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1868-2008 : 150 years of Acoustic propagation in Viscous / Thermal conducting fluids at rest From Fundamentals of Acoustics To Current applications Michel.

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Présentation au sujet: "1868-2008 : 150 years of Acoustic propagation in Viscous / Thermal conducting fluids at rest From Fundamentals of Acoustics To Current applications Michel."— Transcription de la présentation:

1 : 150 years of Acoustic propagation in Viscous / Thermal conducting fluids at rest From Fundamentals of Acoustics To Current applications Michel BRUNEAU Laboratoire d'Acoustique de l'Université du Maine (LAUM), UMR CNRS 6613, Le Mans - France

2 2 Recent short history 1810, J.J. Fourier : heat diffusion 1850, G.G. Stokes then C.L. Navier : viscosity 1868, G. Kirchhoff : Basic equations Inertia, bulk and shear viscosity: Newton's law Compressibility: mass conservation law Thermal diffusion: heat conduction equation Wave motion Infinite medium (plane and spherical waves) Guided plane wave (boundary layers effects) 1948, L. Cremer : impedance-like boundary layers Applications: Small acoustic elements (1908), thermoacoustics (1978), acoustic gyrometry (1988), Boltzmann's constant measurement (1988), non-linear propagation from loudspeaker (1998), among others... Jean-Baptiste Joseph Fourier french, ( ) Claude-Louis Navier french, ( ) George Gabriel Stokes english, ( ) Gustav Kirchhoff german, ( )

3 3 The dynamics of fluid motion Basic equations (inviscid fluid at rest) Viscosity effects Thermal conduction effects Wave motion in thermo-viscous fluid: dissipation Thermo-viscous boundary layers Boundary conditions: boundary layers Tubes and slits, acoustic transmission lines Field/wall interaction: damping and delay Content

4 4 Inviscid fluid at rest 1)Inertia 3)Adiabatic behaviour Euler équation relationship between p et ' div v 0: conservation of mass equation Acoustic wave motion: basic equations Nature of the compressibility 2)Compressibility v dx P (x) P (x+dx) F(x) q d V dVdV ' v(x) v(x+dx) ' 2 0 cp h

5 5 W = 0 E a = 0 0V ~ 0P ~ 0 ~ 0 ~ ~ 0P 0V ~ Adiabatic behaviour P V max min = - max max min /2

6 6 The dynamics of fluid motion Basic equations (inviscid fluid at rest) Viscosity effects Thermal conduction effects Wave motion in thermo-viscous fluid: dissipation Thermo-viscous boundary layers Boundary conditions: boundary layers Tubes and slits, acoustic transmission lines Field/wall interaction: damping and delay Content

7 7 Viscous fluid Navier-Stokes equation with and : shear viscosity coefficient : bulk viscosity coefficient compressional - extentional acoustic wave : 2 equations Shear wave : v v v i.e. with Now

8 8 The dynamics of fluid motion Basic equations (inviscid fluid at rest) Viscosity effects Thermal conduction effects Wave motion in thermo-viscous fluid: dissipation Thermo-viscous boundary layers Boundary conditions: boundary layers Tubes and slits, acoustic transmission lines Field/wall interaction: damping and delay Content

9 9 Non adiabatic behaviour P W > 0 Heat transfer between the particle considered and its neighbouring particles E a < 0 V 0V ~ 0P ~ 0 ~ 0 ~ ~ 0P 0V ~

10 10 Thermal conduction Thermal conduction equation : thermal conduction coefficient s : entropy variation h :heat source Now with(meter) Conservation of mass equation

11 11 Summary Inviscid fluid (outside sources) Thermoviscous fluid (with operation of sources) Euler equation Conservation of mass equation Adiabatic law (thermodynamic behaviour) Stokes-Navier equation Conservation of mass eq. Thermal conduction eq. ou 0 c 0 q 0 C p h F c0c0 1

