Mathematical aspects of thermoacoustics - Pure · Mathematical Aspects of Thermoacoustics PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,

Mathematical aspects of thermoacoustics

in 't Panhuis, P.H.M.W.

DOI:10.6100/IR642908

Published: 01/01/2009

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Mathematical Aspects ofThermoacoustics

Copyright c©2009 by P.H.M.W. in ’t panhuis, Eindhoven, The Netherlands.All rights are reserved. No part of this publication may be reproduced, stored in a re-trieval system, or transmitted, in any form or by any means, electronic, mechanical,photocopying, recording or otherwise, without prior permission of the author.

Printed by Print Service Technische Universiteit Eindhoven

Cover design by Jorrit van Rijt

A catalogue record is available from the Eindhoven University of Technology Library

ISBN 978-90-386-1862-3NUR 919Subject headings: thermoacoustics; acoustics; acoustic streaming; thermodynamics; per-turbation methods; numerical methods; boundary value problems; nonlinear analysis;shock waves.

This research was financially supported by the Technology Foundation (STW), grantnumber ETTF.6668.

Mathematical Aspects ofThermoacoustics

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het Collegevoor Promoties in het openbaar te verdedigen

op donderdag 25 juni 2009 om 16.00 uur

door

Petrus Hendrikus Maria Wilhelmus in ’t panhuis

geboren te Roermond

Dit proefschrift is goedgekeurd door de promotor:

prof.dr. J.J.M. Slot

Copromotoren:dr. S.W. Rienstraenprof.dr. J. Molenaar

To Ik Cuin

Preface

This project is part of a twin PhD program between the Departments of Mathemat-ics and Computer Science and Applied Physics and is sponsored by the TechnologyFoundation (STW), Royal Dutch Shell, the Energy Research Centre of the Netherlands(ECN), and Aster Thermoacoustics. I would like to express thanks to all people whoparticipated in this project. First and foremost I would like to thank my mathematicalsupervisors dr. Sjoerd Rienstra, prof.dr. Han Slot, and prof.dr. Jaap Molenaar for theirexpert guidance and stimulating support. I am also indebted to my physics colleaguesPaul Aben, dr. Jos Zeegers, and prof.dr. Fons de Waele, who helped to broaden anddeepen my understanding of the physics involved. I am also grateful for the manyuseful discussions I have had with the people from ECN, Aster Thermoacoustics, andShell.

My defense committee is formed by prof.dr. Anthony Atchley, prof.dr. Bendiks-JanBoersma, and prof.dr. Mico Hirschberg, together with my supervisors dr. Sjoerd Rien-stra, prof.dr. Han Slot, and prof.dr. Jaap Molenaar. I would like to thank them for thetime invested and their willingness to judge my work. I also want to thank prof.dr. BobMattheij for agreeing to be part of the extended defense committee.

What made these four years especially enjoyable was the great working atmospherewithin CASA and the Low Temperature group. My special thanks go out to my of-fice mate and partner-in-crime Erwin who started and finished his PhD (and Master)at the same time as I did. Many thanks also to all the current and former colleaguesthat I have had the pleasure to work with, in particular the PhD students and postdocs:Aga, Ali, Andriy, Bart, Berkan, Christina, Darcy, Davit, Dragan, Hans, Jurgen, Kakuba,Kamyar, Kundan, Laura, Marco, Maria (×2), Mark, Matthias, Maxim, Michiel, Miguel,Mirela, Nico, Oleg, Patricio, Paul, Remko, Remo, Roxana, Shruti, Sven, Tasnim, Temes-gen, Valeriu, Venkat, Wenqing, Yabin, Yan, Yixin, Yves, Zoran. I fondly think back toour daily lunches at the Kennispoort, the weekly poker games, the road-trips to Den-mark, the regular squash/tennis/football games, and the many nights in town that Ihave enjoyed with so many of you. Our two secretaries Marese and Enna also deservea special word of thanks, for making life of a PhD student so much easier by taking careof all administrative details. I am also thankful to the members of the football teamsPusphaira and Old Soccers, and hope they will have more success without me.

On a more personal level, I would like to thank all my friends and family, for theircontinuous love and support. I especially want to show appreciation to my mom forher unbridled enthusiasm and my dad, who I wish could have been here today. I alsowant to thank my siblings Jos, Hellen, and Dorris and their significant others Marjanne,Joram, and Tonnie. Of course I should not forget to mention my little nephew Sep, who

ii Preface

is getting so big now. Last, but definitely not least, I would like to thank my girlfriendJessey, to whom this thesis is dedicated, for her unlimited love and patience in these lastfew months.

Peter in ’t panhuisEindhoven, May 2009

Nomenclature

General symbols and variables

A [m2] cross-sectional area

A [m2] cross-section

b [ms−2] specific body force field

c [m s−1] speed of sound

Cp [J kg−1 K−1] isobaric specific heat

Cs [J kg−1 K−1] specific heat of stack material

Cv [J kg−1 K−1] isochoric specific heatd [m] diameterf [Hz] frequencyfν viscous Rott functionfk thermal Rott functionfs solid Rott functionFν viscous Arnott functionFk thermal Arnott functionFs solid Arnott functionF Fourier transformG Green’s function

g [ms−2] gravitational accelerationH [W] total power

h [J kg−1] specific enthalpyIm imaginary part

k [m−1] wave number

K [W K−1 m−1] thermal conductivityℓ [m] displacementL [m] typical length

m [kg m−2 s−1] time-averaged mass flux

M [kg m−2 s−1] time-averaged volumetric mass flux

n [mol s−1] molar flow rateP [W] powerp [Pa] pressurepA [Pa] pressure oscillation amplitudepamb [Pa] ambient pressure

iv Nomenclature

Q [W] heat flow per unit time

q [W m−2] heat fluxr [m] radial coordinateRe real part

Rspec [J kg−1 K−1] specific gas constant

R [m] radius

s [J kg−1 K−1] specific entropy

S [J kg−1] entropyS surfaceSu [K] Sutherland’s constantt [s] timeT [K] temperature

U [m s−1] typical fluid speedU [W] internal energy

v = (u, v, w) [m s−1] velocity vector

V [m3] volumeW [W] acoustic powerx = (x, y, z) [m] spatial coordinate

Z [N s m3] impedance

β [K−1] isobaric volumetric expansion coefficientΓ boundary, interfaceδ [m] penetration depth

ǫ [J kg−1] specific internal energyλ [m] wave lengthµ [Pa s] dynamic (shear) viscosityζ [Pa s] second viscosity

ρ [kg m−3] density

τ [N m−2] viscous stress tensorθ [rad] angular coordinate

ω [rad s−1] angular frequency of the acoustic oscillations

Dimensionless numbers

A amplitudeBr blockage ratioCOP coefficient of performanceCOPR relative coefficient of performanceCOPC Carnot coefficient of performanceDr drive ratioFr Froude numberMa acoustic Mach numberNL Lautrec number gasNs Lautrec number solidPr Prandtl numberR reflection coefficient

Nomenclature v

Sk Strouhal number based on δk

Wo Womersley numberWζ second Womersley numberβ coefficient of nonlinearityγ ratio specific heatsδ coefficient of stack dissipation∆ deviation from resonanceη coefficient of weak nonlinearityη efficiencyηR relative efficiencyηC Carnot efficiencyε aspect ratioε driver Mach numberεs stack heat capacity ratioκ Helmholtz numberσ ratio thermal conductivitiesφ porosityµ dimensionless viscosity

Sub- and superscripts and special operators

a dimensionlessa time averaging〈a〉 transverse averaginga per unit timea∗ complex conjugate

are f reference value

a+ top plate

a− bottom platea0 steady zeroth ordera1 first harmonica2,0 steady second order (streaming)a2,2 second harmonicaC coldaH hotag gas, fluidak thermalaL thermalap isobaricaR rightas solid, source, stack centerat outeraτ transverse componentsav isochoricaν viscous

vi Nomenclature

Contents

Preface i

Nomenclature iii

1 Introduction 11.1 A historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The basic mechanism of thermoacoustics . . . . . . . . . . . . . . . . . . . 61.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Thermodynamics 172.1 Laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Thermodynamic performance . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Refrigerator or heat pump . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Prime mover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 The thermodynamic cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Standing-wave phasing . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Traveling-wave phasing . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Bucket-brigade effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Modeling 273.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Small-amplitude and long-pore approximation . . . . . . . . . . . . . . . . 34

4 Thermoacoustics in two-dimensional pores with variable cross-section 374.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Mean temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Integration of the generalized Swift equations . . . . . . . . . . . . . . . . 51

4.4.1 Exact solution at constant temperature . . . . . . . . . . . . . . . . 524.4.2 Short-stack approximation . . . . . . . . . . . . . . . . . . . . . . . 544.4.3 Approximate solution in short wide channels . . . . . . . . . . . . 56

viii Contents

4.5 Acoustic streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.6 Second harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.7 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.7.1 Acoustic power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.7.2 Total power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Thermoacoustics in three-dimensional pores with variable cross-section 735.1 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Mean temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.3 Integration of the generalized Swift equations . . . . . . . . . . . . . . . . 83

5.3.1 Ideal stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.2 Rotationally symmetric pores . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Acoustic streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.5 Second harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.6 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Standing-wave devices 936.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3 A thermoacoustic couple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.1 Acoustically generated temperature differences . . . . . . . . . . . 976.3.2 Acoustic power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4 A standing-wave refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.5 A standing-wave prime mover . . . . . . . . . . . . . . . . . . . . . . . . . 1086.6 Streaming effects in a thermoacoustic stack . . . . . . . . . . . . . . . . . . 112

7 Traveling-wave devices 1157.1 A traveling-wave prime mover . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2.1 Regenerator and thermal buffer tube . . . . . . . . . . . . . . . . . . 1197.2.2 Optimization procedure . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.2 Regenerator efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.3 Geometry optimization . . . . . . . . . . . . . . . . . . . . . . . . . 129

8 Nonlinear standing waves 1338.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.1.1 Kuznetsov’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.1.2 Bernoulli’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.1.3 Perturbation expansion . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.2 Solution away from resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.2.1 Arbitrary excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.2.2 Harmonic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.3 Solution near resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.3.1 Exact solution when δ = ∆ = 0 using Mathieu functions . . . . . . 1438.3.2 Steady-state solution for δ = O(1) . . . . . . . . . . . . . . . . . . . 144

8.3.3 Steady-state solution for δ = O(ν−1) . . . . . . . . . . . . . . . . . 148

Contents ix

8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.4.1 Nonlinear standing waves in a closed tube . . . . . . . . . . . . . . 1558.4.2 Nonlinear standing waves in a thermoacoustic refrigerator . . . . . 157

9 Conclusions and discussion 161

Appendices 165

A Thermodynamic constants and relations 165

B Derivations 167B.1 Total-energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167B.2 Temperature equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

C Green’s functions 169C.1 Fj-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169C.2 Fj,2-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171C.3 Green’s functions for various stack geometries . . . . . . . . . . . . . . . . 172

Bibliography 175

Index 185

Summary 187

Samenvatting 189

Curriculum Vitae 191

x Contents

Chapter 1

Introduction

As the name indicates, thermoacoustics combines the fields of thermodynamics andacoustics and describes the interaction between heat and sound. The term was firstcoined in 1980 by Rott [119] who described its meaning as “rather self-explanatory”. Ac-cording to Rott the most general interpretation of thermoacoustics includes “all effectsin acoustics in which heat conduction and entropy variations of the (gaseous) mediumplay a role”.

Ordinarily a sound wave in a fluid is seen as the combined effect of pressure andvelocity (displacement) oscillations, but as a response to these (isentropic) pressure os-cillations, temperature oscillations may occur as well. In free space the temperaturevariations will be small and the gas parcels will expand and compress adiabatically.However, when the fluid is allowed to interact thermally with solid boundaries, heattransfer between the gas and the solid will take place and a wide range of thermoacous-tic effects may occur.

It has been realized that under the right operating conditions these thermoacousticconcepts can be harnessed and exploited to create two kinds of thermoacoustic devices:the refrigerator or heat pump that turn sound into useful refrigeration or heating, and theprime mover that turns heat into useful (acoustic) work. Typically, such devices are con-structed from straight or looped tubes with a porous medium suitably located inside.

Thermoacoustic devices can be of much practical use, because significant amountof heat and mechanical power can be produced at a reasonable efficiency. Moreover,in contrast to more conventional engines and refrigerators, they can operate withoutcranks and pistons, and usually have no more than one (mechanically) moving part.Because of this and their inherent simplicity, they are very reliable, require little mainte-nance, and can be produced at relatively low cost. Furthermore, thermoacoustic devicesare friendly to the environment, as they use environmentally friendly gases, produceno toxic waste, and are easily adaptable to use solar power [27] or industrial wasteheat [127] as a driving source. Currently, research is also being done on the possibil-ities of using biomass to drive a thermoacoustic stove [112], to be used in developingcountries.

Despite all these advantages there are still some challenges left that need to be re-solved before thermoacoustic devices can be used competitively on a large scale. Firstly,due to the oscillatory nature of the flow and the interaction with solid boundaries all

2 1.1 A historical perspective

kinds of complicated flow patterns may arise, such as vortex shedding, turbulence, orasymmetric flow. Furthermore, the heat transfer is far from ideal and entropy is created.Additionally, thermoacoustic devices often operate at high pressure amplitudes, whichcan give rise to various nonlinear effects, such as the build-up of shock waves. The com-bined effect of all these phenomena can and will degrade the performance and holdsback the development of highly efficient devices. Moreover, as long as these effects arenot understood and modeled systematically, it will be hard to make accurate theoreticalpredictions. The mathematical aspects of oscillatory gas flow with heat transfer to solidboundaries in wide or narrow pores will be the topic of this thesis.

