1868-2008 : 150 years of

Acoustic propagation inViscous / Thermal conducting fluids at

rest

From Fundamentals of AcousticsTo Current applications

Michel BRUNEAULaboratoire d'Acoustique de l'Université du Maine (LAUM), UMR CNRS 6613, Le Mans - France

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, (1768-1830)

Claude-Louis Navier french, (1785-1836)

George Gabriel Stokesenglish, (1819-1903)

Gustav Kirchhoff german, (1824 - 1887 )

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

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

Fpgradt

v00

qvdivt

'00

Nature of the compressibility

2) Compressibility

v

dx

P (x) P (x+dx)F(x)

0dPdˆT

CSd

T00

v

dPdT0

'

T0

pP

P

0

0

0

0

dPd

20c

q dV

dV'

v(x) v(x+dx)

dthSdT

' 20cp

h

5

W = 0

Ea = 0

0V~ 0P

~ 0~

0~ ~

0P 0V

~

Adiabatic behaviour

P

V

0Pdˆ1

TdT

CSd

0

p

Pdˆ1

Td

pP

P

T

T

0

0

0

0

Pdˆ1

Td

pˆ1

maxmin= -max

max

min

/2

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

Viscous fluid Navier-Stokes equation

Fc

1vrotrot'vdivgradpgrad

c

1

t

v

c

1

0vv

000

00v c'

00

v c34

with and

: shear viscosity coefficient : bulk viscosity coefficient

compressional - extentional acoustic wave :

2 equations

Shear wave :

Fvdivgrad3

pgradt

v00

vv

0 vrotrott

v

v

v v

Fvvdivgrad3

pgradt

v00

Fvrotrotvdivgrad3

4pgrad

t

v00

rotrotdivgrad

i.e.

with

vvvv

Now

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

Non adiabatic behaviour

P

W > 0

Heat transfer between the particle considered and its neighbouring particles

Ea < 0

V

0V~ 0P

~ 0~

0~ ~

0P 0V

~

10

Thermal conduction Thermal conduction equation

ht

sT 000

: thermal conduction coefficients : entropy variationh : heat source

Pdˆ1

TdT

CSd

0

p

pP

P

T

T0

psS

S

0

0

0

0

0

0

Pdˆ1

TdT

CSd

pˆ1

T

Cs

0

p

Now

p00h

0 C

h

t

p

c

1ˆ1

tc

1

withp00

h Cc

(meter)

Conservation of mass equation

qvdivt

'00

ˆp

c'

20

qcˆptc

1vdivc 00

000

11

Summary Inviscid fluid (outside sources)

Thermoviscous fluid (with operation of sources)

pˆ1

0pgradc

1

t

v

c

1

000

0t

p

c

1vdivc

000

Euler equation

Conservation of mass equation

Adiabatic law (thermodynamic behaviour)

Stokes-Navier equation

Conservation of mass eq.

Thermal conduction eq.

'cp 20 ou

ˆptc

1vdivc

000

0 c0 q

t

p

c

1ˆ1

tc

1

0h

0

0 Cp

h

vrotrot'vdivgradpgradc

1

t

v

c

1vv

000

F

c0

1

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

Wave equation in thermo-viscous fluidDissipation phenomena

Outside harmonic sources - angular frequency (Kirchhoff, 1868)

0t

p

c

1

tc

111p

2

2

200

hv

0

0 ck

with

i k0 pk20

t,r,0pki1kk hv020

2a

hv00a k

2

i1kk complex

wavenumber

t

h

Ct

qFdiv

t

p

c

1

tc

111p

p02

2

200

hv

pk2a2

ak

it

Wave equation with sources (lower order of the thermo-viscous terms)

Remark : SST1

represents the gap between the isothermal compressibility and the adiabatic compressibility ("amplitude" of the thermal effect)

hv020

2a ki1kk dispersion

equation

vh

10-8 m

0pk 2a

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

Boundary conditions, boundary layersParticle behaviour (1/5)

Without viscosity and without thermal conduction (inviscid fluid)

v = 0p = 0

v = 0p = 0

max / minv max / minp 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

maxv = 0p max

minv = 0p min

v max / minp = 0

16

Boundary conditions, boundary layersParticle behaviour (2/5)

Without viscosity and with thermal conduction (stationary wave)

Thermal boundary layer

thickness

Displacement

Quasiisothermal

Polytropic

Quasiadiabatic

vn = van+vhn = 0

vn = vanvn = van

vn = van+vhnvn = van+vhn

van+vhn = 0

> 0 < 0= 0

local heat flux Temperature of the wall = constant =0 on the wall The normal component van of the acoustic velocity compensate on the wall

the «entropic» velocity vhn (linked to the heat flux) vn , van , vhn depend on the distance between the particle and the wall