12 12 Content The dynamics of fluid motion Basic equations (inviscid fluid at rest) Viscosity effects Thermal conduction effects Wave motion in thermo-viscous fluid: dissipation Thermo-viscous boundary layers Boundary conditions: boundary layers Tubes and slits, acoustic transmission lines Field/wall interaction: damping and delay

13 13 Wave equation in thermo-viscous fluid Dissipation phenomena Outside harmonic sources - angular frequency (Kirchhoff, 1868) with i k 0 pk 2 0 complex wavenumber pk 2 a 2 a k i t Wave equation with sources (lower order of the thermo-viscous terms) Remark : represents the gap between the isothermal compressibility and the adiabatic compressibility ("amplitude" of the thermal effect) dispersion equation vh m

14 14 The dynamics of fluid motion Basic equations (inviscid fluid at rest) Viscosity effects Thermal conduction effects Wave motion in thermo-viscous fluid: dissipation Thermo-viscous boundary layers Boundary conditions: boundary layers Tubes and slits, acoustic transmission lines Field/wall interaction: damping and delay Content

15 15 Boundary conditions, boundary layers Particle behaviour (1/5) Without viscosity and without thermal conduction (inviscid fluid) v = 0 p = 0 v = 0 p = 0 max / min v max / min p max / min In the bulk On the wall Progressive plane wave: pressure and particle displacement in quadrature Stationary wave : pressure and particle displacement in phase In the bulk On the wall max v = 0 p max min v = 0 p min v max / min p = 0

16 16 Boundary conditions, boundary layers Particle behaviour (2/5) Without viscosity and with thermal conduction (stationary wave) Thermal boundary layer thickness Displacement Quasi isothermal Polytropic Quasi adiabatic v n = v an +v hn = 0 v n = v an v n = v an +v hn v an +v hn = 0 > 0 < 0 = 0 local heat flux Temperature of the wall = constant =0 on the wall The normal component v an of the acoustic velocity compensate on the wall the «entropic» velocity v hn (linked to the heat flux) v n, v an, v hn depend on the distance between the particle and the wall

17 17 Boundary conditions, boundary layers Particle behaviour (3/5) Without viscosity and with thermal conduction (stationary wave) Thermal boundary layer thickness Displacement Quasi isothermal Polytropic Quasi adiabatic > 0 < 0 = 0 local heat flux Temperature of the wall = constant t =0 on the wall Heat transfer

18 18 Boundary conditions, boundary layers Particle behaviour (5/5) With viscosity and with thermal conduction (stationary wave) - Si s < refrigerator - Si s > temperature gradient is maintained in the wall, heat exchanges are inverted acoustic generator Thermal and viscous boundary layer thickness T m + ( T m + ( T m + ( T m + s ( s T m + s ( s T m + s ( s Displacement Heat transfer

19 19 Boundary conditions - boundary layers Basic equations (1/2) Summary of the basic equations (thermo-viscous fluid with operation of sources) Stokes-Navier equation Conservation of mass eq. Thermal conduction eq. 0 c 0 q 0 C p h Components of the 1st equation: normal velocity tangent velocity v vuvu v w v w 1 v w 2 u w Boundary (u=s) localy plane perfectly rigid v vuvu v w F c0c0 1

20 20 Boundary conditions - boundary layers Basic equations (2/2) Stokes-Navier equation Conservation of mass equation Heat conduction equation Equations in the frame (u, w 1, w 2 ), harmonic oscillations Boundary conditions (on a wall at u = s) ; ; p : champ source with Boundary (u=s) localy plane Impédance Z u w v vuvu v w with

21 21 The dynamics of fluid motion Basic equations (inviscid fluid at rest) Viscosity effects Thermal conduction effects Wave motion in thermo-viscous fluid: dissipation Thermo-viscous boundary layers Boundary conditions: boundary layers Tubes and slits, acoustic transmission lines Field/wall interaction: damping and delay Content