1.1 A historical perspective

Thermoacoustics has a long history that dates back more than two centuries. The inter-est in thermoacoustics was first piqued in 1816, when Laplace showed that even Newtonwas not infallible. Laplace [72] pointed out that Newton’s approximation of the speedof sound [92] was incorrect because he assumed isothermal compression and expansionof the air and did not compensate for variations in temperature. Correcting for theseeffects, Laplace found a value that was 18% higher.

Hydrogensupply

Flame

(a) Higgins’ singing flame

Wirescreen

Generationof sound

Convectionflow

(b) Rijke tube

Figure 1.1: (a) Higgins’ singing flame: for suitable positions of the flame the tube will start toproduce sound. (b) The Rijke tube: the loudest sound is produced when the heated wire screen ispositioned at one-fourth from the bottom of the pipe.

The first records of heat-driven oscillations are the observations of Higgins [54, 102]in 1777, who experimented with an open glass tube in which acoustic oscillations wereexcited by suitable placement of a hydrogen flame, the so-called “singing flame”. A

Introduction 3

similar, but more famous experiment was performed by Rijke [111] who in his efforts todesign a new musical instrument from an organ pipe, constructed the so-called “Rijketube”. As depicted in figure 1.1, he replaced Higgins’ hydrogen flame by a heated wirescreen and found that when the screen was positioned in the lower half of the opentube spontaneous oscillations would occur, which were strongest when the screen waslocated at one fourth of the pipe. The oscillations would stop if the top of the tube wasclosed, indicating that the convective air current through the pipe was necessary forsound to be produced. Higgins’ and Rijke’s work later led to the birth of combustionscience, with applications in rocket science and weapon industry. For a full reviewon devices related to the Rijke tube we refer to Feldman [41] or more recently Raunet al. [108].

The earliest predecessor of the type of thermoacoustic prime movers considered inthis thesis is the Sondhauss tube, depicted in figure 1.2. It was invented in 1850 by Sond-hauss [126] based on an effect often noticed by glass blowers: generation of loud soundwhen a hot bulb is blown at the end of a cold narrow tube. Sondhauss found that if asteady gas flame was supplied to the closed bulb, the tube would produce a clear soundwhich was characterized by the length of the tube; the larger the bulb or the longer thetube, the lower the frequency of the sound. Unfortunately, Sondhauss did not manageto explain why the oscillations arose. Feldman [42] also reviewed most of the phenom-ena related to the Sondhauss tube as he did for the Rijke tube. An important differencebetween these two devices is that the Sondhauss tube does not require a convective aircurrent for oscillations to occur.

Bulb Tube stem

SoundFlame

Figure 1.2: The Sondhauss tube: sound is generated from the tip of the stem, when heat issupplied to the bulb.

Another early example of a thermoacoustic prime mover is the phenomenon knownas “Taconis oscillations”, often observed in cryogenic storage vessels. Taconis [139]cooled a gas-filled tube from room temperature to a cryogenic temperature by insertingit into liquid helium and observed spontaneous oscillations of the gas. The conditionsfor these type of oscillations have been investigated experimentally by Yazaki et al. [156].

The first qualitative explanation for heat-driven oscillations was given in 1887 byLord Rayleigh. In his classical work “The Theory of Sound” [109], he explains the pro-duction of thermoacoustic oscillations as an interplay between heat fluxes and densityvariations:

4 1.1 A historical perspective

”If heat be given to the air at the moment of greatest condensation (compression)or taken from it at the moment of greatest rarefaction (expansion), the vibration isencouraged”.

Rayleigh’s qualitative understanding turned out to be correct, but a quantitatively ac-curate theoretical description of these phenomena was not achieved until much later.

The reverse process, generating temperature differences using acoustic oscillations,is a relatively new phenomenon. In 1964 Gifford and Longsworth [48] invented thepulse-tube refrigerator, by which they managed to cool down to a temperature of 150K. In their device, depicted in figure 1.3, heat was pumped along the tube wall by sup-plying pressure pulses at low frequencies. Initially it was considered nothing more thanan academic curiosity as it was highly inefficient, but current-day pulse-tube cryocool-ers can reach efficiencies up to 20% of the ideal efficiency and temperatures as low as2 K. In fact, nowadays pulse-tube refrigeration is one of the most favored technologiesfor cryocooling . For a complete history and review of pulse-tube crycooling we re-fer to Radebaugh [104, 105]. Detailed modeling and numerical analysis of pulse-tuberefrigerators can be found in [79–81].

Rotary

valve

VentHigh-pressuresource

Regenerator

Cold end

Pulsetube

Roomtemperature

Figure 1.3: The pulse-tube refrigerator of Gifford and Longsworth. The temperature is cooledfrom room temperature at the hot end to 150 K at the cold end.

Sound-driven cooling was also observed by Merkli and Thomann [88] when theyperformed experiments on cooling in a simple gas-filled piston-driven resonator. To ex-plain these effects an extended acoustic theory was developed which predicted coolingin the tube where the velocity amplitude was at its maximum, in good agreement withthe experiments.

A crucial advance in experimental thermoacoustics came in 1962 when Carter et al.[22] realized that the performance of the Sondhauss tube could be improved by suitableplacement of a porous medium inside, in the form of a stack of parallel plates. Thepresence of a “stack”, with heat transfer from one end to the other, makes it much easierto produce a significant temperature difference and will be the essential ingredient forthe kind of thermoacoustic devices considered in this thesis.

The foundation for theoretical thermoacoustics was laid in 1868 by Kirchhoff [66],

Introduction 5

who investigated the acoustic attenuation in a duct due to oscillatory heat transfer be-tween the isothermal tube wall and the gas inside the tube. His results were generalizedby Kramers [67] for a tube supporting a temperature gradient. The breakthrough camein 1969 when Rott et al. , inspired by the Taconis oscillations, started an impressive se-ries of articles [91, 115–118, 120, 121], in which a linear theory of thermoacoustics wasderived. Rott abandoned the boundary-layer approximation as used by Kirchhoff andKramers, and formulated the mathematical framework for small-amplitude dampedand excited oscillations in wide and narrow tubes with an axial temperature gradient,only assuming that the tube radius was much smaller than the length of the tube. In1980 Rott summarized his results in a review work [119], which became the cornerstonefor the subsequent intensified interest in thermoacoustics.

In the eighties a very intensive and successful research program was started at theLos Alamos National Laboratory by Wheatley, Swift, and coworkers [133, 151, 152]. Us-ing Rott’s theory of thermoacoustic phenomena they started to design and build practi-cal thermoacoustic devices. Important was Hofler’s invention of a standing-wave ther-moacoustic refrigerator [55, 56], which proved that Rott’s theoretical analysis was cor-rect. Hofler’s refrigerator, shown in figure 1.4, used a loudspeaker to drive a closedresonator tube with a stack of plates positioned near the speaker. At the other end ofthe tube a resonator sphere was attached to simulate an open ending, so that effectivelyone can speak of a quarter-wave-length resonator. Inside the refrigerator a standing-wave is maintained by the speaker, generating a temperature difference across the stacksuch that heat is absorbed at the low temperature or waste heat is released at the hightemperature.

Hot heat exchanger

Cold heat exchanger

Driver Stack

Resonatorsphere

Figure 1.4: Hofler’s standing-wave refrigerator. The hot end of the stack is thermally anchoredat room temperature and the standing wave generates cooling at the cold end of the stack.

A whole new branch of thermoacoustic devices started in 1979 with Ceperley’s real-ization [23, 24] that thermoacoustic devices based on the Stirling cycle [21] with idealheat transfer, could reach much higher efficiencies than devices based on standing-wave modes of operation. His idea was to design machines that allow a traveling waveto pass through a dense porous medium (the regenerator) using a toroidal geometry.Yazaki et al. [155] managed to build a traveling-wave prime mover based on these prin-ciples, but at very low efficiency due to large viscous losses. Finally, Backhaus andSwift [14] managed to overcome these problems by designing a traveling-wave primemover (shown in figure 1.5) that combines the toroidal geometry with a resonator tubeto reduce the velocities in the loop.

Swift was the first to give a comprehensive analysis of thermoacoustic devices in his

6 1.2 The basic mechanism of thermoacoustics

Resonator

Load

Jet pump

Regenerator

Figure 1.5: Schematic drawing of the traveling-wave prime mover of Backhaus and Swift. Thesound produced by the regenerator is absorbed by an acoustic load that is attached to the regen-erator.

review article [131] based on Rott’s work. He also gives a detailed description of thethermodynamics involved, a complete historical overview, experimental results, and hetreats several types of devices. Since then Swift and others have contributed much to thefurther development and analysis of thermoacoustic devices. Most of the literature hasbeen collected and summarized in Garrett’s review work [45]; in particular we mentionSwift’s textbook [135], which provides a clear introduction into thermoacoustics. Lastly,we note that several articles have been written as well [15, 47, 133, 150, 153], aimed atreaders new to the field of thermoacoustics, while various educational animations canbe found at the website of Los Alamos National Laboratory [130].

1.2 The basic mechanism of thermoacoustics

The thermoacoustic principles can be understood best by following a given parcel offluid as it oscillates near a solid boundary. We start by considering fluid parcels oscillat-ing far away from the wall in a closed tube supporting a standing wave or in a infinitetube supporting a traveling wave, as depicted in figure 1.6. Note that under isentropicconditions the pressure oscillations are accompanied by temperature oscillations, whichis used in figures 1.6(c) and 1.6(d) to draw schematically the temperature-position cyclethat a fluid parcel undergoes during one oscillation.

Consider a fluid parcel oscillating in the closed tube, far away from the tube wall. Inthe center of the tube we have a pressure node and a velocity antinode. As a result theparcel will undergo large displacement without temperature variations. On the otherhand, near the ends of the tube we have a pressure antinode and a velocity node, and theparcel will almost stand still and undergo large temperature variations, giving a verysteep temperature gradient. If the parcel is oscillating away from a velocity or pressurenode, then a finite nonzero temperature gradient will arise. Within thermoacoustics theslope of these adiabatic temperature variations is known as the “critical temperaturegradient”.

In the infinite tube the situation is somewhat different, since the pressure and ve-locity oscillations are in phase for all positions in the tube. As a result the parcels al-ternately move to the right with a high temperature and then to the left with a lowertemperature, giving circle-like temperature-position cycles. It follows that when a smalltraveling-wave component is added to a standing wave, or vice versa, the thermody-

Introduction 7

(a) (b)

T

x

(c)

T

x

(d)

T

x

(e)

Figure 1.6: Gas parcels oscillating (a) with standing-wave phasing in a closed tube and (b) withtraveling-wave phasing in an infinite tube. The pressure oscillations are indicated by dashedlines, the velocity oscillations by solid lines, and the arrows give the direction of oscillation. (c)The temperature of the gas parcel during one oscillation as a function of its position relative to thestanding wave. (d) Temperature-position diagram for traveling wave. (e) Temperature-positiondiagram when a small traveling-wave component is added to the standing wave.

namic cycles will turn into tilted ellipses as shown in figure 1.6(e).Suppose we look at the same parcels, but now oscillating near a wall supporting a

temperature gradient in axial direction. We consider two cases:

(a) The tube wall supports a much smaller temperature gradient than the critical tem-perature gradient of the fluid parcel. As a result heat will be transported from thegas parcel into the wall at the hot end and from the wall into the gas parcel at thecold end (figure 1.7(a)). The transport of heat from the low temperature to the hottemperature will require the input of acoustic work. This is the condition for arefrigerator or heat pump.

(b) The tube wall supports a much larger temperature gradient than the critical tem-perature gradient of the fluid parcel. As a result heat will be transported from thewall into the gas parcel at the hot end and from the gas parcel into the wall at thecold end (figure 1.7(b)). The transport of heat from the high temperature to thelow temperature will produce acoustic work as output. This is the condition for aprime mover.

This is in a nutshell the basic mechanism of thermoacoustics. In the next chapter wewill analyze the combined effect of oscillations and heat transfer in more detail using“Brayton” [21] and “Stirling” cycles [145]. Specifically we will show how the distance to

8 1.3 Classification

the solid and the phasing of pressure and velocity affect the heat transfer between gasand solid.

T

position parcel

δQ

δQ

parcel

wall

(a) Refrigerator

T

position parcel

δQ

δQ

parcel

wall

(b) Prime mover

Figure 1.7: The temperature-position diagram for the adiabatic parcel-temperature oscillations(—) and the wall temperature (- -). In (a) we consider a refrigerator and apply a small tempera-ture difference and in (b) we consider a prime mover with large temperature difference across thewall. The arrows show the transport of heat δQ from the wall to the gas parcel and vice versa.

