Auto-oscillateurs thermoacoustiques : effets non linéaires et

Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion

Auto-oscillateurs thermoacoustiques effets

non lineacuteaires et comportement dynamique

Guillaume Penelet

Laboratoire drsquoAcoustique de lrsquoUniversiteacute du Maine UMR CNRS 6613avenue Olivier Messiaen 72085 Le Mans cedex 9 France

guillaumepeneletuniv-lemansfr

Preacuteambule de quelle thermoacoustique parle-t-on

Theacuteorie lineacuteaire de la thermoacoustique

Exemple le tube de Sondhauss

Effets de saturation non lineacuteaires

Dynamique des auto-oscillateurs thermoacoustiques

Forccedilage des auto-oscillateurs thermoacoustique

Conclusion

G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 2 85

Conclusion

Quelle thermoacoustique

Thermoacoustique = interaction thermiqueacoustique

rArr de quelle thermoacoustique parle-t-on ici

Tomographie thermoacoustique

Haut-parleurs thermoacoustiques

Effet piston

Instabiliteacutes de combustion

Machines thermoacoustiques (pompes agrave chaleur et moteurs thermoacoustiques)

Tomographie Thermoacoustique

Tomographie thermoacoustiqueMeacutethode drsquoimagerie meacutedicale baseacutee sur la reacuteponse de certainstissus agrave une impulsion EM

chauffage =gt deacutetente thermoeacutelastique =gt onde acoustique

Scheacutema de principe de la tomographie TA

Effet piston

Machines thermoacoustique

Image 3D drsquoun rein drsquoagneauobtenu par tomographie TA

(CD) compareacutee agrave celleobtenue par IRM Drsquoapregraves [1]

[1] Kruger RA et al Radiology 216 279-283 2000

Haut-parleur TA

Haut-parleurs thermoacoustiqueschauffage fluctuant drsquoune surface solide

=gt rayonnement acoustique

Haut-parleur thermoacoustique Drsquoapregraves [1]

Effet piston Instabiliteacutes de combustion Machines thermoacoustiques

Reacuteponse en freacutequence drsquoun HP TADrsquoapregraves [1]

Ecouteurs thermoacoustiques Drsquoapregraves [2]

[1] Nyskacen et al Appl Phys Let 95 163102 2009 [2] US Patent n 8625822 Tsinghua University (Beijing)

Effet piston

Tomographie thermoacoustique Haut-parleurs thermoacoustiques

Effet pistonUn 4e mode de transport de la chaleur

Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ

Instabiliteacutes de combustion Machines thermoacoustiques

Mesure des ondes TA dans du CO2 aupoint critique1

Reacuteponse ∆ρρ agrave un pulse de chauffage1

∆ρ(x t) au cours du chauffage1

[1] Y Miura et al Phys Rev E 74 010101 2006

Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston

Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion

Machines thermoacoustiques

Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston Instabiliteacutes de combustion

Machines thermoacoustiqueMachine thermique cyclique faisant usage drsquoun travailmeacutecanique de nature acoustique

Congeacutelateur thermoacoustique reacutealiseacute agrave lrsquouniversiteacute de Penn

State pour le glacier ben amp Jerryrsquos1

Machine TA tritherme (moteur+frigo) reacutea-

liseacute au Los Alamos National Laboratory2

[1] httpwwwacspsueduthermoacousticsrefrigerationbenandjerryshtm [2] G W Swift J J WollanGasTIPS Volume 8 Number 4 pages 21-26 (Fall 2002)

Elementary sources of sound emission unsteady heat release

The linearized equations of motion for an inviscid and non-heat-conducting idealgas

parttρprime + ρ0divv = Q(r t) Q mass addition

pulsating sphere

ρ0parttv + gradp = f(r t) f force

oscillating sphere

ρ0Cpparttτ minus parttp = q(r t) q Heat release

sphere with oscillating temperature

The resulting inhomogeneous wave equation (using p = RT0ρprime + Rρ0τ) is

∆p minus1

ttp = minuspartt q

minus ρ0parttQ + divf

Contrarily to Q and f q is difficult to derive in realistic cases because The heat conductivity of the gas should be considered Generally q = q (p v) which may lead to self-sustained oscillations(eg Rijke tube etc )

The Rayleigh criterium

The Rayleigh criterium (int

pqdt gt 0)

ldquo at the phase of greatest condensation heat is receivedby the air and at the phase of greatest rarefaction heat isgiven up from it and thus there is a tendency to maintainthe vibrations rdquo JW Strutt (Lord Rayleigh) laquo The theory ofsound raquo Dover New York 2nd edition sect 322i p 231 1945)

TA Laser Rijke tube

WoutWout

q = f (dxT p vx stack ) q = f (Twire vx V0 )

⋆ not acoustically compact ⋆ acoustically compact(q asymp q(t)δ(x minus xwire ))

⋆ amplification of the first mode ⋆ amplification of the first mode R

pqdt gt 0 =gt dxT lt 0R

pqdt gt 0 =gt xw lt L2

Common features⋆ Autonomous oscillators driven by heat (which both fill the ldquoRayleigh criteriumrdquo)⋆ Simple in terms of geometry but complicated operation above threshold