22 22 Tubes, slits, acoustic transmission lines (1/3) Plane wave, hypotheses near the wall : ; general solution of the homogeneous equation ; Stokes-Navier (Poiseuille) equation + boundary conditions Conservation of mass equation Heat conduction equation + boundary conditions Boundary(u=s) u w v w u v u w u h Shear movement inside the boundary layer

23 23 Tubes, slits, acoustic transmission lines (2/3) Wave equation (mean value over the section), (Kirchhoff, 1868) 0 KvKv KhKh Outside the sources: Elementary wave equation with a complex wavenumber Boundary(u=s) u w v w u v u w u h p

24 24 Tubes, slits, acoustic transmission lines (3/3) Acoustic transmission lines: mean value over the section Tubes and slits (z-axis), outside the sources Stokes-Navier (Poiseuille) equation + boundary conditions Conservation of mass equation Heat conduction equation + boundary conditions z v z = v p KhKh KvKv vzvz 0 vzvz

25 25 The dynamics of fluid motion Basic equations (inviscid fluid at rest) Viscosity effects Thermal conduction effects Wave motion in thermo-viscous fluid: dissipation Thermo-viscous boundary layers Boundary conditions: boundary layers Tubes and slits, acoustic transmission lines Field/wall interaction: damping and delay Content

26 26 Field/wall interaction Thermo-viscous admittance-like of the wall Wave equation inside the thermo-viscous boundary layers (outside the sources) u s du u s du with for (u-s) v,h with Specific admittance-like effects of the thermoviscous boundary layers on the reflection of the acoustic field (Cremer, 1948) and (extinction of the shear and the entropic modes for (u-s) v,h ) with u w boundary layers v vuvu v w s

27 Small Acoustic elements Thermo-viscous boundary layer effects LAUM - LNE

28 28 Results An Annular slit Thickness: 71.2 µm ± 6 µm Length: 3.8E3 ± 6 µm

29 29 4 open tubes Ø: 449 ±1µm Length: 3.80 ±0.01mm New results

30 Microphones Thermo-viscous boundary layer effects LAUM - LNE

31 31

32 32

33 Acoustic gyrometer Inertial viscous boundary layer effect Michel BRUNEAU Henri LEBLON LAUM SEXTANT-AVIONIQUE

34 34 Acoustic gyrometer Demonstrator

35 35 Introduction Applications : - Transportation, Navigation - Guidance - Robotics... Advantages of the acoustic gyros : - Lower manufacturing cost - Lower power consumption - Smaller dimension, even miniaturisation - Higher reliability - Improved lifetime - Short transient response - High dynamic range

36 36 Device, mechanisms involved 1000 (°/s) AS /ACAS /AC [Herzog et al.] dB (°/s) ACAC [Herzog et al.] C = A C J 1 (k 10 r) cos S = A S J 1 (k 10 r) sin

37 37 The wave equation Flow induced perturbation Radial density variation effect Angular acceleration Coriolis acceleration Centrifugal acceleration r/R =0°/s =20000°/s =40000°/s =60000°/s =80000°/s

38 38 Acoustic gyrometer Experimental gyrometer Gyrometer Rotating table

39 39 Miniaturised acoustic gyrometer Loudspeaker and microphones on silicon chips

40 40 Conclusion Transient regime - Short transient response (less than 50 ms). - Much shorter than the stabilization time of the unsteady circular flow (several seconds) - Linear response (output as function of the rotation rate) High rotation rates - Non linear behaviour of the phenomena involved - Linear response - High dynamic range (from ° /s up to 10 5 ° /s)

41 Boltzmann constant measurement Thermo-viscous boundary layer effects LAUM/GDF INM/LNE/CNAM

42 42

43 43

44 Machines thermoacoustiques Modélisation analytique Machines thermoacoustiques (Roth, ) : systèmes multiphysiques (acoustique, vibratoires, thermique...), systèmes multi-échelles ( =, L=dimensions guide d'onde), systèmes sièges de processus physiques couplés et complexes. => modélisation numérique d'une machine complète : non envisageable pour l'heure => modélisation analytique : complémentaire de la modélisation numérique Au LAUM, depuis 1995, approche essentiellement analytique et expérimentale Intérêt linéaire et non linéaire régimes transitoires et stationnaires aide au dimensionnement estimation des performances (par la classification des principaux effets non-linéaires qui contrôlent le comportement de ces machines). 44