1.3 Classification

We consider thermoacoustic devices of the type shown in figure 1.8, that is, a possiblylooped duct containing a fluid (usually a gas) and a porous solid medium, if necessarywith neighboring heat exchangers. In addition loudspeakers or other sources of soundmay be attached to the ends of the tubes. The porous medium is modeled as a collectionof narrow arbitrarily shaped pores aligned in the direction of sound propagation. Typ-ical examples are parallel plates and circular or rectangular pores. As will be discussedbelow, thermoacoustic devices can be divided [45] into several categories: heat-drivenversus sound-driven devices, standing-wave versus traveling-wave devices, or stack-based versus regenerator-based devices.

Prime mover vs. refrigerator

We distinguish heat-driven and sound-driven devices. A thermoacoustic prime moverabsorbs heat at a high temperature and exhausts heat at a lower temperature whileproducing acoustic work as an output. A refrigerator or heat pump absorbs heat at a lowtemperature and requires the input of acoustic work to exhaust more heat to a highertemperature. The only difference between a heat pump and a refrigerator is whetherthe purpose of the device is to extract heat at the lower temperature (refrigeration) or toreject heat at the higher temperature (heating). Therefore, in this thesis we will use theterm refrigerator loosely to refer to either of these devices.

Introduction 9

(a) straight duct (b) looped duct

Figure 1.8: Schematic view of two possible duct configurations: straight or looped.

In the literature the term thermoacoustic engine is used as well, either to indicate athermoacoustic prime mover or as a general term to describe all thermoacoustic devices.To avoid confusion we will refrain from using this term.

Stack-based devices vs. regenerator-based devices

A second classification depends on whether the porous medium used to exchange heatwith the working fluid is a “stack” or a “regenerator”. Typically inside a regeneratorthe pore size is much smaller than inside a stack. Garrett [45] uses the so-called Lautrecnumber NL to indicate the difference between a stack and regenerator. The Lautrecnumber is defined as the ratio between the half pore size and the thermal penetration

depth1. Gas parcels that are separated from the wall by a distance much larger than thethermal penetration depth, will have no thermal contact with the wall. If NL & 1 theporous medium is called a stack and the gas parcels in the stack will have imperfectthermal contact with the solid. If NL ≪ 1, then the porous medium is called a regen-erator, and the gas parcels inside the regenerator will have perfect thermal contact withthe solid. Whenever the pore size is unknown or irrelevant, the porous medium will bereferred to as stack.

Standing-wave devices vs. traveling-wave devices

Lastly, thermoacoustic devices can also be categorized depending on the phase shiftbetween the pressure and velocity oscillations at the location of the stack. In a closedand empty resonator a pure standing-wave can be maintained and the pressure andvelocity oscillations will be exactly 90 degrees out of phase. In an empty infinite tube(or a loop) a pure traveling-wave can be maintained, so that the pressure and velocityoscillations are exactly in phase. As soon as we insert a stack in either of these tubes,the phasing between pressure and velocity will change because of partial reflection atthe stack interfaces. Moreover, if we consider a looped tube of the type depicted in

1The thermal penetration depth is the distance heat can diffuse through within a characteristic time

10 1.4 Applications

figure 1.8(b), with a resonator tube attached to it, the phasing will be affected evenmore. However, in practise both the straight geometry depicted in figure 1.8(a) and thelooped geometry depicted in figure 1.8(b) can be chosen such that the sound field at thestack is predominantly a standing wave or a traveling wave.

In Section 2.3 we will show that it is beneficial to use a stack inside a standing-wavedevice and a regenerator inside a traveling-wave device. For this reason thermoacousticdevices are usually classified as either standing-wave stack-based devices or traveling-wave regenerator-based devices.

1.4 Applications

Over the years thermoacoustic devices have found applications in areas like food in-dustry, defense industry, spacecraft, telecommunication, electronics, energy sectors, andconsumer products. Some of these devices are heat-driven, some are sound-driven, andothers combine the two effects. In this section we will treat a few examples that havebeen or will be used commercially. Most of these applications are motivated by thequest for reliable, cheap, or environmentally friendly sources of energy.

Down-well power generation

The natural-gas industry use sensors to measure properties of the gas that streamsthrough subterranean gas pipes. These sensors are located far below ground and needelectrical power, which has to be delivered via batteries or long cables. The reliabilityof such equipment is usually quite poor, requiring costly repairs or replacements due todifficult accessibility and extreme operating conditions. Since most wells are used formany years, there is a necessity for a cheap and reliable alternative for power genera-tion, which thermoacoustics can provide.

stack

main flow

Figure 1.9: Schematic drawing of the side-branch system. A standing wave is generated in theside branches due to the interaction of the main flow, supplied by the pipeline, with the edges ofthe side branch. If a stack is suitably positioned in the standing wave, a temperature differencecan be generated.

The answer lies in a technique suggested by Slaton and Zeegers [123–125], whichavoids the use of moving parts and uses part of the main-flow energy in the gas pipes

Introduction 11

to generate aero-acoustic sound. In their experimental set-up, shown in figure 1.9, theyattach two side branches to the main pipeline. Due to the interaction of the main flowwith the edges of the side branch, vortices will be created. By adjusting the lengthsof the side branch to the flow speed, it is possible to match the frequency of the vor-tex shedding with the fundamental standing-wave mode of the side branch, and astanding-wave sound field is created in the side branch. Finally, by suitable placementof a stack a temperature difference can be generated, which can be used to produceelectrical power with thermoelectric elements. The flow patterns and vortex sheddingin such side-branch systems have been visualized experimentally and numerically byKriesels et al. [68] and Dequand et al. [36].

Upgrading of industrial waste heat

Another important potential application for thermoacoustics is the upgrading of indus-trial waste heat. Huge amounts of heat produced by industry remain unused becauseof small power outputs or temperatures that are too small. The Energy research Centreof the Netherlands (ECN) has developed a thermoacoustic system that uses part of thewaste heat to power a prime mover that drives a heat pump to upgrade the temperatureof the remainder. The apparatus is shown in figure 1.10.

Prime MoverHeat Pump

Resonator

Figure 1.10: Upgrading of industrial waste heat using a combination of a prime mover and heatpump. Part of the waste heat is used to power the prime mover, which generates acoustic powerto drive the heat pump and heat the remaining part of the waste heat.

Thermoacoustic cryocooling

Thermoacoustics is the only technology that can cool to temperatures close to the abso-lute zero without using moving parts and is therefore very interesting for applicationsrequiring crycooling. One such application is the liquefaction of natural gases whichrequires very low temperatures. At the Los Alamos National Laboratory (LANL) aheat-driven thermoacoustic refrigeration system [134, 138] has been designed capableof liquefying natural gases. Their system, depicted in figure 1.11, uses a toroidal geom-etry attached to a long resonator tube, with a prime mover located in the toroidal partand a refrigerator located near the end of the resonator. Part of the natural gas is burnedto supply heat which is converted into acoustic power by the prime mover. The acousticpower is then provided to the refrigerator and subsequently used to cool the remainderof the natural gas until it is liquefied.

Thermoacoustic cyrocooling is also applied to the cooling of electronics. The Amer-ican Navy used a Shipboard Electronics ThermoAcoustic Chiller (SETAC) to cool elec-tronics on board of one of their destroyers [86]. A similar application found its way into

12 1.5 Thesis overview

PrimeMover

Refrigerator

Figure 1.11: Schematic drawing of a heat-driven thermoacoustic refrigeration system. By burn-ing part of the natural gases at the prime mover, the remainder can be cooled to liquefaction atthe refrigerator.

spacecraft, when in 1992 the Space ThermoAcoustic Refrigerator (STAR) was launchedon the Space Shuttle Discovery. It was the answer [46] to the need for reliable, com-pact, and long-lived spacecraft cryocooling for the cooling of sensors aboard the shut-tle. Other applications of thermoacoustics within spacecraft concern the developmentof thermoacoustic systems suitable for electricity generation on space missions [16,140].

Food refrigeration and airconditioning

Thermoacoustics refrigerators can also be used to replace conventional food refriger-ators or airconditioning, without the harmful polluting side-effects. One well-knownexample is the collaboration [101] between Pennsylvania State University and Ben andJerry’s Homemade Ice Cream, that led to the development of a thermoacoustic in-storeice-cream cabinet, capable of keeping ice cream at -18 ◦C. The aim was to find a cost-competitive and reliable alternative to the use of polluting refrigerants and thereby re-duce the emission of global-warming gases.

Cooking stove

Recently the SCORE (Stove for Cooking, Refrigeration and Electricity) project has started,an international research program led by the University of Nottingham [112]. The aimis to develop a biomass-powered thermoacoustic system, to be used as an affordable,safe, and efficient alternative for the energy needs of third-world countries. Currentsources of energy, like open fires, are highly inefficient and can produce serious healthhazards. The SCORE stove will serve as a versatile domestic appliance, being a cooker,a refrigerator, and an electricity regenerator all in one.

1.5 Thesis overview

In this thesis we will derive a weakly nonlinear theory of thermoacoustics, applicableto wide and narrow pores with arbitrarily shaped cross-sections that may vary slowlyin axial direction. We will use dimensional analysis and small-parameter asymptoticsso that (quadratically) nonlinear terms can be systematically included. The use of adimensionless framework has two main advantages:

Introduction 13

• Nondimensionalization allows us to analyze limiting situations in which param-eters differ in orders of magnitude, so that we can study the system as a functionof dimensionless parameters connected to geometry, heat transport, and viscouseffects.

• We can give explicit conditions under which the theory is valid. Furthermore, wecan clarify under which conditions additional assumptions or approximations arejustified.

In the end we will apply this theory to model, analyze, and simulate both standing-wave and traveling-wave devices and we will show how this approach can be used asan aid for optimizing their design.

The main ingredients of this thesis are therefore the choice of geometry for the stackpores, the use of dimensionless parameters, the inclusion of quadratic nonlinearities,and the modeling and numerical simulation of complete devices. In Section 1.6 we willdiscuss briefly the relevant literature on each of these topics, but first we will give achapter-by-chapter overview of the contents of this thesis.

In Chapter 2 the thermodynamics of thermoacoustics will be discussed and a mea-sure for the thermodynamic performance is introduced. Then in Chapter 3 we will givethe governing equations and we will set the mathematical framework for our small-parameter asymptotics. Chapter 4 is concerned with the systematic development of thetheory of thermoacoustics in slowly-varying two-dimensional pores. We will show howthe leading-order, first-order, and second-order terms with respect to the acoustic Machnumber can be derived, which includes the derivation of the second harmonics andthe streaming terms. We will also show how under certain simplifying assumptionsanalytic expressions can be obtained. In Chapter 5 the results from Chapter 4 will begeneralized to arbitrary three-dimensional slowly-varying pores. Next in Chapter 6 thethermoacoustic equations are implemented and validated experimentally for severaltypes of standing-wave devices: a thermoacoustic couple, a thermoacoustic refrigera-tor, and a thermoacoustic prime mover. We will show how the performance is affectedby the operating conditions and parameters connected to geometry, stack material, andworking gas. Finally we will also show what kind of streaming velocity profiles canoccur. In Chapter 7 a specific type of traveling-wave prime mover is modeled and im-plemented numerically. An optimization routine is developed that for given systemparameters computes the ideal geometry. Moreover, we we will show how the perfor-mance is affected by variation of various parameters. Then in Chapter 8 an evolutionequation will be derived that predicts the development of shock waves near resonance,both in a closed tube and in interaction with a thermoacoustic stack. Lastly, in Chapter9 we give some conclusions, discuss our results, and give some suggestions for futurework.

1.6 Literature review

While Rott’s theory of thermoacoustics [91,115–121] was still limited to two-dimensionalpores or three-dimensional cylindrical pores, it was Arnott et al. [5] who first extendedRott’s theory to arbitrary three-dimensional cross-sections, although nonlinear termswere not yet included. We have extended their results by allowing a slow variation in

14 1.6 Literature review

the pore cross-section in axial direction. Furthermore, we also allow temperature de-pendence of speed of sound, specific heat, viscosity, and thermal conductivity, and wehave eliminated the restriction of steady pore-wall temperature. Additionally we havealso incorporated quadratic nonlinearities such as the second harmonics and streamingterms into the analysis.

There are numerous articles that have analyzed more specific choices for the stackpore geometry, shown in figure 1.12. The parallel-plate geometry has been investigatedmost extensively, in particular by Swift [131] and Rott [115], who also investigated cylin-drical cross-sections. Stinson [128] and Roh et al. [113] derived simultaneously expres-sions for rectangular cross-sections and Stinson and Champoux [129] used results ofHan [53] to solve the equations for triangular cross-sections. Lastly, we mention Swiftand Keolian [136] who calculated the thermoacoustic behavior for so-called pin-arraystacks, consisting of a hexagonal array of pins aligned in axial direction.

(a) Parallel plates (b) Circular (c) Rectangular (d) Triangular (e) Pin array

Figure 1.12: Possible stack pore cross-sections.

Previous treatments with variable cross-sections have always been restricted to widelyspaced pores [98, 117], whereas in this thesis no such assumption is made. Althoughvariable cross-sections occur mostly within the main resonator tube, they may also occurin the narrow stack pores. A more general formulation would allow stack geometriesother than collections of arbitrarily shaped pores. We mention Roh et al. [114], who in-troduce tortuosity and viscous and thermal dynamic shape factors to extend single-porethermoacoustics to bulk porous medium thermoacoustics. Furthermore, there exists avast amount of papers on flow through porous media with random or stochastic prop-erties, that could also be applied to thermoacoustic configurations. Auriault [13] givesa clear overview of various techniques that can be used:

• Statistical modeling [69];

• Self-consistent models [157];

• Volume-averaging techniques [103];

• Method of homogenization for periodic structures [12].