The TA Laser a prototypical example

Heat input =gt sound output (self-sustained acoustic oscillations at the frequency of themost unstable mode(s))

Keywords heat engine (prime-movers and heat pumps) autonomous oscillator Paradigmatic of the challenges that need to be taken up to understand more deeply the

processes controlling the operation of TA engines

Auto-oscillateur thermo-acoustique contexte et probleacutematique

Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique

regenerator

flow straighteners

⋆ Application drsquoun nablaT le long drsquoun mateacuteriau poreux placeacute dans un reacuteseau de guidedrsquoonde rArr auto-oscillations agrave la freacutequence du mode le plus instable

⋆ Geacuteomeacutetrie plusmn simple mais fonctionnement complexe au delagrave du seuil (pompage θac ldquovent acoustiquerdquo perte de charge singuliegravere propa NL )

rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo

Conclusion

Basic principles

The TA Laser a conceptual very simplified approach

⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0

Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)

Q Qrsquo

21 3 4 1τp

ξ max2

1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction

=gt Work production if |dxT |ξmax gt τmax and dxT lt 0

⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact

=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p

2κωis the (frequency-dependent) thermal boundary layer thickness

⋆ Threshold of thermoacoustic instability (for a given mode)

=gt Sound amplification must be larger than losses

Basic equations

⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient

⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities

the ldquobucket brigaderdquo

C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)

Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow

P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp

T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|

S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988

Basic equations

⋆ Equations fondamentales lineacuteariseacutees

ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )

parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0

ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )

dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0

dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p

⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0

vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]

τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]

ρprime = 1ω2

[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p

s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]

Fνκ(r) =J0(kνκr)J0(kνκR)

kνκ = 1minusiδνκ

δν =q

2νω δκ =

q2κω

fνκ = 〈Fνκ〉 = 2kνκR

J1(kνκR)J0(kνκR)

0 05 1 15 20

(1minusFν)(1minusfν)

R=100δν

R=10δν

R=δν

Le point de vue thermodynamique

⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo

p(x) = Pmax cos (k0x)

ρprime = 1ω2

1 minus PrFνminusFκPrminus1 +

idxT0T0

dx p + 1c20

[1 + (γ minus 1)Fκ] p

d = δκ3 d = δκ d = 3δκ

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km

Sound amplification

⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2

([w2 ] = Wmminus3)

w2 = partx(p lt vx gt)

⋆ After some calculations3

w2 = wκ + wν + wsw + wtw

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

|1 minus fν |2|〈vx〉|

wsw = minusimage(h)dxT0

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

image (h)image (fκ)

[3] A Tominaga Cryogenics 1995

Heat transport by sound

⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)

q2 = ρ0T0(〈svx 〉) s =CpT0

τ minus 1

ρ0T0p

⋆ After some calculations

q2 = qsw + qtw + λacdxT0

qsw = minusimage(g)1

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

image (Prf lowastν minus fκ)

|1 minus fν |2|〈vx〉|2

g =f lowastν minus fκ

(1 + Pr)(1 minus f lowastν )

Amplification of a standing wave

w2 = wκ + wν| z

losses

propminusimage(h)z|wsw +

propreal(h)z|wtw

| z production

h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better

[4] J C Wheatley et al J Acoust Soc Am 1983

How to build an intrinsically reversible Thermoacoustic engine

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx

⋆ [CeperleyJASA1979]

regenerator

=gt closed-loop resonator

bull p vx prop eminusi(kxminusωt)

bull Z = pvx

= ρ0c0

bull f asymp c0L=gt does not singbecause wtw lt wν

⋆ [Yazaki et al PRL 1998]

bull Z = pvx

= ρ0c0

bull f asymp c0L=gt does sing but employs a stack

⋆ [Backhaus et al Nature1999]

regenerator

=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well

Examples

A Standing wave engine [Swift JASA 1992]

bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )

A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]

bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )

dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )

Examples

Advantages of θac engines (and heat pumps)

⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements

Limitations of θac engines (and heat pumps)

⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies

Potential applications of θac engines (and heat pumps)

⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling

regenerator

flow straighteners

Sketch of a thermo-acousto-electric

generator Such kind of engine has

been demonstrated to achieve a glo-

bal efficiency of 18 6

[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004

Design of thermoacoustic engines

⋆ The basic principles of design tools may be summarized as

A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation

An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2

⋆ A simplistic illustration would be for instance as follows

T (x)0

|p(x)|

1 Assign QH and set PA = 0 (and thusQ2 = 0)

2 Solve heat transfer =gt get T0(x)

3 Solve acoustics =gt get spatial distributionp(x) and working frequency f

4 Fix arbitrary amplitude PA

5 Calculate thermoacoustic heat transport Q2

6 Solve heat transfer =gt get T0(x)(accounting for Q2)

7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2

Limitations of the existing design tools

⋆ Up to now the design of TA engines is always realized from the linear TA theory

⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9

⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines

⋆ These tools are restricted to the linear (or weakly nonlinear) regime

⋆ These tools are restricted to a 1-D description of the phenomena

⋆ They predict performance in steady-state

⋆ The user need to be experienced in the fieldlowast

[9] WC Ward et al J Acoust Soc Am 1994

lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008

Conclusion

Le systegraveme consideacutereacute

⋆ Systegraveme consideacutereacute sim tube de Sondhauss1

volume V

⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )

hypothegravese BF + analogies eacutelectriques

Approximation (fausse) de son fonctionnement par une DDE

Reacutesolution analytique par la meacutethode des eacutechelle multiples

Modeacutelisation de la saturation NL par pompage thermoacoustique

C Sondhauss Ann Phys(Leipzig) 79 11850

Analogie eacutelectroacoustique

⋆ portion de stack de longueur dx ≪ λ

dp = minus iωρ0dxφS

11minusfν

du = minus iωφSdxγP0

[1 + (γ minus 1)fκ] p +(fκminusfν )

(1minusfν )(1minusσ)

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

⋆ Analogie eacutelectro-acoustique

dp = minus(iωM + Rν)u

du = minus (iωC + 1Rκ) p + Gu

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

(1 + (γ minus 1)real(fκ))

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

ωφSdximage(minusfκ)P0

G = fκminusfν(1minusfν )(1minusσ)

⋆ Approximation ldquoquasi adiabatiquerdquo

fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)

M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ

prop ωǫκ G prop (1 minus i)ǫκdT0T0

isin C

Scheacutema eacutelectrique eacutequivalent

⋆ Scheacutema eacutelectrique eacutequivalent

volume V

~ uC p~

~(1+G)uMstackcavite tuyau

⋆ Equation caracteacuteristique

(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

Approximation par une DDE

⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient

minus (iω)32 f (TH) asymp 12 ω0

radicω0|f |eminusiω0t1 minus iω times 3

2radic

ω0|f |eminusiω0t2

avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π

4 minus Arg(f )

⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque

ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12

de sorte que (sous reacuteserve drsquoune grossedagger approximation)

Fminus1h

minus (iω)32 f u(ω)i

asymp ω320 |f |

2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)

rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard

d2ttu(t) minus 3

2radic

ω0|f |dtu(t minus t2) + 12 ω0

radicω0|f |u(t minus t1) + ω2

0u(t) asymp 0

dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques

daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard

Reacutesolution numeacuterique

p minus 32radic

ω0|f |p(t minus t2) + 12 ω0

radicω0|f |p(t minus t1) + ω2

0p asymp 0

rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa

volume V

Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau

viscositeacute micro = 184 10minus5Pas longueur L = 12cm

conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2

masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm

coef polytropique γ = 14 rayon pore Rs = 3mm

caviteacute porositeacute φ = 095

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

Reacutesolution analytique meacutethode des temps multiples

⋆ Changement de variable θ equiv ω0t rArr

d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

≪ 1 (ζ prop (∆TTm)(q

2νmω0

Rs) ≪ 1)

⋆ On cherche une solution sous la forme

p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ

dθp asymp partτp + ζpartsp

d2θθp asymp part

2ττ p + 2ζpart

p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +

dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)

⋆ Ordre ζ0

part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C

⋆ Ordre ζ1

part2ττ p1 + p1 =

minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo

| z =0 condition de solvabiliteacute

e iτ + cc

Reacutesolution analytique meacutethode des temps multiples (2)

dsA = 12Aldquo

3eminusiθ2 + ieminusiθ1rdquo

A(s) = R(s)e iΦ(s)rArr

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

2 [cos θ1 minus 3 sin θ2]

p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)

Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

prop (∆TTm)(δκRs) ≪ 1

θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

Saturation par pompage thermoacoustique

T =cteC

T (t)H

~ uC p~

QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց

rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire

⋆ Amplification TAp(t ζt) asymp P(ζt) cos

`Ωprime(ζt)t

⋆ Equation de la thermocineacutetique

ρsCsVsdtTH = QH + Qc + Q2

Qc = minusλsS∆TLs

Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs

avec λac (ζt) prop P2(ζt)

⋆ BilanDagger

dtP = σ(TH )P

ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s

0 15 30 450

temps (min)∆

Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =

18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1

Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH

Autres meacutecanismes de saturation

dtP = σ(TH )P

∆T |infin = ∆T |seuil

Pinfin =

bdquoQHL2

sVs∆T|infin minus λs

Ajoutons arbitrairement un terme de saturation quadratique

prime = σ(T primeH )Pprime minus αPprime2

ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2

et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin

0 15 30 450

temps (min)

Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En

rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4

Conclusion

Autonomous oscillators =gt nonlinear saturation processes

⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified

⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described

Thermoacoustic heat transport by sound

A second-order effect which is described by the linear thermoacoustic theory

2realraquo

f lowastν minus fκ

p〈vlowastx 〉ndash

+ρ0Cp

2ω(1 minus Pr2)

|1 minus fν |2|〈vx〉|2dxT0

but actual stackregenerators employ materials of complicated geometry

mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section

and finally the problem is the same for the evaluation of w2

rArr Measure the sound scattered by the TAC

to evaluate both f 1minus4νκ (with QH = 0)

to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

Sketch of the experimental set-up used by Bannwart

et al6 to measure the T-matrix of the TAC as afunction of QH

[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014

Higher harmonics generation

⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1

parttp + (c + vx ) partxp = 0 An extensively studied topic23

t + t0 ∆t0

(c+v ) tx ∆

⋆ Impact on the operation of TA engines Outside the stack

Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic

ωWithin the stack

Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it

Outside the stack Solve NL propagation5

Determine optimal shape of the resonator5

Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T

rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering

Transient regime in a SW TA engineunder an assigned ∆T from [7]