17

Boundary conditions, boundary layersParticle behaviour (3/5)

Without viscosity and with thermal conduction (stationary wave)

Thermal boundary

layer thickness

Displacement

Quasiisothermal

Polytropic

Quasiadiabatic

> 0 < 0= 0

local heat flux Temperature of the wall = constant t =0 on the wall

Heat transfer

18

Boundary conditions, boundary layersParticle 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

Thermaland viscous boundary

layer thickness

Tm + (

Tm + (

Tm + (

Tm + s

(sTm + s

(sTm + s

(s

Displacement

Heat transfer

19

Boundary conditions - boundary layersBasic 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.

ˆptc

1vdivc

000

0 c0 q

t

p

c

1ˆ1

tc

1

0h

0

0 Cp

h

Components of the 1st equation: normal velocity

tangent velocityvvu

v w

v w1v w2

u

w

Boundary (u=s) localy plane

perfectly rigid

v

vu

v w

vrotrot'vdivgradpgradc

1

t

v

c

1vv

000

F

c0

1

20

Boundary conditions - boundary layersBasic equations (2/2)

u

p

c

1vvdiv

u'vdiv

u

v

uv

c

i

00uwwwvww

uvu

0

i00

wuwi

wuwvww

uwvw

0 w

p

c

1v

w

v

u

vgrad'vdiv

u

vgradv

c

iii

i

iii

p00h

0 Cc

hp

c

iˆ

1

c

i

qˆpc

i

cvdiv

u

v

000ww

u

Stokes-Navier equation

Conservation of mass equation

Heat conduction equation

Equations in the frame (u, w1, w2), harmonic oscillations 00ckit

Boundary conditions (on a wall at u = s)

0sviw pgrad

1uvi

ii w0

vw

0s pˆ

1u h

;

;

0pkp 2a p : champ source

spc

vsv00

0u

Z/1c00

withBoundary (u=s) localy plane

Impédance Z

u

w

v

vu

v w

withh0hh k2k2

v0vv k2k'2

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

v

0v

k

2

i1k

h

0h

k

2

i1k

Tubes, slits, acoustic transmission lines (1/3)

Plane wave, uww vvandgraduiihypotheses near the wall:

wpgradi

1v

uk

11 w

0w2

2

2v

pˆ

1u h

p2

2

2h Ci

hp

ˆ1

uk

11

0svw

; pgrad1

uviii w

0vw

sk

uk1pgrad

i

1v

vv

vvw

0w

general solution of the homogeneous equation

0s ;

sk

uk1

Ci

hp

ˆ1

hh

hh

p

qˆpc

i

cvdiv

u

v

000ww

u

Stokes-Navier (Poiseuille) equation + boundary conditions

Conservation of mass equation

Heat conduction equation + boundary conditions

Boundary(u=s)

uw

v w

u v

uw

u h

Shear movement inside the boundary layer

23

Tubes, slits, acoustic transmission lines (2/3)

Wave equation (mean value over the section), (Kirchhoff, 1868)

q

C

hip

sk

uk11

cp

sk

uk1

u

vi

p0

hh

hh20

2

wvv

vvu0

0 KvKh

q

C

hipK11kpK1

p0h

20wv

Outside the sources:

0pK1

K11k

v

h20w

Elementary wave equation with a complex wavenumber

Boundary(u=s)

uw

v w

u v

uw

u h

p

24

Tubes, slits, acoustic transmission lines (3/3) Acoustic transmission lines: mean value over the section

Tubes and slits (z-axis), outside the sources

sk

uk1

Ci

hp

ˆ1

hh

hh

p

qˆpc

i

cvdiv

u

v

000ww

u

Stokes-Navier (Poiseuille) equation + boundary conditions

Conservation of mass equation

Heat conduction equation + boundary conditions

z

vz = < vw >

v

p

Kh

sk

uk1pgrad

i

1v

vv

vvw

0w

Kv

vz

z

v

000 vK1

cki

z

zp

0vz

zpK11c

ki

z

vh

00

0z

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

,pc

skiukik

1skiuki

k

1

k

ksvuv

200

hhhhh

vvvvv

20

2w0

uu

Field/wall interaction Thermo-viscous admittance-like of the wall

Wave equation inside the thermo-viscous boundary layers (outside the sources)

qC

hp

c

i

sk

uk1

sk

uk

ku

p

i

1

u

v

p200hh

hh

vv

vv20

w2

2

0

u

u

sdu

u

s

du

0u

p

i

1u

s0

usuv

with

pc

sv00

u

for (u-s)v,h pcc

uv00

v

00u

hv20

2w

000

v

k

11

k

1

k

kk

cwith

Specific admittance-like effects of the thermoviscous boundary layers on the reflection of the acoustic field (Cremer, 1948)