In addition Auriault [13] gives a short explanation of how the method of homogeniza-tion can be applied to analyze heat and mass transfer in composite materials. A detaileddiscussion of methods and results from the theory of homogenization and their appli-cations to flow and transport in porous media can be found in [58]. Another approachis demonstrated by Kaminski [63], who combines the homogenization approach witha stochastic description of the physical parameters to analyze viscous incompressibleflow with heat transfer.

Introduction 15

In our analysis, we assume a steady-state situation in which the variables oscillateat (integer multiples of) the fundamental frequency (cf. [5, 119, 131]), which is not al-ways the case. Prime movers require a sufficiently high temperature difference before itgoes into self-oscillation. Atchley et al. have given a standing-wave analysis of this phe-nomenon and measured the complete evolution of the quality factor of a prime moverfrom below, through, and above onset of self-oscillation [6–9].

The use of a dimensionless framework is not very wide-spread. Olson and Swift [97]use dimensionless parameters to analyze thermoacoustic devices, but without trying toconstruct a complete theory of thermoacoustics. Instead dimensionless numbers areused to reduce the number of independent parameters in their experiments and forscaling purposes.

There have been many observations [14, 100, 132] demonstrating that at high am-plitudes measurements deviate significantly from predictions by linear theory. Stream-ing, turbulence, transition effects, higher harmonics, and shock waves are mentioned asmain causes for these deviations. Streaming [75,95] refers to a steady mass-flux densityor velocity, usually of second order, that is superimposed on the larger first-order oscil-lations. With the addition of a steady non-zero mean velocity, the gas moves throughthe tube in a repetitive ”102 steps forward, 98 steps backward” manner as describedby Swift [135]. Streaming is important as a nonzero mass flux can seriously affect theperformance of thermoacoustic devices. It can cause convective heat transfer, whichcan be a loss, but it can also be essential to transfer heat to and from the environment.Backhaus and Swift [14], in their analysis of a traveling-wave heat prime mover, showhow streaming can cause significant degradation of the efficiency.

The concept of mass streaming has been studied by many authors, (see e.g. [17, 50,52,98,117,148]), but restricted to simple geometries such as straight two-dimensional orcylindrical pores, although Olson and Swift [98] do allow slowly varying cross-sectionsin the tube. Moreover, they show that variable cross-sections can occur in practicalgeometries and can be used to suppress streaming; a suitable asymmetry in the tubecan cause counter-streaming that balances the existing streaming in the tube. Baillietet al. [17], Rott [117], and Olson and Swift [98] also take into account the temperaturedependence of viscosity and thermal conductivity, although the latter two only considerwidely spaced pores.

Higher harmonics oscillate at integer multiples of the fundamental frequency, andcan become quite important at high amplitudes. Atchley and Bass [10], noticed exper-imentally that the generation of higher harmonics can cause highly nonlinear wave-forms that degrade the performance significantly. The impact of the harmonics is great-est when excited near resonance, but Atchley and Gaitan [43] analyzed that this canbe suppressed by careful tuning of the resonator. There is no literature that systemat-ically includes the higher harmonics into the analysis, which Swift [135] describes as“a formidable challenge” and Rott [118] as “a hopeless undertaking”. Although a com-plete extension would require going up to fourth order in the asymptotic expansion,we have shown here that the second-harmonic pressure and velocity oscillations can beexpressed in terms of the first-harmonic oscillations.

It has been noticed that higher harmonics can interact together to form shock waves[59, 60]. Neglecting the nonlinear sound field in the stack, Gusev [49] analyzed shockformations using a nonlinear evolution equation. Modeling the stack by a reflectioncoefficient, we will derive a new evolution equation that predicts the development of

16 1.6 Literature review

shock waves when the tube is excited near resonance.Turbulence arises at high Reynolds numbers, where the assumption of laminar flow

is no longer valid. It disrupts boundary layers and can negatively affect the heat trans-fer. Turbulence may also arise due to abrupt changes in the shape or direction of thechannels, which leads to the shedding of vortices [2, 89, 158] and can cause significantlosses. Swift [44, 135] gave some suggestions on how to include turbulence into themodeling, but they are by no means complete. It is therefore still a challenge to includesystematically the effects of turbulence into the analysis.

Finally we note that there is a long list of publications on the numerical simulation ofheat-driven and sound-driven thermoacoustic devices, focusing on both standing-waveand traveling-wave modes of operation. This list is too long to go into here, but it shouldbe mentioned that a large part of the thermoacoustic community uses the DeltaEC (orDeltaE) code [146, 147] which was developed at the Los Alamos National Laboratorybased on Swift’s linear theory of thermoacoustics [135].

Chapter 2

Thermodynamics

In this chapter we will explain the basic thermodynamics concepts that are at the basisof thermoacoustics. We will start by giving the fundamental laws of thermodynam-ics and show how they can be used to derive a thermodynamic efficiency and coefficientof performance that indicate how well a thermoacoustic prime mover or refrigerator per-form. Next we will shed some more light on the mechanisms behind the thermoacousticproduction of sound or heat by analyzing the thermodynamic cycle a gas parcel experi-ences.

2.1 Laws of thermodynamics

Thermodynamically speaking a thermoacoustic system is completely characterized bythe flows of heat and work, as shown in figure 2.1. Let TH be the temperature of a hotreservoir and TC the temperature of a cold reservoir. In a refrigerator acoustic work W isused to generate a heat flow against the temperature gradient, removing heat QC at thelow temperature and releasing heat QH at the high temperature. In a prime mover workW is produced by transporting heat from the high to the low temperature, removingheat QH at the high temperature and releasing heat QC at the low temperature.

The energy flows within a thermoacoustic system are governed by the first and sec-ond law of thermodynamics. The first law concerns the conservation of energy anddescribes the rate of change of the internal energy U of a system with volume V, theheat and enthalpy flows into the system, and the work done by the system [35]:

U = ∑ Q + ∑ nHm − pV + P. (2.1)

Here n represents the molar flow rate entering the system, Hm represents molar en-thalpy, Q represents heat power, and P represents other forms of power done on thesystem. The summation is used to account for all the different sources of sound andmass that are in contact with the system and plus and minus signs are used to indicateflows into or out of the system, respectively.

The second law of thermodynamics states that any process that occurs will tend toincrease the total entropy of the system. Mathematically this can be expressed by the

18 2.1 Laws of thermodynamics

Refrigerator

TH

TC

QH

W

QC

(a) Refrigerator or heat pump

Prime mover

TH

TC

QH

W

QC

(b) Prime mover

Figure 2.1: The flows of work W and heat Q inside (a) a thermoacoustic refrigerator or heatpump and (b) a thermoacoustic prime mover. The arrows are used to indicate the exchange ofheat and work between the thermoacoustic system and the environment.

inequalitySi ≥ 0, (2.2)

where Si represents the entropy production in the system. The second law can also beformulated using an equality [35],

S = ∑ Q

T+ ∑ nSm + Si. (2.3)

It states that the rate of change of entropy S of a thermodynamic system is equal tothe sum of the entropy change due to heat flows Q with temperature T, due to massflows nSm, and due to the irreversible entropy production in the system Si. Again thesummation signs are used to allow for a multitude of sound or mass sources.

In the thermoacoustic systems considered there is no mass flow into or out of thesystem, the volume of the system is constant, and the only work performed on thesystem is the acoustic work W. As a result, we find that the laws of thermodynamicsreduce to

U =

{QC − QH + W, for a refrigerator or heat pump,

−QC + QH − W, for a prime mover,(2.4)

and

s =

QC

TC

− QH

TH

+ Si, for a refrigerator or heat pump,

− QC

TC

+QH

TH

+ Si, for a prime mover.

(2.5)

In our analysis we consider a (time-averaged) steady-state situation, so that we can putU = 0 and S = 0. It follows that

QC − QH + W = 0, (2.6)

Thermodynamics 19

for all thermoacoustic devices. In addition, since the entropy generated by the systemSi ≥ 0, we find

QC

TC

≤ QH

TH

, for a refrigerator or heat pump, (2.7)

QC

TC

≥ QH

TH

, for a prime mover, (2.8)

where equality can only be reached in the ideal situation when there are no irreversibleprocesses.

2.2 Thermodynamic performance

2.2.1 Refrigerator or heat pump

The performance of a refrigerator or heat pump is measured by the so-called coefficientof performance COP. Since a refrigerator and a heat pump each have a different goal,cooling versus heating, the coefficients of performance are defined differently also.

When analyzing a refrigerator we are interested in maximizing the cooling powerQC extracted at temperature TC, while at the same time minimizing the net requiredacoustic power W. On the other hand, a heat pump aims at maximizing the heatingpower QH at temperature TH, while minimizing W. Therefore, the coefficients of per-formance COPre f (refrigerator) and COPhp (heat pump) are defined as the ratios of thesequantities,

COPre f :=QC

W, (2.9)

COPhp :=QH

W. (2.10)

The second law of thermodynamics limits the interchange of heat and work. Inparticular it follows from (2.6) and (2.7) that

COPre f =QC

QH − QC

≤ TC

TH − TC

=: COPCre f , (2.11)

COPhp =QH

QH − QC

≤ TH

TH − TC

=: COPChp. (2.12)

The quantity COPC is called the Carnot coefficient of performance , and gives the max-imal performance for all refrigerators or heat pumps. Using COPC we can introduce arelative coefficient of performance COPR as

COPR :=COP

COPC. (2.13)

Note that although COP and COPC may attain any nonnegative value, the relative co-efficient COPR will always be between 0 and 1.

20 2.3 The thermodynamic cycle

2.2.2 Prime mover

The performance of a prime mover is measured by the so-called efficiency η. A primemover uses heating power QH to produce as much acoustic power W as possible andthus the efficiency is defined as

η =:W

QH

. (2.14)

Using equation (2.6), we can write

η =QH − QC

QH

. (2.15)

Applying equation (2.8), we obtain

η ≤ TH − TC

TH

= 1 − TC

TH

=: ηC . (2.16)

By definition both the efficiency η and the Carnot efficiency ηC are between zero andone. The same holds for the relative efficiency ηR which is defined as

ηR :=η

ηC

. (2.17)

Of course the efficiency is not the only criterion for a good thermoacoustic primemover. For certain applications one might be primarily interested in maximizing thepower output with the efficiency only of minor interest. Other competing criteria canbe cost, size, reliability, available materials, safety, and the complexity of the design.Naturally, the same holds for a refrigerator or heat pump.

2.3 The thermodynamic cycle

In Section 1.2 we gave an intuitive explanation for the basic mechanism of thermoa-coustics. In this section we will give a more detailed analysis, describing all the steps afluid parcel undergoes while it oscillates in a narrow pore. In particular we will discussthe time phasing of the thermodynamic effect and its relation to the pore size, basedon the analysis of Swift [131] and Ceperley [23]. Swift showed that for standing-wavedevices the fluid-parcel movements are very accurately described by the Brayton cy-cle [21]. Ceperley suggested to design thermoacoustic devices such that the thermody-namic cycle would match the ideal Stirling cycle [145] to reduce irreversible effects toa minimum. Liang [73] analyzed Ceperley’s concepts further using a sinusoidal modelthat describes the thermodynamic cycles of gas parcels oscillating in a regenerator.

2.3.1 Standing-wave phasing

Suppose gas parcels oscillate with standing-wave phasing at a distance y from a solidplate supporting a temperature gradient. As shown in figure 2.2 the parcels will ex-perience a build-up of pressure (compression) and drop of pressure (expansion) while

Thermodynamics 21

A B C D At

︷︸︸︷compression

displacement

︷︸︸︷expansion

displacement

Figure 2.2: Velocity (—-) and pressure (– – –) as a function of time in a gas supporting astanding wave. The cycle consists of a compression/displacement step (A-B) and an expan-sion/displacement step (C-D). Depending on the distance between the gas and the solid heatingand cooling may occur as well.

undergoing displacement. Depending on the size of y relative the thermal penetrationdepth δk, heat transfer may occur as well. We consider three cases:

• No thermal contact (y ≫ δk)When there is no thermal contact between the gas and the plate the gas parcelswill expand and compress adiabatically and reversibly and no heat transfer willtake place.

• Perfect thermal contact (y ≪ δk)In the first step the gas parcels are compressed and displaced towards the hot endof the plate and the parcel. At the same time the parcels will be heated if a largetemperature gradient is imposed (prime mover) and cooled if a small temperaturegradient is imposed (refrigerator). Because there is perfect thermal contact thecompression and heat exchange will take place simultaneously. In the next stepthe gas parcels are displaced back towards the cold end and the reverse effect takesplace. The parcels will be cooled for a prime mover and heated for a refrigerator.

• Imperfect thermal contact (y ∼ δk)Because of the distance between the gas and the plate, there is a time delay be-tween motion and heat transfer. As a result the gas parcels execute a four-stepcycle. In case of a refrigerator the parcels are compressed and displaced (A-B),cooled (B-C), expanded and displaced back (C-D), and heated (D-A). In case of aprime mover the heating and cooling steps are interchanged.