An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]

[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002

Higher harmonics generation shock waves

⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed

TW engine=gt shock wave along x uarr

SW engine=gt no shock waves

damped SWvs

amplified TW

within the TACComplex processes

Entrance effects

⋆ Geometrical singularities =gt Vorticity

ldquoMinorrdquo losses1minus4

Basic modelingωt le π ∆pout(t) = minus 1

2 Koutρ0u2(t)

ωt ge π ∆pin(t) = 12 Kinρ0u

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

but thermal effects as well Abrupt transition from polytropic to adiabatic

rArr higher harmonics in τ5minus6

Heat transport accompanying vortex shedding

polytropic process adiabatic process

Vorticity field at the edge of a stack obtainedwith PIV (from [4])

Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])

rArr Are established results for steady flows applicable to oscillating flows

[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011

Acoustic streaming

⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets

ν0part2yyvx2 = 1ρ0partxp2 + partx

+ party`vx1vy1

acuteminus party

ν1partyvx1

⋆ A steady velocity vx2 generated by acoustic oscillations

- because of the convective derivative

+ party`vx1vy1

- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))

ν1partyvx1

- and actually many other sources of streaming

ν prop T1+βR 21 νdt gt

R 32 νdt

net drift

inner streaming

Acoustic streaming

Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including

⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))

⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)

⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p

MSh asymp 1)

⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

closed loop traveling wave engine standing wave engine

but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )

⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment

[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013

Acoustic streaming

⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p

MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl

insulatingfoam

Experimental setminusup of Thompson et al

LDV measurement window

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)

driven in phase

axial position

out of phaseπ2

Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)

nllow Re andor stack close to a velocity antiminusnode

nlRe =4

high Re andor stack close to a velocity nodenl

velocity antinode

Acoustic streaming

Acoustic streaming from the engineering standpoint

⋆ Acoustic streaming well-described only in simple devices at low amplitudes

=gt Empirical solutions to remove acoustic streaming

=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT

taperedtube

∆p2 right∆p2 left

∆p2 left∆p2 right

steady flow

Jet Pump elastic membraneor

⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines

[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997

Turbulence

Transition to turbulence extensively studied but in steady flows

rArr Only a few works in oscillating flows1minus8

Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers

St = ωDU amp Re = UD

vxvxvx

weakly turbulentlaminar conditionaly turbulentcenter of the duct

close to the wall

tω tωtω

HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])

10minus2

10minus1

transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

Merkli et al Moreau et al

others

iminusst

quasistationarity

loss of

Compressibility

And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc

[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006

Turbulence

Evaluation of losses due to turbulence (DELTA-EC)

=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows

steady flows

∆p = fMLD

12 ρ0〈vx〉2

〈wνturb〉 = ∆p〈vx〉L =

ρ0fM〈vx 〉32D

oscillating flows (Re(t) = 〈vx (t)〉Dν )

〈wνturb〉 =ρ0fM |〈vx 〉|3

fM(Re(t)) asymp fM +dfMdRe

(|Re(t)| minus Remax)

laminar turbulent

where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )

Avoiding turbulence from the engineering standpoint

polish internal surfaces of ducts

use of ldquoflow straightenersrdquo

flow straightener(mesh grids)

Preliminary conclusion

⋆ Linear theoryDesign tools are available but

- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation

⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines

TA Heat pumping significant

NL acoustics not much

Streaming significant

Edge effects significant

Turbulence

experience

⋆ How to fit experiments and theory =gt adjust any poorly known parameter

⋆ How to dissociate the role of each NL process =gt study the transient regime

Conclusion

Pourquoi eacutetudier le transitoire

Hysteresis6

compeacutetition de modes reacutegimes QP8

Switch onoff24

ldquoFishbone-like instabilityrdquo7

[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013

regenerator

flow straighteners

Description of the TAO in the time domain

⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem

part2ppartx2 minus 1

part2ppartt2

= q(x t)

⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1

part2p

partt2minus part

asymp minus 2c20radic

Cpartminus12

parttminus12

part2p

partx2

| z thermoviscous effects

+ (C + CT )1

partminus12

parttminus12

bdquopartp

| z thermoacoustic amplification

where the fractional derivatives

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )

rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )

[1] N Sugimoto J Fluid Mech 658 89-116 2010

TA wave amplification

⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12

bdquop(L)u(L)

= M3 times M2 times M1 timesbdquo

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2

=gt Muu = 0

Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that

Muu (ω T0(x) ) = 0

Quasi-steady state assumption (ie ǫ ≪ Ω)

dtasymp ǫ [T0(x)] P

where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)

[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012

Unsteady Heat Transfer

⋆ Unsteady heat transfer1(simplified and summarized)