p0000

20

2w

0v Cc1

ck

kk

2

i1and

and2ww k uki

h,vh,veu

(extinction of the shear and the entropic modes for (u-s) v,h )

with

u

w

boundary layers

v

vu

v w

s

Small Acoustic elementsThermo-viscous boundary layer

effects

LAUM - LNE

28

Results

• An Annular slitThickness: 71.2 µm ± 6 µmLength: 3.8E3 ± 6 µm

29

• 4 open tubesØ: 449 ±1µmLength: 3.80 ±0.01mm

New results

MicrophonesThermo-viscous boundary layer

effects

LAUM - LNE

31

32

Acoustic gyrometerInertial viscous boundary layer

effect

Michel BRUNEAU Henri LEBLONLAUM SEXTANT-AVIONIQUE

34

Acoustic gyrometer

Demonstrator

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

Device, mechanisms involved

1000

(°/s)

-10000.1

0.2

0.3

0.4

-0.1-0.2

-0.3

-0.4AS /AC

[Herzog et al.]

100 200 500 1000

-1

dB

(°/s)

AC

[Herzog et al.]

C = AC J1(k10r) cos

S = AS J1(k10r) sin

37

qp)c1( t0

2tt2

0

prc

2r

220

p)( t0c220

.0

The wave equation

Flow induced perturbation

Radial density variation effect

tor.v)v(tor.)v(div v

Angular acceleration

)v( 1te

Coriolis acceleration

Centrifugal acceleration

v2c

v1tta

0 0.2 0.4 0.6 0.8 10.992

0.994

0.996

0.998

1

r/R

=0°/s

=20000°/s

=40000°/s

=60000°/s=80000°/s

)R(P/)r(P 00

38

Acoustic gyrometer

Experimental gyrometer

Gyrometer

Rotating table

39

Miniaturised acoustic gyrometer

Loudspeaker and microphones

on silicon chips

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 10-2 °/s up to 105 °/s)

Boltzmann constant measurementThermo-viscous boundary layer

effectsLAUM/GDF

INM/LNE/CNAM

42

43

Machines thermoacoustiquesModélisation 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

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), 1998Références

- H. Bailliet, doctorat, Univ. du Maine, 1998

Génération d’harmoniques (en résonateur sans stack) : modèle de Burgers généralisé

Contrôle actif : signal HP en source d'énergie) et 2(opposition aux effets NL)

=> augmentation de l’amplitude de pression jusqu'à 50% (résultat expérimental)

45

Exemples de résultats

Régime transitoire d'établissement du champ de température dans un réfrigérateur thermoacoustique (coll. LMFA)

ThotTcold

0 xc

x

Description analytique du régime transitoire - flux de chaleur dû à l’effet thermoacoustique, - conduction thermique retour dans l’empilement, - chaleur générée par effets visqueux dans l’empilement, - fuites thermiques en parois du résonateur et aux extrémités de l’empilement

Référence P. Lotton, P. Blanc-Benon , M. Bruneau , V. Gusev , S. Duffourd , M. Mironov , G. Poignand, International Journal of Heat and Mass Transfer, 52, 4986-4996, 2009 46

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):642-659, 2007.- B. Lihoreau, doctorat, Univ. du Maine, 2002.- G. Poignand, doctorat, Univ. du Maine, 2006.

- M. Bruneau, P. Lotton, Ph. Blanc-Benon, V. Gusev, E. Gaviot, S. Durand, Brevet FR 03 05982 (Univ. du Maine et CNRS) juin 2003 (étendu PCT WO2004402084).

47

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 .....

stack échangeur

adiaba

tiq

uepo

lytro

piq

uepo

lytro

piq

ue

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), 2001.Références

- Gusev V., Lotton P., Bailliet H., Job S., Bruneau M., J. Sound Vib., 235(5), 711-726, 2000.

49

Exemples de résultats

Générateurs d'ondes thermoacoustiques : déclenchement et effets NL de saturation

DéclenchementModélisation : solution analytique exacte de l'équation de la thermoacoustique linéaire prise sous forme d'une équation intégrale de Volterra de seconde espèceSaturation

expérience modèleRéférences- Gusev V., Job S., Bailliet H., P. Lotton, M. Bruneau, J. Acoust. Soc. Am.,

110, p.1808, 2001- S. Job, doctorat, Univ. du Maine, 2001- G. Penelet, V. Gusev, P. Lotton, M. Bruneau, Phys. Let. A, 351, 268-273, 2006

- 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