The acoustic power produced or absorbed by a gas parcel can be found from thearea

∮p dV in the pressure-volume diagrams. In figure 2.3 we see that heat can only be

converted in acoustic power or vice versa if there is a time delay between the compres-sion or expansion and the heat exchange. This thermodynamic cycle is what is knownas the idealized Brayton cycle. This is the reason why in standing-wave based devices


volume

pre

ssu

re

y ≫ δk

volume

pre

ssu

re

y ≪ δk

volume

pre

ssu

re

y ∼ δk

A

B C

D

(a) prime mover

volume

pre

ssu

re

y ≫ δk

volume

pre

ssu

re

y ≪ δk

volume

pre

ssu

re

y ∼ δk

D

C B

A

(b) refrigerator

Figure 2.3: Schematic pressure-volume cycle executed by the gas parcel in (a) a standing-waveprime mover and (b) a standing-wave refrigerator. Only when the gas parcels are about a thermalpenetration depth away from the plate, pressure and volume variations are out of phase and network is performed.

stacks are used with stack pores that have radii of several penetration depths. The poorthermal contact is crucial for the thermoacoustic effect to occur, but it also gives rise toirreversible heat transfer and friction, which has a negative effect on the efficiency. Wewill show below that in traveling-wave devices we can have perfect thermal contact andthus potentially higher efficiencies.

2.3.2 Traveling-wave phasing

Suppose now the gas parcels oscillate with traveling-wave phasing at a distance y ≪ δk

from a solid plate supporting a temperature gradient. As shown in figure 2.4 the parcelswill undergo a cycle consisting of four steps:

A-B First the gas parcel is compressed at constant temperature.

B-C Then the parcel is displaced towards the hot end at constant volume and simulta-neously it is heated (prime mover) or cooled (refrigerator).

C-D Next the parcel is expanded at constant temperature.

Thermodynamics 23

A B C D At

︷︸︸︷compression

︷︸︸︷expansion

︸︷︷︸cooling/heating

displacement

︸︷︷︸heating/cooling

displacement

Figure 2.4: Velocity (—-) and pressure (– – –) as a function of time in a gas supporting a trav-eling wave in perfect contact with the solid. The gas undergoes a Stirling cycle of compression,cooling/heating, expansion, and heating/cooling.

D-A Finally, the parcel is displaced back towards the cold end at constant volume andsimultaneously it is cooled (prime mover) or heated (refrigerator).

Note that there is a continuous exchange of heat between the gas and the solid duringthe displacement steps.

The acoustic power produced or absorbed by the gas parcel can be found from thearea

∮p dV in the pressure-volume diagrams. In figure 2.5 we see that heat is con-

verted into acoustic power and vice versa, even though y ≪ δk (cf. figure 2.3). Thisthermodynamic cycle is what is known as the idealized Stirling cycle.

volume

pre

ssu

re

y ≪ δk

D

C

BA

(a) refrigerator

volume

pre

ssu

re

y ≪ δk

A

B

CD

(b) prime mover

Figure 2.5: Schematic pressure-volume cycle executed by the gas parcel in (a) a traveling-waverefrigerator and (b) a traveling-wave prime mover at perfect thermal contact with the wall. Onlywhen the gas parcels are about a thermal penetration depth away from the plate, pressure andvolume variations are out of phase and net work is performed.


This is the reason why in traveling-wave based devices regenerators can be usedthat have perfect thermal contact between gas and solid. As there is no irreversible heattransfer much higher efficiencies can be obtained. It should be mentioned that becauseof the very narrow pores, large viscous losses may occur, which is why traveling-wavedevices are usually designed such that in the regenerator the velocities are small.

In figure 2.6 we have summarized the (idealized) four-step cycles the fluid parcelsundergo as they oscillate along the walls of the stack pores of a standing-wave primemover or refrigerator and the regenerator pores of a traveling-wave prime mover orrefrigerator.

1 - Compression

2 - Cooling

δQ1

δW1

3 - Expansion

4 - Heating

δQ2

δW2

(a) Standing-wave re-frigerator

1 - Compression

2 - Heating

δQ1

δW1

3 - Expansion

4 - Cooling

δQ2

δW2

(b) Standing-wave txprime mover

1 - Compression

2 - Cooling

δQ1

δW1

3 - Expansion

4 - Heating

δQ2

δW2

(c) Traveling-wave re-frigerator

1 - Compression

2 - Heating

δQ1

δW1

3 - Expansion

4 - Cooling

δQ2

δW2

(d) Traveling-wave txprime mover

Figure 2.6: Typical fluid parcels, near a stack plate, executing the four steps (1-4) of the thermo-dynamic cycle in (a) a stack-based standing-wave refrigerator, (b) a stack-based standing-waveprime mover, (c) a regenerator-based traveling-wave refrigerator and (d) a regenerator-basedtraveling-wave prime mover.

Thermodynamics 25

2.3.3 Bucket-brigade effect

Usually the displacement of one fluid parcel is small with respect to the length of theplate. Thus there will be an entire train of adjacent fluid parcels, each confined to a shortregion of length 2δx and passing on heat as in a bucket brigade (figure 2.7). Althougha single parcel transports heat δQ over a very small interval, δQ is shuttled along theentire plate because there are many parcels in series.

QC QH

TC TH

δx

δQ

︸︷︷︸

W

Figure 2.7: Work and heat flows inside a thermoacoustic refrigerator. The oscillating fluidparcels work as a bucket brigade, shuttling heat along the stack plate from one parcel of gas tothe next. As a result heat is transported from the left to the right, requiring acoustic power W.Inside a prime mover the arrows will be reversed, i.e. heat is transported from the right to the leftand acoustic power W is produced.


Chapter 3

Modeling

3.1 Geometry

As mentioned in the introduction we consider devices of the form shown in figure3.1. The porous medium is modeled as a collection of narrow arbitrarily shaped poresaligned in the direction of sound propagation as shown in Fig. 3.2. Furthermore, weallow the pore boundary to vary slowly (with respect to the channel radius) in the di-rection of sound propagation, as shown in figure 3.2(b). For the analysis here we focuson what happens inside one narrow channel, which can represent a stack pore, but alsothe main resonator tube.

(a) straight duct (b) looped duct

Figure 3.1: Schematic view of two possible duct configurations: straight or looped.

In our analysis of the thermoacoustic effect we restrict ourselves to a pore and itsneighboring solid, where the system (x, r,θ) forms a cylindrical coordinate system. LetAg(x) denote the gas cross-section and As(x) the half-solid cross-section at position x.Let Γg denote the gas-solid interface and Γt the outer boundary of the half-solid. Wechoose Γt and As such that there is no heat flux through Γt (for periodic structures Γt isthe centerline of the solid). The remaining stack pores and solid can then be modeled

28 3.2 Governing equations

by periodicity. On Γg we write

Sg(x, r,θ) := r −Rg(x,θ) = 0,

with Rg(x,θ) the distance of Γg to the pore centerline at position x and angleθ. Similarlyon Γt we write

St(x, r,θ) := r −Rt(x,θ) = 0,

where Rt(x,θ) := Rg(x,θ) +Rs(x,θ) denotes the distance of Γt to the origin at positionx and angle θ.

(a) transverse cut of stack

solid

gasrx

θ

Rg(x,θ)

Rs(x,θ)

Γg

Γt

(b) longitudinal cut of one stack pore

Figure 3.2: Porous medium modelled as a collection of arbitrarily shaped pores.

3.2 Governing equations

The general equations describing the thermodynamic behavior in the gas are the well-known conservation laws of mass, momentum and energy [85]. Introducing the con-vective derivative

d f

dt=

∂ f

∂t+ v ·∇ f ,

these equations can be written in the following form:

mass :dρ

dt= −ρ∇ · v, (3.1)

momentum : ρdv

dt= −∇p +∇ · τ + ρb, (3.2)

energy: ρdǫ

dt= −∇ · q − p∇ · v + τ :∇v. (3.3)

Here ρ is the density, v the velocity vector, p the pressure, ρb the body forces (gravity),q the heat flux, ǫ the specific internal energy, and τ the viscous stress tensor. For q and

Modeling 29

τ we have the following relations

Fourier’s heat flux model: q = −K∇T, (3.4)

Newton’s viscous stress tensor: τ = 2µD +ζ(∇ · v)I , (3.5)

where T the absolute temperature, K is the thermal conductivity, D = 12 (∇v + (∇v)T)

the strain rate tensor, and µ and ζ the dynamic (shear) and second viscosity.In general the viscosity and thermal conductivity will depend on temperature. For

example, Sutherland’s formula [74] can be used to model the dynamic viscosity as afunction of the temperature

µ = µre f 1 + Su/Tre f

1 + Su/T

(T

Tre f

)1/2

,

where µre f is the value of the dynamic viscosity at reference temperature Tre f and Su

is Sutherland’s constant. According to [26] the variation of thermal conductivity is ap-proximately the same as the variation of the viscosity coefficient. The thermodynamicparameters Cp, Cv, β, and c may also depend on temperature (see Appendix A). Thetemperature dependence of µ and K is particularly important, as it allows investigationof Rayleigh streaming: forced convection as a result of viscous and thermal boundary-layer phenomena.

As shown in Appendix B.2 (eq. (B.9)), the equations above can be combined and

rewritten to yield the following equation1 for the temperature in the fluid

ρCp

(∂T

∂t+ v · ∇T

)= βT

(∂p

∂t+ v ·∇p

)+∇ · (K∇T) + τ :∇v, (3.6)

where Cp is the isobaric specific heat and β the volumetric expansion coefficient as de-fined in Appendix A.

In the solid we only need an equation for the temperature. We have that the temper-ature Ts in the solid satisfies the diffusion equation

ρsCs

∂Ts

∂t= ∇ · (Ks∇Ts) , (3.7)

where Ks, Cs and ρs are the thermal conductivity, the isobaric specific heat and the den-sity of the solid, respectively.

Another useful equation, that can replace (3.3), describes the conservation of energyin the following way:

∂∂t

(1

2ρ|v|2 + ρǫ

)= −∇ ·

[v

(1

2ρ|v|2 + ρh

)− K∇T − v · τ

]+ ρv · b, (3.8)

where h is the specific enthalpy. The details of the derivation leading to this equationcan be found in Appendix B.1.

The equations (3.1), (3.2), (3.6), and (3.7) can be used to determine v, p, T, and Ts.

1The double-dot product results in a scalar: A:B = ∑i, j Ai jB ji.


Additionally one should add an equation of state that relates the density to the pressureand temperature in the gas. For the analysis here it is enough to express the thermody-namic variables ρ, s, ǫ and h in terms of p and T using the thermodynamic equations(A.7)-(A.10) given in Appendix A. For the numerical examples given in Chapters 6 and7 it is necessary to make a choice and we choose to impose the ideal gas law

p = ρRspecT,

where Rspec = Cp −Cv is the specific gas constant. However, the analysis presented hereis also valid for non-ideal gases and nowhere will we use this relation.

To distinguish between variations along and perpendicular to sound propagationwe will use a τ in the index to denote the transverse vector components. For example

the transverse gradient ∇τ and transverse Laplace operator ∇2τ are defined by

∇ = ex

∂∂x

+ ∇τ , ∇2 =∂2

∂x2+ ∇2

τ .

The equations will be linearized and simplified using the following assumptions:

• the temperature variations along the stack are much smaller than the typical ab-solute temperature;

• time-dependent variables oscillate with fundamental frequencyω.

At the gas-solid interface we impose the no-slip condition

v = 0 if Sg = 0. (3.9)

The temperatures in the solid and in the gas are coupled at the gas-solid interface Γg bycontinuity of temperature and heat fluxes:

T = Ts if Sg = 0, (3.10a)

K∇T · n = Ks∇Ts · n if Sg = 0, (3.10b)

where n is a vector outward normal to the surface. The boundary Γt, with outwardsurface normal n′, is chosen such that no heat goes through, i.e.

∇Ts · n′ = 0 if St = 0. (3.11)

Note that since ∇Sg and ∇St are vectors normal to the surfaces we can also replace thetwo flux conditions by

K∇T · ∇Sg = Ks∇Ts · ∇Sg if Sg = 0, (3.12)

∇Ts · ∇St = 0 if St = 0. (3.13)

Modeling 31

3.3 Scaling

To make the equations above dimensionless we apply a straightforward scaling proce-dure. First we scale the spatial coordinate by a typical pore radius Rg:

x = Rgx.

Note that because we consider narrow stack pores, the typical length scale L of radiusvariation is much larger than a radius. We therefore introduce the small parameter ε asthe ratio Rg/L between a typical radius and this length scale. At Γg we write

Sg(X, r,θ) = r − Rg(X,θ) = 0, Rg(X,θ) = RgRg(x,θ), X = εx =x

L.

Similarly at Γt we have

St(X, r,θ) = r − Rt(X,θ) = 0, Rt(X,θ) = RgRt(x,θ).

Our formal assumption of slow variation has now been made explicit in the slow vari-

able X. As a typical time scale we use L/cre f , and we scale

t =L

cre ft.

Secondly, we rescale the remaining variables as well using characteristic values

u = cre f u, vτ = εcre f vτ , p = ρre fg (cre f )2 p, τ =

µre f cre f

Rg

τ , (3.14a)

ρ = ρre fg ρ, T =

(cre f )2

Cre fp

T, ρs = ρre fs ρs , Ts =

(cre f )2

Cre fp

Ts, (3.14b)

h = (cre f )2h, ǫ = (cre f )2ǫ, s = Cre f

p s, b = gb, (3.14c)

β =Cre f

p

(cre f )2β, Cp = Cre f

p Cp, Cs = Cre fs Cs, c = cre f c, (3.14d)

K = Kre f K, µ = µre fµ, ζ = ζ

re fζ . (3.14e)

Thirdly, we observe that the system contains 19 parameters, in which 6 physical dimen-sions are involved. The Buckingham π theorem [20] implies that the 19 parameters canbe combined into 13 independent dimensionless parameters. In Table 3.1 a possiblechoice is presented. Here the parameters δν, δζ , δk and δs are the viscous penetrationdepths based on dynamic and second viscosity and the thermal penetration depths forthe fluid and solid, respectively.