8T =Twall

Heating Q(t)

thermoacoustic heat transport x

Heat exchange with walls

x isin [0 xc ] cup [xh L]partT0partt = 1

ρf Cf

partpartx

λfpartT0partx

minus T0minusTinfinτf

x isin [xc xh]partT0partt = 1

ρsCspart

λspartT0partx

minus T0minusTinfinτs

minus 1ρsCs

partϕacpartx

x = xc λfpartT0partx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

x = xh λspartT0partx

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x isin [xc xh] ϕac = 12

ρf T0πR2 real (〈s vlowast

x 〉)

Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)

[1] M Guedra et al J Appl Phys 2014

Summary

⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

coupled to that describing wave amplification

dt= ǫ [T0(x)] P

=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme

⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area

Comparison with experiments (1)

The device

L = 05m length of the ductLs = 005m length of the stack

rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)

Qin heating powert = 0 time at which heating is switched on

Comparison theoryexperiments

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

(a) (d=20cmQin=233 W)(d=20cmQin=186 W)

(b) (d=14cmQin=233 W)(d=20cmQin=233W)

M Guedra et al J Appl Phys 2014

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack

T3 temperature 25 cm far from the stack

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

Interfeacuteromeacutetrie

⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct

rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl

rArr vent acoustique pompage thermoacoustique effets de bords

⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

Principe de la mesure

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

milieu drsquoetude n(xyzt)

camera CCD

λ∆ϕref (x y)

nref (x y z) minus ninfini

dz + 2R

Lv[nv (x y z) minus ninfin] dz

λ∆ϕ(x y t)

L [n(x y z t) minus ninfin] dz + 2R

=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =

n(x y z t) minus nref (x y z)i

rArr accegraves agrave 〈n〉 = (1L)R

ndz puis 〈ρ〉 = (1L)R

ρdz via la relation de Gladstone-Dalesect

r = 23

1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm

sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)

paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde

Monteacutee en teacutempeacuterature

Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)

Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct

Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W

pas drsquoeffet de chromatisme

mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement

mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire

Deacuteclenchement simple masse volumique instantaneacutee

xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

pression acoustique instantaneacutee lors du deacuteclen-chement

masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas

Quelques estimations (au plus fort des auto-oscillations )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

alors mecircme que Pmax cos(k0xs)P0 asymp 07

dac asymp Pmax

ωc0sin(k0xs ) asymp 14mm

Deacuteclenchement simple analyse spectrale

⋆ Analyse spectrale des signaux de masse volumique instantaneacutee

Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)

Amplitude des harmoniques en fonction de la zonedrsquoobservation

rArr Effets de bords ldquothermiquesrdquolowastlowast

Passage abrupte drsquoune zone polytropiquevers une zone adiabatique

Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]

lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011

Deacuteclenchement simple moyenne glissante

⋆ moyenne glissante ρ = (1T )R

ρdt (T asymp 20 peacuteriode acoustiques)

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12

rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)

⋆ Explication possibles

Reacutechauffement localiseacute (dissipation visqueusevent acoustique )

Pertes de charges singuliegraveres ρ = f (T P)

Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]

Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

moyenne glissante ρm dans les zones 369et 12

Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

The transient regime of TA Oscillators the big deal

What is the mechanism responsible for the switch onoff process

1D assumption for both T and p

Entrance effects impacting T

Acoustic streaming Everything together

and how can we claim understanding TA engines if we cannot reproduce such an effect

Conclusion

The TAO as an interesting dynamical system

The ThermoAcoustic Oscillator as an interesting dynamical system

⋆ An autonomous oscillator driven by heat

which is almost out of control above threshold

and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )

⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives

revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)

investigate the active control of TA engine to ր their efficiency

[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo

Cambridge University Press NY 2001

Synchonization Phenomena

Synchronization phenomena

⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

A movie made at Lancaster University (UK) which can be easily found on youtube

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

⋆ Use or observation of synchronization phenomena are nowadays abundant biology medicine(singing crickets circadian rythm cardiac pacemaker ) electronics engineering(triode generators for radio communications ) mechanics(clocks organ pipes ) physicschemistry(lasers BndashZ reaction ) social life(clapping audience)

⋆ An example in the field of acoustics synchronization of organpipes

Former study by Lord Rayleigh mutual synchronizationof two organ pipes and the quenching effect (oscillationdeath)

but even recent studies12

Sketch of the exp by Abel et al2

[1] Abel et al J Acoust Soc Am 119 2467 2006 [2] Abel et al Phys Rev Let 103 114301 2009

Synchronization of a TAO by a loudspeaker experiment

Thermoacoustic oscillator

loudspeaker enclosure

BampK Nexus 2692C

conditioning amplifier

to data acquisition cardNI PCI 6143

amplifier

power Frequency generator

Agilent 33120 A

Agilent 54624 A

Oscilloscope Audio spectrum analyzer

Stanford Research SR785

NI BNC 2110

DC Power SupplyMCP Lab electronic

M10minusTPminus305minusC

connector block

Nichrome wire

Loudspeaker enclosure

(a) Photograph of the complete experimental set-up (c) Sketch of the experimental set-up