45 Exemples de résultats Contrôle actif des non linéarités Réfrigérateur thermoacoustique « classique » - Gusev V., Bailliet H., Lotton P., Job S., Bruneau M., J. Acoust. Soc. Am., 103(6), 1998 Références - H. Bailliet, doctorat, Univ. du Maine, 1998 Génération dharmoniques (en résonateur sans stack) : modèle de Burgers généralisé Contrôle actif : signal HP en source d'énergie) et 2 (opposition aux effets NL) => augmentation de lamplitude de pression jusqu'à 50% (résultat expérimental) 45

46 Exemples de résultats Régime transitoire d'établissement du champ de température dans un réfrigérateur thermoacoustique (coll. LMFA) T hot T cold 0 xcxc x Description analytique du régime transitoire - flux de chaleur dû à leffet thermoacoustique, - conduction thermique retour dans lempilement, - chaleur générée par effets visqueux dans lempilement, - fuites thermiques en parois du résonateur et aux extrémités de lempilement Référence P. Lotton, P. Blanc-Benon, M. Bruneau, V. Gusev, S. Duffourd, M. Mironov, G. Poignand, International Journal of Heat and Mass Transfer, 52, ,

47 Exemples de résultats Nouvelle architecture de réfrigérateur thermoacoustique (miniaturisation) Champ acoustique optimal pour le flux de chaleur thermo-ac. (modélisation analytique). Réfrigérateur compact à 2 sources : maquette à champ acoustique optimal Proto. (échelle centimétrique) Stack (plaques Kapton © ) Plaque avec jonctions thermo- élect. (mesure du champ de températures) Références - G.Poignand, B. Lihoreau, P. Lotton, E. Gaviot, M. Bruneau, V. Gusev, Appl. Ac., 68(6): , B. Lihoreau, doctorat, Univ. du Maine, G. Poignand, doctorat, Univ. du Maine, M. Bruneau, P. Lotton, Ph. Blanc-Benon, V. Gusev, E. Gaviot, S. Durand, Brevet FR (Univ. du Maine et CNRS) juin 2003 (étendu PCT WO ). 47

48 48 Exemples de résultats Extrémité stack / échangeur de chaleur : effets de bords thermiques (coll. LMFA) Modélisation analytique des transferts thermiques (harmoniques de température) Réflexions sur la distance optimale stack-échangeurs..... stackéchangeur adiabatiqu e polytropiqu e Variation brusque de la nature des échanges thermiques => génération d'effets non linéaires thermiques - Gusev V., Bailliet H., Lotton P., Job S., Bruneau M., J. Acoust. Soc. Am., 109(1), Références - Gusev V., Lotton P., Bailliet H., Job S., Bruneau M., J. Sound Vib., 235(5), , 2000.

49 49 Exemples de résultats Générateurs d'ondes thermoacoustiques : déclenchement et effets NL de saturation Déclenchement Modélisation : solution analytique exacte de l'équation de la thermoacoustique linéaire prise sous forme d'une équation intégrale de Volterra de seconde espèce Saturation expériencemodèle Références- Gusev V., Job S., Bailliet H., P. Lotton, M. Bruneau, J. Acoust. Soc. Am., 110, p.1808, S. Job, doctorat, Univ. du Maine, G. Penelet, V. Gusev, P. Lotton, M. Bruneau, Phys. Let. A, 351, , G. Penelet, doctorat, Univ. du Maine, 2004 Régime transitoire : modélisation analytique (déclenchement saturation NL) : - vent acoustique - génération d'harmoniques - pompage thermoacoustique - pertes de charges singulières


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