δν =

√√√√ 2µre f

ωρre fg

, δζ =

√√√√ 2ζre f

ωρre fg

, δk =

√√√√ 2Kre fg

ωρre fg Cre f

p

, δs =

√2Kre f

s

ωρre fs Cre f

s

.

32 3.3 Scaling

Symbol Formula Description

γ Cre fp /Cre f

v specific heats ratio

η (cre f Rg/UL)2ε

2/M2a

ε Rg/L stack pore aspect ratio

φ Rs/Rg porosity

σ Kre fs /Kre f

g ratio thermal conductivities

κ ωL/cre f Helmholtz number

Br Rg/Rt blockage ratio

Dr pA/pamb drive ratio

Ma U/c Mach number

Fr U2/(gL) Froude number

NL Rg/δk Lautrec number fluid

Ns Rs/δs Lautrec number solid

Pr δ2ν/δ

2k Prandtl number

Sk ωδk/U Strouhal number based on δk

Wo

√2Rg/δν Womersley number based on µ

Wζ

√2Rg/δζ Womersley number based on ζ

Table 3.1: Dimensionless parameters

Note that 16 dimensionless numbers are introduced. Since at most 13 can be indepen-dent, we have at least three dependent numbers. For example, Pr, Sk, and η can beexpressed in the other parameters as follows

Pr =2N2

L

W2o

, η =ε

2

M2a

, Sk =κε

NLMa

. (3.15)

The remaining dimensionless numbers are independent and can be chosen arbitrarily.In the analysis below we will use explicitly that Ma ≪ 1 (small velocity amplitudes)

and ε ≪ 1 (long pores). Other important parameters in thermoacoustics are κ, Sk andNL. κ is a Helmholtz number and is therefore a measure for the relative length of thestack with respect to the wavelength; short stacks imply small Helmholtz numbers. InSection 4.7.2 it is shown that Sk is a measure for the contribution of the thermoacousticheat flux to the total heat flux in the stack and can be both small and large depending onthe application. As mentioned previously, the Lautrec number distinguishes betweenthe porous medium being a stack (NL ∼ 1) or a regenerator (NL ≪ 1). These limitingcases are described schematically in Table 3.2. Note also that the gravity is present viathe Froude number Fr. The gravitational terms only appear in the momentum equations(3.17) and (3.18), and in the energy equation (3.22), as second or third-order term in Ma.Therefore, as long as the Froude number does not become too small, the gravity will

Modeling 33

NL Ma ε κ Sk

≪ 1 regenerator small velocities long pores short stackthermoacoustic heat

flow dominates

∼ 1 stack large velocities short pores long stack -

≫ 1 resonator - - -heat conduction

dominates

Table 3.2: Parameter ranges for some important dimensionless numbers

have little effect on the thermoacoustic behaviour. It follows from the definition of theFroude number that in large machines, or at very small velocity amplitudes, gravity canbecome more important. This should be kept in mind when upscaling thermoacousticdevices to a larger size.

From here on we will use dimensionless variables, but omit the tildes for conve-nience. Note that

∇ = εex

∂∂X

+∇τ , v = uex + εvτ .

Substituting (3.14) into equations (3.1)-(3.3) we arrive at the following set of dimension-less equations:

∂ρ∂t

= − ∂(ρu)

∂X−∇τ · (ρvτ ), (3.16)

ρ

(∂u

∂t+ u

∂u

∂X+ vτ ·∇τu

)= − ∂p

∂X+κ

W2o

∇ · (µ∇u) +M2

a

Fr

ρbx

+ ε2κ

∂∂X

[(µ

W2o

+ζ

W2ζ

)(∂u

∂X+ ∇τ · vτ

)], (3.17)

ρ

(∂vτ∂t

+ u∂vτ∂X

+ vτ · ∇τvτ

)= − 1

ε2∇τ p +

κ

W2o

∇ · (µ∇vτ ) +M2

a

εFr

ρbτ

+κ∇τ[(

µ

W2o

+ζ

W2ζ

)(∂u

∂X+∇τ · vτ

)], (3.18)

ρ

(∂ǫ∂t

+ u∂ǫ∂X

+ vτ · ∇τǫ)

= − p

(∂u

∂X+∇τ · vτ

)− κ

2N2L

∇ · (K∇T) +κΣ, (3.19)

where

Σ =µ

W2o

[(∂u

∂y+ε2 ∂v

∂X

)2

+

(∂u

∂z+ ε2 ∂w

∂X

)2

+ ε2

(∂w

∂y+

∂v

∂z

)2

+2ε2

([∂u

∂X

]2

+

[∂v

∂y

]2

+

[∂w

∂z

]2)]

+ε

2ζ

W2ζ

(∂u

∂X+∇τ · vτ

)2

.

34 3.4 Small-amplitude and long-pore approximation

Equations (3.16) and (3.17) show that the gravitational term ρb occurs as a second-ordereffect in the Mach number Ma. The equations (3.6) and (3.7) for the fluid and solidtemperature transform into

ρCp

(∂T

∂t+ u

∂T

∂X+ vτ · ∇τT

)= βT

(∂p

∂t+ u

∂p

∂X+ vτ · ∇τ p

)

+κ

2N2L

∇ · (K∇T) +κΣ, (3.20)

ρsCs

∂Ts

∂t=κφ

2

2N2s

∇ · (Ks∇Ts) . (3.21)

Furthermore the conservation of energy (3.8) transforms into

∂∂t

(1

2ρ|v|2 + ρǫ

)= −∇ ·

[v

ε

(1

2ρ|v|2 + ρh

)− κ

2N2L

∇T − κ

W2o

v · τ

]+

M2a

Fr

ρv · b.

(3.22)Finally we have the following boundary conditions at the gas-solid interface. First

for the velocity we have

v = 0 if Sg = 0. (3.23a)

Secondly, for the temperature we can write

T = Ts if Sg = 0, (3.23b)

K∇T ·∇Sg = σKs∇Ts ·∇Sg if Sg = 0, (3.23c)

∇Ts ·∇St = 0 if St = 0. (3.23d)

3.4 Small-amplitude and long-pore approximation

We assume a small maximal amplitude A of acoustic oscillations, i.e. the velocity, pres-sure, density and temperature fluctuations are small relative to their mean value, whichcan be used to linearize the equations given above. Since all variables are dimension-less, we can use the same A to linearize all variables, e.g. the relative pressure or velocityamplitude. Let f be any of the fluid variables (p, v, T, etc.) with stationary equilibriumprofile f0. Expanding f in powers of A using a Fourier series we write

f (x, t) = f0(x) + ARe[

f1(x)eiκt]

+ A2 f2,0(x) + A2Re[

f2,2(x)e2iκt]+ · · · , A ≪ 1.

Furthermore, for the velocity we assume v0 = 0. As the velocity was scaled with thespeed of sound, the obvious choice for A then becomes the acoustic Mach number Ma,

Modeling 35

and we will therefore use the following asymptotic expansion:

f (x, t) = f0(x) + MaRe[

f1(x)eiκt]

+ M2a f2,0(x) + M2

a Re[

f2,2(x)e2iκt]+ · · · ,

Ma ≪ 1. (3.24)

The first index is used to indicate the corresponding power of Ma, and the index behindthe comma is used to indicate the frequency of the oscillation (cf. [135]). The fluid vari-ables are built up from steady terms and harmonic modes that are integer multiples ofthe fundamental frequency. We assume that only the first harmonic f1 is excited exter-nally. It then follows from equation (3.26) below that the first-order oscillations excitethe second-order fluid variables by the combined effect of the second harmonic f2,2, thatoscillates at twice the fundamental frequency, and the steady streaming part f2,0. In thenext chapters we will show how both the streaming variables and the second harmonicscan be expressed in terms of the first harmonics and the equilibrium values.

Note that for the subsequent analysis we will only use the leading-order approxima-tion of the thermodynamic parameters c, β, Cp, and Cv. Their indices will therefore beomitted.

Additionally, since we also assumeε≪ 1, we may expand the perturbation variablesfi again to include powers of ε as well. However this would lead to messy derivations.Instead we will assume that ε ∼ Ma, so that the geometric and streaming effect can beincluded at the same order. In the end, when all the analysis has been performed, onecan still consider the limits ε≪ Ma or ε≫ Ma and neglect certain terms.

We will use an overbar to indicate time-averaging and brackets to indicate transverseaveraging in the gas or solid, i.e.

f (x) =κ

2π

∫ 2π/κ

0f (x, t) dt,

〈 f 〉(X, t) =1

Ag(X)

∫

Ag(X)f (x, t) dA, 〈 f 〉s(X, t) =

1

As(X)

∫

As(X)f (x, t) dA,

where Ag and As are the cross-sectional area at position X of the gas and solid, respec-

tively. The time-average of a harmonic variable always yields zero, since Re[ae jiκt] ≡ 0,for any j ∈ N, a ∈ C. However the time-average of the product of two harmonic vari-ables is in general not equal to zero, as

Re[

f1eiκt]

Re[

g1eiκt]

=1

2Re [ f1g∗

1] =1

2Re [ f ∗1 g1] . (3.25)

Here the superscript “∗” denotes complex conjugation. In addition we can show thatthe product of two harmonic variables yields a second harmonic plus the constant time-averaged term from (3.25)

Re[

f1eiκt]

Re[

g1eiκt]

=1

2Re[

f1g1e2iκt]+

1

2Re [ f1g∗

1] . (3.26)

From this equation it follows that the first-order acoustics can only excite second har-monics and steady terms, unless externally excited. This motivates why we only in-

36 3.4 Small-amplitude and long-pore approximation

cluded f2,0 and f2,2 as second-order terms in the expansion (3.24).

Chapter 4

Thermoacoustics intwo-dimensional pores withslowly varying cross-section

For certain types of geometries it suffices to use a two-dimensional representation. Specif-ically we will consider narrow two-dimensional pores with slowly varying cross-sectionsas shown in figure 4.1, where we use plus-superscripts to denote parameters and vari-ables in the upper plane and minus-superscripts in the bottom plane. In particular R+

g

and R−g give the distances between the centerline and the top plate and bottom plate,

respectively, and R+s and R−

s denote the thicknesses of the top and bottom plate, re-spectively. Additionally we define

Rg :=R+

g +R−g

2, Rs :=

R+s +R−

s

2R+

t := R+g +R+

s , R−t := R−

g +R−s .

Two dimensional pores can be used to model a geometry where the three-dimensionalpore arises from an extension of a two-dimensional channel into the third-coordinate di-rection. The most common example is the parallel-plate stack, consisting of various par-allel plates stacked on top of each other. Moreover, often a two-dimensional model candescribe in good approximation the behavior of a more complicated three-dimensionalshape.

Due to the assumption of slow variation we will be able to treat the axial varia-tion separately from the transverse variation. In particular we will determine the X-dependence of the variables as solvability conditions for the (higher-order) y-dependence.The method adopted here is also referred to as the method of slow variation [85, 143].


y

X

Π

R+g

R−g

Γ+g

Γ+t

Γ−g

Γ−t

Figure 4.1: Three-dimensional plates as an extension of a narrow two-dimensional channel withslowly varying cross-section. At position X = εx, the plate separation is given by 2Rg(X) :=

R+g (X) + R−

g (X) and the cross-sectional area of the gas is given by Ag(X) = 2ΠRg(X).


In two dimensions the equations (3.16)-(3.22) simplify into

∂ρ∂t

+∂(ρu)

∂X+

∂(ρv)

∂y= 0, (4.1)

ρ

(∂u

∂t+ u

∂u

∂X+ v

∂u

∂y

)= − ∂p

∂X+

κ

W2o

[ε

2 ∂∂X

(µ

∂u

∂X

)+

∂∂y

(µ

∂u

∂y

)]+

M2a

Fr

ρbx

+ε2κ

∂∂X

[(µ

W2o

+ζ

W2ζ

)(∂u

∂X+

∂v

∂y

)], (4.2)

ρ

(∂v

∂t+ u

∂v

∂X+ v

∂v

∂y

)= − 1

ε2

∂p

∂y+

κ

W2o

[ε

2 ∂∂X

(µ

∂v

∂X

)+

∂∂y

(µ

∂v

∂y

)]

+M2

a

εFr

ρby +κ∂

∂y

[(µ

W2o

+ζ

W2ζ

)(∂u

∂X+

∂v

∂y

)], (4.3)

ρCp

(∂T

∂t+ u

∂T

∂X+ v

∂T

∂y

)= βT

(∂p

∂t+ u

∂p

∂X+ v

∂p

∂y

)

+κ

2N2L

[ε

2 ∂∂X

(K

∂T

∂X

)+

∂∂y

(K

∂T

∂y

)]+κΣ, (4.4)

ρsCs

∂Ts

∂t=κφ

2

2N2s

[ε

2 ∂∂X

(K

∂Ts

∂X

)+

∂∂y

(K

∂Ts

∂y

)], (4.5)

Thermoacoustics in two-dimensional pores with variable cross-section 39

∂∂t

(1

2(ρ|u|2 +ε2|v|2) + ρǫ

)= − ∂

∂X

[1

2ρu(|u|2 +ε2|v|2) + ρuh −ε2 κK

2N2L

∂T

∂X

]

− ∂∂y

[1

2ρv(|u|2 +ε2|v|2) + ρvh − κK

2N2L

∂T

∂y− κµ

W2o

u∂u

∂y

]+ ∇ · T +

M2a

Fr

ρv · b, (4.6)

where

Σ =µ

W2o

[(∂u

∂y+ε2 ∂v

∂X

)2

+ 2ε2

([∂u

∂X

]2

+

[∂v

∂y

]2)]

+ε

2ζ

W2ζ

(∂u

∂X+

∂v

∂y

)2

,

T = ε2κ

µ

W2o

(u

∂u

∂X+ ε2v

∂v

∂X+ v ·∇u

)+

ζ

W2ζ

u∇ · v

µ

W2o

(v

∂v

∂y+ v · ∇v

)+

ζ

W2ζ

v∇ · v

.