[ Penelet amp Biwa Am Journ Phys 2013]

Synchronization phase locking phase trapping beating death

Synchronization by a sound source experimental protocol

⋆ Experimental protocol Fix heat power Q=226 W (Q gt Qonset ) Fix both d and ds

Proceed to measurementsdaggerdagger by varying the driving frequencyf and the loudspeaker voltage Urms

Q=226 W

⋆ Signal processing Make both FFT and Hilbert transforms of p(t) and U(t) Quantities of interest for data analysis - Frequency spectra p(f ) and U(f )- Amplitude modulation Ap(t)- Instantaneous phase difference Ψ(t) = Φp(t) minus ΦU(t)

⋆ Different possible states(PS) (IPL) (QP) (BD)

Perfect Synchronization Imperfect Phase Locking QuasiPeriodicity Beating Death

fTAO = f (or fTAO = fn ) fTAO 6= f (or fTAO 6= f

n ) fTAO 6= f (or fTAO 6= fn )

Ap (t) = cte Ap (t) 6= cte Ap (t) 6= cte Ap (t) = cte

Ψ(t) = cte Ψ(t) 6= cte but bounded Ψ(t) 6= cte (unbounded)

daggerdaggerEach set of measurements sim 12-24 hours of total duration

Synchronization the obtained Arnolrsquod Tongues

⋆ Stability diagram as a function of Uloudspeaker and f for d=1 mm ds=8 cm and Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

time (s) time (s) time (s)

Stability diagram

Transition QP=gtBD

Transition PS=gtIPL=gtQP

synchronization 31

Quenching

synchronization 21

synchronisation 11

Some universal aspects of synchronization are retrieved by this experimentDaggerDagger

Some aspects are intrinsic to the TAO itself =gt why are the Arnold tongues asymetric =gt why does quenching occurs around f asymp fTAO2

DaggerDaggerOnly the 2nd experimental evidence of the phase trapping phenomenon

External forcing of relaxational oscillations

Other phenomena related to the external forcing of a TAO

⋆ What if the TAO is operated as a relaxa-tional oscillator

Q=245 W

d=4 cmd =25 cmLoudpseaker off =gt ldquointegrate and fire regi-merdquo with fTAO asymp 1771 Hz

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

⋆ Experimental procedure1- Settle f to 1769 Hz (6= fTAO)2- Switch the loudspeaker on at t=100 s3- Switch the loudspeaker off at t=150 s4- Repeat steps 1-3 with another driving vol-tage

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

interpretation =gt Simply quenching or more complicated processes =gt does external forcing lead to the transformation of a bistable regime (with periodic switch

between two stable states) into a stable one

Synchronization mutuelle de 2 TAO

Active Control of a TAO the idea

And what about forcing a TAO to increase its efficiency

Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1

- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase

thermoacoustic core

Loudspeakers

A new proof of concept study on a larger TA engine2

- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop

[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014

Active Control of a TAO the device

⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Auxiliary source S (WLS)

Microphone micro

Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)

⋆ The control parameters

The heat input Qh The phase shift φ

The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output

Feedback loop switched off

The output power Wel0

The efficiency η0 = Wel0Qh

The temperature difference ∆T0

as a function of Qh

Feedback loop switched on

Wel and ∆Wel = Wel minus Wel0

η = Wel ldquo

Qh + WLS

and ηη0

∆T and ∆T minus ∆T0

as a function of Qh φ and G

Active Control of a TAO the results(1)

Feedback loop switched off (G=0)

η0(Qh) (bull) ∆T0(Qh) ()

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Feedback loop switched on (G 6= 0)

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

rArr The electroacoustic feedback loop impacts the operation of the TA engine

Possibility to ր η (ie to have η gt η0)

Here φopt asymp 270

Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)

The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

rArr η gt η0 while ∆T gt ∆T0

The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

∆Wel () and WLS (bull) as a function of G for Qh = 70W

∆Wel () and WLS (N) as a function of G for Qh = 150W

rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )

Conclusion

⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)

Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport

⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines

Turbulence

⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies

⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines

RemerciementsLrsquoOR thermoacoustique du LAUM

Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)

Les anciens doctorants

Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART

Autres collegravegues du LAUM

M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART

ou drsquoailleurs

T BIWA (Tohoku Univ Japan)

merci de mrsquoavoir eacutecouteacute



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples




Analogies eacutelectroacoustique




Reacutesolution analytiquemeacutethode des temps multiples

Reacutesolution analytiquemeacutethode des temps multiples (2)







Entrance effects

Acoustic streaming

Turbulence





ThermoAcoustic wave amplification


Summary











Bilan




Synchronization of a TAo by a loudspeaker experiment

Synchronization phase locking phase trapping beating death hellip

Synchronization the obtained Arnold Tongues


Active control of a TAOthe idea

Active control of a TAOthe device

Active control of a TAO the results(1)



Conclusion

Conclusion

Effet piston

Haut-parleur TA

Effet piston

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

Effet piston

Haut-parleur TA

Effet piston

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Effet piston

Haut-parleur TA

Effet piston

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Effet piston

Haut-parleur TA

Effet piston

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Haut-parleur TA

Effet piston

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Effet piston

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

pulsating sphere

oscillating sphere

∆p minus1

ttp = minuspartt q

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

pqdt gt 0)

