Post on 03-Apr-2015
transcript
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