The boundary conditions (3.23) reduce to

v = 0 if y = ±R±g , (4.7a)

T = Ts if y = ±R±g , (4.7b)

K∂T

∂y−σKs

∂Ts

∂y= ±ε2 ∂R±

g

∂X

(K

∂T

∂X−σKs

∂Ts

∂X

)if y = ±R±

g , (4.7c)

Ks

∂Ts

∂y= ±ε2Ks

∂R±t

∂X

∂Ts

∂Xif y = ±R±

t . (4.7d)

4.2 Acoustics

In this section we will derive two ordinary differential equations in X, with coefficientsdepending on T0, for the acoustic pressure p1 and the averaged acoustic velocity 〈u1〉.Moreover, it will be shown that p0 is constant and that p1 is independent of y. Finally,explicit expressions will be derived, expressing u1, v1, T1, and ρ1 in terms of T0, p1, and〈u1〉.

We start from the momentum equation in which we relate ε to Ma by putting ε2 =

ηM2a (see equation (3.15)). Expanding the variables in powers of Ma according to (3.24),

substituting the expansions into the y-component of the momentum equation (4.3), andkeeping terms up to first order in Ma we find

∂p0

∂y+ Ma

∂p1

∂y= o(Ma).

This equation should be valid for any Ma ≪ 1. Therefore, the coefficients of M0a and

M1a are equal to zero, so we find that ∂p0/∂y = ∂p1/∂y = 0, so that p0 and p1 are

independent of y.In the same way it can be shown that T0 does not depend on y either. Collecting the

40 4.2 Acoustics

leading-order terms from the temperature equations (4.4) and (4.5), we find

∂∂y

[K0

∂T0

∂y

]= 0,

∂∂y

[Ks0

∂Ts0

∂y

]= 0.

The only solution, satisfying the boundary conditions given in (3.23), is given by Ts0=

T0 = T0(X). As a result ρ0, ρs0, K0, Ks0

, and µ0 are independent of y as well. Theexact X-dependence of T0 will be derived in Section 4.3 as a solvability condition for thetemperature T2,0.

We now turn to the X-component of the momentum equation (4.2). Keeping termsup to first order in Ma, we find that (4.2) reduces to

iMaκρ0u1 = −dp0

dX− Ma

dp1

dX+ Ma

κ

W2o

∂∂y

(µ0

∂u1

∂y

)+ o(Ma).

Thus to leading order we find that dp0/dX = 0 and p0 must be constant. Collecting theterms of first order in Ma, we find that u1 satisfies

u1 =i

κρ0

dp1

dX+

1

α2ν

∂2u1

∂y2, (4.8)

where

αν = (1 + i)

√ρ0

2µ0

Wo. (4.9)

Integrating (4.8) subject to v|Γg= 0, we find that u1 and 〈u1〉 satisfy

u1 =iFνκρ0

dp1

dXand 〈u1〉 =

i(1 − fν)

κρ0

dp1

dX. (4.10)

where Fν is chosen such that it vanishes on Γ+g and Γ−g ,

Fν := 1 −sinh(ανR+

g ) + sinh(ανR−g )

sinh(αν(R+g +R−

g ))cosh(ανy)

+cosh(ανR+

g ) − cosh(ανR−g )

sinh(αν(R+g +R−

g ))sinh(ανy), (4.11)

and

fν := 1 − 〈Fν〉 =tanh(ανRg)

ανRg

, Rg :=R+

g +R−g

2, (4.12)

consistent with the notation of Arnott et al. [5], Rott [115], and Swift [131]. In particularwhen R+

g = R−g = Rg, we find the familiar expression

Fν = 1 − cosh(ανy)

cosh(ανRg). (4.13)

Note thatαν, fν, and Rg are X-dependent.


Next we go back to the temperature equation. Substituting our expansions into (4.4)and (4.5), substituting p0 is constant and T0 = T0(X), and collecting the first-orderterms, we find for the gas temperature

Maρ0Cp

(iκT1 + u1

dT0

dX

)= iMaκβT0 p1 + Ma

κK0

2N2L

∂2T1

∂y2+ o(Ma), (4.14)

and for the solid temperature

iκMaρs0CsTs1

= Ma

κφKs0

2N2s

∂2Ts1

∂y2+ o(Ma). (4.15)

Setting the coefficient of Ma equal to zero in (4.14) and (4.15), and substituting (4.10) foru1, we find after some manipulation that T1 and Ts1

can be obtained from

T1 −1

α2k

∂2T1

∂y2= − 1

κ2ρ0

dT0

dX

dp1

dXFν +

βT0

ρ0Cp

p1, (4.16a)

Ts1− 1

α2s

∂2Ts1

∂y2= 0, (4.16b)

where analogous to (4.9) we define

αk := (1 + i)

√ρ0

K0Cp

NL, (4.17)

αs := (1 + i)φ

√ρs0

Ks0Cs

Ns. (4.18)

Collecting the first-order terms, the boundary conditions given in (4.7b, 4.7d) can bewritten as

T1 = Ts1, y = ±R±

g , (4.19a)

∂Ts1

∂y= 0, y = ±R±

t . (4.19b)

(4.19c)

Before we can solve (4.16), we first need to introduce some auxiliary functions. For thegas we define

F+k := 1 −

sinh(αkR+g )

sinh(αk(R+g +R−

g ))cosh(αky) +

cosh(αkR+g )

sinh(αk(R+g + R−

g ))sin(αky), (4.20)

F−k := 1 −

sinh(αkR−g )

sinh(αk(R+g +R−

g ))cosh(αky)−

cosh(αkR−g )

sinh(αk(R+g + R−

g ))sin(αky), (4.21)

Fk := F+k + F−

k − 1. (4.22)

42 4.2 Acoustics

and in the solid (with y+ = R+t − y, y− = −R−

t + y)

F+s := 1 − cosh(αsy+)

cosh(αsR+s )

, (4.23)

F−s := 1 − cosh(αsy−)

cosh(αsR−s )

, (4.24)

chosen such that Fk|Γ±g = 0, F+k |Γ+

g= 1, F+

k |Γ−g = 0, F−k |Γ+

g= 0, F−

k |Γ−g = 1, F±s |Γ±g = 0, and

∂F±s /∂y|Γ±t = 0. Additionally we also define

fk := 1 − 〈Fk〉 =tanh(αkRg)

αkRg

, f ±k := 1 − 〈F±k 〉 =

1

2fk, (4.25)

f +s := 1 − 〈F+

s 〉+s =tanh(αsR+

s )

αsR+s

, f −s := 1 − 〈F−s 〉−s =

tanh(αsR−s )

αsR−s

. (4.26)

Using these auxiliary functions we can integrate (4.16) subject to (4.19), so that

T1 =βT0Fk

ρ0Cp

p1 −Fk − PrFν

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX+ T+

b1(1 − F+

k ) + T−b1

(1 − F−k ), (4.27a)

T±s1

= T±b1

(1 − F±

s

), (4.27b)

where T+b1

= T1|Γ+g

and T−b1

= T1|Γ−g are yet to be determined. To leading order, the

remaining boundary condition (4.7c) can be written as

K0

∂T1

∂y

∣∣∣∣y=R+

g

= −Ks0σ

∂T+s1

∂y+

∣∣∣∣∣y+=R+

s

, K0

∂T1

∂y

∣∣∣∣y=−R−

g

= Ks0σ

∂T−s1

∂y−

∣∣∣∣∣y−=R−

s

. (4.28)

Substituting the expressions for T1 and Ts1, we find the following two coupled equations

for T−b1

and T−b1

:

σKs0R+

s α2s f +

s T+b1

= K0Rgα2k fk

(βT0 p1

ρ0Cp

− 1 − fν/ fk

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX+

T+b1

2 sinh2(αkRg)

−(1 + coth2(αkRg))T−

b1

2

), (4.29)

and

σKs0R−

s α2s f −s T−

b1= K0Rgα

2k fk

(βT0 p1

ρ0Cp

− 1 − fν/ fk

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX− (1 + coth2(αkRg))

T+b1

2

+T−

b1

2 sinh2(αkRg)

). (4.30)


Here we used thatα2k = Prα

2ν and

∂Fk

∂y

∣∣∣∣y=±R±

g

= ∓Rgα2k fk,

∂Fν∂y

∣∣∣∣y=±R±

g

= ∓Rgα2ν fν,

∂F±k

∂y

∣∣∣∣y=±R±

g

= ±1

2

Rgα2k fk

sinh2(αkRg),

∂F±k

∂y

∣∣∣∣y=∓R±

g

= ±1

2(1 + coth2(αkRg))Rgα

2k fk,

∂F+s

∂y+

∣∣∣∣∣y+=R+

s

= −R+g α

2s f +

s ,∂F−

s

∂y−

∣∣∣∣y−=R−

s

= −R−g α

2s f −s .

Introducing the stack heat-capacity ratios

ε±s :=

1

σ

K0Rgα2k fk

Ks0R±

s α2s f ±s

, (4.31)

we can rewrite (4.29) and (4.30) as

[1 − ε

±s

2 sinh2(αkRg)

]T±

b1+ε±s

2

[1 + coth2(αkRg)

]T∓

b1

= ε±s

(βT0 p1

ρ0Cp

− 1 − fν/ fk

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX

). (4.32)

Solving for T−b1

and T+b2

, we obtain

T+b1

=ε

+s

1 +ε+s

(1 + T +)

(βT0 p1

ρ0Cp

− 1 − fν/ fk

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX

), (4.33a)

T−b1

=ε−s

1 +ε−s(1 + T −)

(βT0 p1

ρ0Cp

− 1 − fν/ fk

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX

), (4.33b)

where

T + =−(ε+

s −ε−s ) cosh(2αkRg)

(1 +ε+s )(1 +ε−s + (ε−s − 1) cosh(2αkRg) + (ε+

s −ε−s ) cosh(2αkRg), (4.34a)

T − =−(ε−s −ε+

s ) cosh(2αkRg)

(1 +ε−s )(1 +ε+s + (ε+

s − 1) cosh(2αkRg) + (ε−s −ε+s ) cosh(2αkRg)

. (4.34b)

Note that T ± = 0, when R+s = R−

s .In an ideal stack the solid has sufficient heat capacity to keep the stack plates isother-

mal, so that the stack heat-capacity ratioεs is equal to zero. Figure 4.2 shows the absolutevalue of εs for a helium-filled parallel-plate stack for various porosities and plate mate-rials and a fixed hydraulic radius. The graphs show that only when the stack plates arethick enough (Br ≪ 1) the stack can be considered ideal.

Using relations (A.5) and (A.8) and substituting T1 we can derive the following rela-

44 4.2 Acoustics

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

|εs|

Br

stainless steelaluminaglassmylarpolystyrene

Figure 4.2: The absolute value ofεs plotted as a function of the blockage ratio Br for various platematerials. We consider a helium-filled parallel-plate stack at room temperature and fix NL = 1.The stack is ideal for small blockage ratios (thick plates).

tion for the acoustic density fluctuations:

ρ1 =1

c2[1 + (γ− 1)(1 − Fk)] p1 +

Fk − PrFν

κ2(1 − Pr)

βdT0

dX

dp1

dX− ρ0βT+

b1(1 − F+

k )

− ρ0βT−b1

(1 − F−k ) (4.35)

=1

c2

[1 + (γ − 1)

(1 − Fk −

ε+s (1 + T +)

1 +ε+s

(1 − F+k )− ε

−s (1 + T −)

1 +ε−s(1 − F−

k )

)]p1

+

[Fν +

Fk − Fν(1 − Pr)

+ε+s (1 + T +)

(1 − fν/ fk)(1 − F+k )

(1 − Pr)(1 +ε+s )

+ε−s (1 + T −)(1 − fν/ fk)(1 − F−

k )

(1 − Pr)(1 +ε−s )

]β

κ2

dT0

dX

dp1

dX. (4.36)

As a result we find

〈ρ1〉 =1

c2

[1 +

γ− 1

1 +εs

fk

]p1

+

[1 − fν +

fν − fk

(1 − Pr)(1 +εs)

]β

κ2

dT0

dX

dp1

dX. (4.37)

where εs is the effective stack heat-capacity ratio, given by

1

1 + εs

=1

2

(1 −ε+

s T +

1 +ε+s

+1 − ε−s T −

1 +ε−s

)(4.38a)

= 1 − 1

2

(ε

+s (1 + T +)

1 +ε+s

+ε−s (1 + T −)

1 +ε−s

). (4.38b)

Finally, we turn to the continuity equation (4.1). Expanding the variables in powers


of Ma and keeping terms up to first order in Ma we find

iMaκρ1 + Ma

∂∂X

(ρ0u1) + Maρ0

∂v1

∂y= o(Ma), (4.39)

We can use equation (4.39) to express v1 in terms of ρ0, u1 and p1,

v1 = − 1

ρ0

∫ y

−R−g

(iκρ1 +

∂∂X

(ρ0u1)

)d y. (4.40)

where we applied the boundary condition that v1 vanishes at Γ−g .