TA Laser Rijke tube

WoutWout

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

regenerator

flow straighteners

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Basic principles

Q Qrsquo

21 3 4 1τp

ξ max2

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Basic equations

ρprime = 1ω2

δν =q

2νω δκ =

q2κω

0 05 1 15 20

R=100δν

R=10δν

R=δν

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

ρprime = 1ω2

idxT0T0

dx p + 1c20

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

minus1

minus05

ρρmax

0 02 04 06 08 1minus1

minus05

ρρmax

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Sound amplification

([w2 ] = Wmminus3)

wκ =1

(γ minus 1)ω

image (fκ) |p|2

wν =1

2ωρ0

image(fν)

2image (p〈vlowast

x 〉)

wtw = real(h)dx T0

2real (p〈vlowast

x 〉)

h =fκ minus fν

(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

δκRh

minus04

minus02

real (h)

image (fν)

|1minusfν|2

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

τ minus 1

ρ0T0p

2image`p〈vlowast

x 〉acute

qtw = real(g)1

2real`p〈vlowast

x 〉acute

λac =ρ0Cp

2ω(1 minus Pr2)

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

w2 = wκ + wν| z

losses

propreal(h)z|wtw

| z production

Q Qrsquo

CYLINDER

SQUARE

R = 3 rh i

PINminusARRAY

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

w2 = wκ + wν| z

losses

propminusimage(h)z|

propreal(h)z|

wtw| z

production

Qrsquo Q

Stirling Cycle

regenerator

bull Z = pvx

= ρ0c0

bull Z = pvx

= ρ0c0

regenerator

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Examples

regenerator

flow straighteners

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

T (x)0

|p(x)|

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

volume V

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

11minusfν

T0 T +dT0 0

p+dp~ ~

u+du~ ~u~

M =ρ0dxφS

1minusreal(fν )

|1minusfν |2

C = φSdxγP0

Rν =ωρ0dx

φSimage(minusfν )

|1minusfν |2

Rν~u(x)

p(x)~ ~p(x+dx)

~u(x+dx)

C Rκ ~ u(x)

= γminus1γ

isin C

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

volume V

~ uC p~

ω20(TH)

M +ρ0Ls

|f | =2radic

bdquoTm

laquoβ2radic

1 +radic

1 + LsφL

bdquoradicPr

1 +radic

rdquo Ls

laquo2

Arg(f ) = arctan

bdquoradicPr

1 +radic

rdquo TC

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

2radic

ω0|f |eminusiω0t2

4 minus Arg(f )

Fminus1h

asymp ω320 |f |

d2ttu(t) minus 3

2radic

0u(t) asymp 0

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

p minus 32radic

0p asymp 0

volume V

volume V = 1l

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

0 005 01 015 02 025

minus1

minus05

temps s

TH = 300K TH = 460K TH = 470K

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

avec θ12 = ω0θ12 et ζ = 12

|f |radicω0

2νmω0

Rs) ≪ 1)

d2θθp asymp part

2ττ p + 2ζpart

⋆ Ordre ζ0

⋆ Ordre ζ1

part2ττ p1 + p1 =

e iτ + cc

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

dsA = 12Aldquo

dsR =`

32 cos θ2 + 1

2 sin θ1acute

RdsΦ = 1

Ωprime = ω0ˆ1 +

`12 cos θ1 minus 3

2 sin θ2acute

ζ˜ et

ζ(TH ) = 12

|f |radicω0

300 350 400 450 500 550 6001005

Ωrsquoω

(K)300 350 400 450 500 550 600

minus001

minus0005

Ωrsquorsquo

minus1

minus05

analyt

0 005 01 015 02 025 03 035 04 045 05

minus1

minus05

temps (s)

analyt

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

T =cteC

T (t)H

~ uC p~

`Ωprime(ζt)t

⋆ BilanDagger

dtP = σ(TH )P

0 15 30 450

temps (min)∆

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

dtP = σ(TH )P

Pinfin =

bdquoQHL2

0 15 30 450

temps (min)

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

2realraquo

+ρ0Cp

2ω(1 minus Pr2)

PZE buzzer

Ac Imp Sensor

Ref planeMic2Mic1

ThermoAcoustic Core

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

t + t0 ∆t0

(c+v ) tx ∆

ωWithin the stack

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

damped SWvs

amplified TW

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Entrance effects

2 Koutρ0u2(t)

Wminor propR π

ω0 ∆pudt +

R 2πω

∆pudt prop U3

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Acoustic streaming

+ party`vx1vy1

acuteminus party

ν1partyvx1

+ party`vx1vy1

ν1partyvx1

R 32 νdt

net drift

inner streaming

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Acoustic streaming

MSh asymp 1)

