases and liquids, in the audible frequency range, this ra-tio is smaller than one, provided the assumpra-tions of continuum mechanics are valid.

I The thermal wavenumber, equals:

kT =

r

ωCpρ∞

λ . (1.44)

This wavenumber can also be expressed as the product of the shear wavenum-ber and the square root of the Prandtl numwavenum-ber3 , so that: kT = kv

√

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Figure 1.2: Visco-thermal profile close to a no-slip/isothermal boundary. Real and imag-inary parts are plotted and the three characteristic lengths: δ, λ and λ0are indicated on the position axis.

1.5 Physical length scales

The three previously introduced wave numbers account for three different length scales. The length scale of the isentropic acoustic wave is defined as the wavelength λ0= 2π/k0. Usually, for air in audible frequency range and standard conditions, the

isentropic wavelength covers the range from a centimeter to ten or so meters. The length scales associated to viscous and thermal waves can be defined using three different conventions. In fact, because of the diffusive nature of the momentum and energy equations, the complex number √−i has to be taken into account in the wave-length estimation. A typical visco-thermal distribution or visco-thermal profile in function of the normal distance to a rigid wall is plotted in figure 1.2. One can notice that the profile does not increase monotonically from the wall up to the bulk. As a matter of fact, the profile follows the evolution of a damped sine function around the bulk value. This damped sine oscillates within an exponential envelope that is used to determine three length scales4_{introduced by Kampinga [77].}

Mathematically, those three lengths can be defined as follow:

δq = − 1 Im √−ikq , λq = 2π Re √−ikq and λq= 2π kq , (1.45a,b,c)

where q is a dummy variable which can be replaced either by v or T .

The first length scale (1.45a) corresponds to the boundary layer thickness δ com-monly defined in the literature. The second length λ defines the distance where the visco-thermal profile touches its lower envelope, while the last value (1.45c) can be

3_{The Prandtl number measures the ration between momentum di}_{ffusivity (or kinematic viscosity) to}

thermal diffusivity and is defined as Pr =µCp

λ

4_{In his work, Kampinga includes the imaginary unit into the definition of the shear and thermal}

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1.6 Order of magnitude analysis 23

determined in between those two previous lengths.

It is worth noting that the three conventions do not induce significant changes of the associated visco-thermal length scales. For air in the audible frequency range and standard conditions, the order of magnitude of the length scale λ is between 0.1 mil-limeter and several milmil-limeters. The viscous and thermal length scales have mutually comparable orders of magnitude in such a fluid.

One can notice the order of magnitude difference occurring between the acoustic scale and the visco-thermal scales. For a given frequency of excitation, the isentropic acoustic wavelength is always much larger than the visco-thermal wavelengths. This essential statement drives most of the approximations used for the derivation of the reduced visco-thermal acoustic models presented in this thesis.

1.6 Order of magnitude analysis

The approximations required to establish reduced models are introduced here. A proper inspection of the order of magnitude of some terms of the LNSF equations is done here. The gradient operator is transformed into its dimensionless equivalent using the acoustic length scale. Subsequently a convenient dimensionless form of the visco-thermal wave numbers is introduced:

ˆ

∇= k−1₀ ∇ and ˆkq =

kq

k0

where q is either v or T . (1.46)

The hat notation denotes the dimensionless parameters.

The viscous and thermal boundary layer thicknesses are much smaller than the isen-tropic acoustic wavelength. This statement can be written as:

kq k0 (1.47)

or equivalently as:

ˆkq 1. (1.48)

Almost everywhere, the pressure and the term linked to its gradient are of compa-rable magnitude and remain smooth. This smoothness assumption comes from the boundary layer theory and can be written as:

ˆ

∇ ¯p= O( ¯p), (1.49)

In addition, the temperature waves evolve in phase with the pressure. Similarly, the velocity waves are in phase with the pressure gradient, and:

¯

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Let us assume that the pressure is the cause of the temperature changes; and ana-logously, that the pressure gradient is the cause of the shear velocity changes. This enables us to easily decouple the viscous and thermal waves from the acoustic waves. From a thermodynamic point of view, there is no action of the acoustic part on heat conduction and viscous dissipation. The latter do not have significant influence on the local pressure or pressure gradient, although they slow down and locally damp the acoustic waves.

Because of the large variations of temperature and shear velocity within the boundary layers, the viscous and thermal fields are first order large, that is:

ˆ

∇ ¯T = O(ˆkT¯p) and ∇¯vˆ = O(ˆkv∇ ¯p).ˆ (1.51a,b)

Subsequently, the Laplacians are second order large compared to the pressure and pressure gradients:

ˆ

∆ ¯T = O(ˆk2

T¯p) and ∆¯v = O(ˆkˆ 2v∇ ¯p).ˆ (1.52a,b)

Using the continuity equation, the divergence of the velocity and the pressure are known to be of the same order of magnitude, as well as the temperature:

ˆ

∇ ·¯v= O( ¯p), (1.53)

and the gradient of the divergence finally reads as a first order large term:

ˆ

∇( ˆ∇ ·¯v)= O(ˆkv¯p). (1.54)

Comparing this last equation (1.54) with equation (1.52b) leads to a meaningful ap-proximation in the momentum equation (1.36b). The volumetric viscosity term con-taining ¯ξ appears to be first order smaller and vanishes with respect to the pressure gradient and the shear viscosity term.

Finally, the approximated momentum equation can be written:

¯ ρ0 " i¯v+ 1 k₀ (M0· ∇¯v+ ¯v · ∇M0) # + 1 k₀ρM¯ 0· ∇M0− 1 k2v ∆¯v = − 1 k0γ ∇ ¯p+ ¯ρ0¯qb, (1.55)

1.7 Boundary conditions

Any system of partial differential equations has to be completed with appropriate boundary conditions in order to find a non-trivial solution corresponding to the phys-ical problem. Viscous and thermal boundary layers are determined by the prescribed boundary conditions.

Refined damped equivalent fluid models for acoustics