u2 ρ1u1ρ + 00

u20 ρ1u1ρ + 0

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Acoustic streaming

insulatingfoam

laser beam

uncontrolled

insulated

isothermal

u (xr=0) Re =102 nl

u (xr=0) Re =202 nl

u (xr=0) Re =402 nl

2 nlu (L4r) Re =40

2 nlu (L2r) Re =40

2u (3L4r) Re =40nl

Re =3nl

Re =13nl

Re =59nl

Re =78nl

Re =132nl

measurement window

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Acoustic streaming

driven in phase

axial position

out of phaseπ2

nlRe =4

velocity antinode

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Acoustic streaming

taperedtube

steady flow

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Turbulence

vxvxvx

close to the wall

tω tωtω

10minus2

10minus1

10minus2

10minus1

103 104 105 106

~minusReδ 280

Reδ minus~500

others

iminusst

quasistationarity

loss of

Compressibility

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Turbulence

steady flows

∆p = fMLD

12 ρ0〈vx〉2

ρ0fM〈vx 〉32D

laminar turbulent

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Turbulence

experience

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Hysteresis6

Switch onoff24

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

regenerator

flow straighteners

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

part2ppartt2

= q(x t)

part2p

partt2minus part

Cpartminus12

parttminus12

part2p

partx2

+ (C + CT )1

partminus12

parttminus12

bdquopartp

partminus12f (x t)

parttminus12=

1radic

minusinfin

f (x τ )radic

t minus τdτ

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

bdquop(L)u(L)

p(0)u(0)

raquoMpp Mpu

Mup Muu

timesbdquo

p(0)u(0)

T 8T 8

T (x)0

xC xH L0

=gt Muu = 0

Muu (ω T0(x) ) = 0

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

8T =Twall

Heating Q(t)

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉)

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Summary

ρf Cf

partpartx

λfpartT0partx

ρsCspart

λspartT0partx

minus 1ρsCs

partϕacpartx

˛˛˛xminusc

= λspartT0partx

˛˛˛x+c

minus ϕac (x+c )

˛˛˛xminush

= λfpartT0partx

˛˛˛x+h

+ ϕac (xminush

) + Q(t)πR2

x 〉) = f (P2 x)

dt= ǫ [T0(x)] P

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

The device

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

0 50 100 150 200 250 300minus15

minus1

minus05

time (s)

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

100 600 700 800 900 1000minus1

minus05

time (s)

200 210 220 230 240 250 260 270 280 290 300minus1

minus05

time (s)

0 100 200 300 400 500 600 700 800 900 100020

time (s)

200 400 600 1000 1200 1400minus1

minus05

time (s)

400 420 440 460 480 500 520 540 560 580 600minus1

minus05

time (s)

0 500 1000 150020

time (s)

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

CameraSpeed High

Tri CCCCamera

laser 660nm

microphone

laser 532nmM0

dichroic plate

5050 cube

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

milieu reference n8

verre nv

∆ϕAcces a y

faisceau objet

faisceau reference

camera CCD

λ∆ϕref (x y)

dz + 2R

λ∆ϕ(x y t)

r = 23

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12

d(2dac )

()8 28 48 68 88 108 128 148 168 188 208 228

ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09

dac asymp Pmax

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

minus3 )

0 05 1 15 2 25 3 35 4070809

temps (s)

minus3 )

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

200 210 220 230 240 250 260 270 280minus800

minus600

minus400

minus200

Temps (s)

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

minus3 )

0 2 4 6 8 10 12 14 1608

temps (s)

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

ρ m (

minus3 )

0 2 4 6 8 10 12 14 16

temps (s)

ρ m (

minus3 )

zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

coupling beam

f 1 f 2 1f f 2

(a) (b)

Mg2tα ( )

1tα ( )

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

BampK Nexus 2692C

amplifier

Agilent 33120 A

Agilent 54624 A

NI BNC 2110

connector block

Nichrome wire

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Q=226 W

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

frequency (Hz)

Stability diagram

Transition QP=gtBD

synchronization 31

Quenching

synchronization 21

synchronisation 11

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Q=245 W

minus1000

minus500

minus1000

minus500

minus1000

minus500

ON OFF

minus1000

minus500

minus1000

minus500

0 25 50 75 100 125 150 175 200 225 250 275minus1000

minus500

1000p(

time (s)

=105mV

=175mV

=380mV

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

thermoacoustic core

Loudspeakers

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Regenerator

HHX (Qh)

Coupling Duct

Alternator (Wel)

0 20 cm

Microphone micro

as a function of Qh

η = Wel ldquo

Qh + WLS

and ηη0

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

60 80 100 120 140 1600

Qh [W]

60 80 100 120 140 160400

∆T0[K

Qh=75 W ηη0(G φ)

0 45 90 135 180 225 270 315 3600

φ [deg]

G = 22

G = 46

G = 10

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

0 45 90 135 180 225 270 315 3600

0 45 90 135 180 225 270 315 360minus10

φ [deg]

∆Tminus

∆T0[K

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

0 05 1 15 210minus4

10minus3

10minus2

10minus1

gain G

WLS∆W

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Conclusion

Turbulence

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

ou drsquoailleurs



Haut-parleur TA

Effet piston







Basic principles

Basic equations


Sound amplification




Examples
















Entrance effects

Acoustic streaming

Turbulence







Summary











Bilan













Conclusion

Documents

Auto-oscillateurs thermoacoustiques : effets non linéaires et