Note that v also vanishes at the boundary y = R+g . Therefore, substituting (4.10) for

u1, multiplying with −iκ, averaging (4.39) over a cross-section, and using the Leibnizrule, we obtain the following equation as a consistency relation for v1:

κ2〈ρ1〉 = − 1

Rg

d

dX

(Rg(1 − fν)

dp1

dX

)− iκ

2Rg

dR+g

dXρ0u1

∣∣∣R+

g

− iκ

2Rg

dR−g

dXρ0u1

∣∣∣−R−

g

= − 1

Rg

d

dX

(Rg(1 − fν)

dp1

dX

). (4.41)

After substitution of (4.37) we obtain

κ2

c2

[1 +

γ − 1

1 +εs

fk

]p1 + (1 − fν)β

dT0

dX

dp1

dX+

fν − fk

(1 − Pr)(1 + εs)β

dT0

dX

dp1

dX

+1

Rg

d

dX

(Rg(1 − fν)

dp1

dX

)= 0. (4.42)

Then using

ρ0

d

dX

(1

ρ0

)= − 1

ρ0

dρ0

dX= − 1

ρ0

∂ρ0

∂T0

dT0

dX= β

dT0

dX, (4.43)

we obtain a reduced wave equation for p1 valid for slowly varying cross-sections,

κ2

c2

[1 +

γ − 1

1 +εs

fk

]p1 +

fν − fk

(1 − Pr)(1 +εs)β

dT0

dX

dp1

dX

+ρ0

Rg

d

dX

(Rg

1 − fνρ0

dp1

dX

)= 0. (4.44)

Combining (4.10) and (4.44) we can derive the following two coupled ordinary differ-ential equations for p1 and 〈u1〉

d(Rg〈u1〉)dX

= κa3 p1 + a4

dT0

dX〈u1〉, (4.45a)

dp1

dX= κa5〈u1〉, (4.45b)

46 4.3 Mean temperature

where

a3 := − i

ρ0c2

[1 +

γ − 1

1 + εs

fk

], (4.46a)

a4 := − ( fν − fk)β

(1 − Pr)(1 − fν)(1 + εs), (4.46b)

a5 := − iρ0

1 − fν. (4.46c)

To solve these equations it only remains to determine T0 and apply appropriate bound-ary conditions in X. In the next section the system of equations will be completed, whenan ordinary differential equation for the mean temperature T0 is derived. Given T0, p1

and 〈u1〉, we can compute u1, T1, Ts1, ρ1, and v1 from (4.10), (4.27), (4.35), and (4.40),

respectively.

4.3 Mean temperature

In this section we will the conservation of energy as given by (4.6) to determine T0.When calculating T2,0 we will determine T0 as a solvability condition from the assump-tion of slow variation.

Putting ε2 = ηM2a , we can rewrite (4.6) as follows:

κ∂∂t

(1

2ρ(|u|2 + ηM2

a |v|2) + ρǫ

)= − ∂

∂X

(1

2ρu(|u|2 + ηM2

a |v|2) + ρvh − M2a

κηK

2N2L

∂T

∂X

)

− ∂∂y

(1

2ρv(|u|2 + ηM2

a |v|2) + ρvh − κK

2N2L

∂T

∂y− κµ

W2o

u∂u

∂y

)+

M2a

Fr

ρubx + o(M4a).

(4.47)

Suppose there is an average heat flux to the boundary given by F. Then averaged intime, the left hand side of this equation will reduce to F. Consequently, on expandingin powers of Ma and keeping terms up to second order, we find

∂∂X

[M2

aρ0u1h1 + M2a h0m2 − M2

a

κηK0

2N2L

dT0

dX

]+

∂∂y

[M2

aρ0v1h1 + M2a h0n2

−M2a

κK0

2N2L

∂T2,0

∂y− M2

a

κ

2N2L

K1

∂T1

∂y− M2

a

κµ0

W2o

u1

∂u1

∂y

]= M2

a F2 + o(M2a),

where m2 and n2 are the components in X and y directions of the second-order time-averaged mass flux m2,

m2 := ρ0(v2,0 + Re[

v2,2e2iκt]) + Re

[ρ1eiκt

]Re[v1eiκt

]= ρ0v2,0 +

1

2Re [ρ1v∗

1] .


Using (3.25) and rearranging terms we find

K0

∂2T2,0

∂y2+

1

2

∂∂y

Re

[K∗

1

∂T1

∂y

]+ η

d

dX

(K0

dT0

dX

)=

N2L

κ

∂∂X

(ρ0Re [u∗1h1] + 2h0m2)

+N2

L

κ

∂∂y

(ρ0Re [v∗

1h1] + 2h0n2 −κµ0

W2o

Re

[u∗

1

∂u1

∂y

])− 2N2

L

κF2. (4.48)

Similarly we can show that (4.5) reduces to

Ks0

∂2T±s2,0

∂y±2+

1

2

∂∂y± Re

[K∗

s1

∂T±s1

∂y±

]+ η

d

dX

(Ks0

dT0

dX

)= 0. (4.49)

We will now use the flux condition (4.7c) and (4.7d) to derive a differential equation for

T0. Putting ε2 = σM2a , averaging in time, expanding in powers of Ma and collecting the

second order terms in Ma, we find

K0

∂T2,0

∂y+

1

2Re

[K∗

1

∂T1

∂y

]±σKs0

∂Ts2,0

∂y± ± σ2

Re

[K∗

s1

∂Ts1

∂y±

]= ±η

(K0 −σKs0

) ∂Rg

∂X

dT0

dX,

if y = ±R±g , y± = R±

s , (4.50)

and

Ks0

∂Ts2,0

∂y± +1

2Re

[K∗

s1

∂Ts1

∂y±

]= −ηKs0

∂R±t

∂X

dT0

dX, if y± = 0. (4.51)

We will combine (4.48) and (4.49) with (4.50) and (4.51). First we integrate (4.49)over its cross-section, multiply with σ , and add it to (4.48), again after integrating overits cross-section, while using the Leibniz rule. This leads to

(K0

∂T2,0

∂y+

1

2Re

[K∗

1

∂T1

∂y

])y=R+g

y=−R−g

+σ

(Ks0

∂Ts2,0

∂y+ +1

2Re

[K∗

s1

∂Ts1

∂y+

])y+=R+s

y+=0

+σ

(Ks0

∂Ts2,0

∂y+ +1

2Re

[K∗

s1

∂Ts1

∂y−

])y−=R−s

y−=0

+ 2ηRg

d

dX

(K0

dT0

dX

)+ 2ησRs

d

dX

(Ks0

dT0

dX

)

=N2

L

κ

∂∂X

(2ρ0RgRe [〈u∗

1h1〉] + 4h0 M2

)− 4N2

L

κRg F2, (4.52)

where M2 = Rg〈m2〉. After inserting the flux conditions (4.50) and (4.51) and rearrang-ing terms, we can simplify (4.52) further into

dH2

dX=

d

dX

[h0 M2 +Rg

ρ0

2Re [〈u∗

1h1〉]−S2

k

2κ(K0Rg +σKs0

Rs)dT0

dX

], (4.53)

where we put Rg F =: dHdX . The quantity H is the time-averaged total power (or energy

flux) along X. In steady state, for a cyclic refrigerator or prime mover without heat flows


to the surroundings H will be constant as F = 0.Combining the thermodynamic expressions (A.7) and (A.9) we find

dh = Tds +1

ρdp = CpdT +

1

ρ(1 −βT)dp. (4.54)

Moreover, it follows thath0 = Cp(T0 − Tre f ), (4.55)

where Tre f is a reference temperature. Substituting (4.54) and (4.55) into (4.53), we find

H2 = M2Cp(T0 − Tre f ) +1

2ρ0CpRgRe [〈T1u∗

1〉] +Rg

2(1 −βT0)Re [p1〈u∗

1〉]

−(

K0Rg +σKs0Rs

) S2k

2κ

dT0

dX, (4.56)

where the constant of integration has been chosen such that H2 = 0 when there is noexchange of heat with the environment.

Before we substitute T1 and u1 we will first prove the following relationships:

〈F±k F∗

ν 〉 =1 − f ±k + Pr(1 − f ∗ν ) − 1

2 f ∗ν1 + Pr

, (4.57a)

〈FkF∗ν 〉 =

1 − fk + Pr(1 − f ∗ν )

1 + Pr

, (4.57b)

〈|Fν|2〉 = 1 − Re [ fν] . (4.57c)

We will only prove the first statement as it is most general; the other two equalities canbe proven in the same way. Define

I1 := 〈F±k F∗

ν 〉, (4.58)

I2 :=

⟨∂F±

k

∂y

∂F∗ν

∂y

⟩. (4.59)

We have the following relationships for Fν and F±k :

Fν = 1 +1

α2ν

∂2Fν

∂y2Fν , (4.60)

F±k = 1 +

1

α2k

∂2F±k

∂y2. (4.61)

Integrating by parts, using (4.60), substituting (α∗ν)

2 = −α2ν, and using the boundary


conditions for F±k and Fν, we can rewrite I1 as follows:

I1 =1

2Rg

∫ R+g

−R−g

F±k

(1 − 1

α2ν

∂2F∗ν

∂y2

)dy

= 1 − f ±k − 1

2Rgα2ν

∫ R+g

−R−g

F±k

∂2F∗ν

∂y2dy

= 1 − f ±k − 1

2Rgα2ν

F±k

∂F∗ν

∂y

∣∣∣∣∣

R+g

−R−g

+1

2Rgα2ν

∫ R+g

−R−g

∂F±k

∂y

∂F∗ν

∂ydy

= 1 − f ±k − 1

2f ∗ν +

I2

α2ν

. (4.62)

In the same way, substituting (4.61) rather than (4.60), we find

I1 = 1 − f ∗ν −I2

α2k

. (4.63)

Eliminating the common term I2 from equations (4.62) and (4.63), we find

I1 =1 − f ±k + Pr(1 − f ∗ν )− 1

2 f ∗ν1 + Pr

, (4.64)

as claimed in (4.57a). Replacing F±k by Fk or Fν, we can repeat the same analysis to prove

(4.57b) and (4.57c).Substituting expressions (4.10) and (4.27a) for u1 and T1, while using relations (4.57)

and (4.38b), we find

〈T1u∗1〉 =

βT0

ρ0Cp

〈FkF∗ν 〉

1 − f ∗νp1〈u1

∗〉+ i〈FkF∗

ν 〉 − Pr〈|Fν|2〉κ(1 − Pr)|1 − fν|2

dT0

dX|〈u1〉|2

+ T+b1〈u1

∗〉〈F∗ν 〉 − 〈F+

k F∗ν 〉

1 − f ∗ν+ T−

b1〈u1

∗〉〈F∗ν 〉 − 〈F−

k F∗ν 〉

1 − f ∗ν

=βT0 p1〈u1

∗〉ρ0Cp

(1 − fk − f ∗ν

(1 + Pr)(1 +εs)(1 − f ∗ν )

)+

i(1 − Pr + PrRe ( fν))

κ(1 − Pr)|1 − fν|2dT0

dX|〈u1〉|2

− i|〈u1〉|2

κ(1 − Pr)|1 − fν|2dT0

dX

(f ∗ν +

( fk − f ∗ν )(1 +εs fν/ fk)

(1 + Pr)(1 +εs)

).

Taking the real part, we get

Re [〈T1u∗1〉] =

βT0

ρ0Cp

Re

[(1 − fk − f ∗ν

(1 + Pr)(1 +εs)(1 − f ∗ν )

)p1〈u1

∗〉]

+|〈u1〉|2

κ(1 − Pr)|1 − fν|2dT0

dXIm

[f ∗ν +

( fk − f ∗ν )(1 + εs fν/ fk)

(1 + Pr)(1 +εs)

]. (4.65)


After substitution of (4.65) into (4.56) we arrive at

H2 = M2Cp

(T0 − Tre f

)+

1

2RgRe

[p1〈u∗

1〉(

1 −βT0

fk − f ∗ν(1 + Pr)(1 + εs)(1 − f ∗ν )

)]

+Rg

ρ0Cp|〈u1〉|2

2κ(1 − Pr)|1 − fν|2dT0

dXIm

[f ∗ν +

(1 +εs fν/ fk)( fk − f ∗ν )

(1 + Pr)(1 + εs)

]

−(

K0Rg +σKs0Rs

) S2k

2κ

dT0

dX. (4.66)

Given H2, we can solve (4.37) for dT0/dX

dT0

dX= κ

2H2 − 2M2Cp

(T0 − Tre f

)−RgRe [a1 p1u∗

1]

Rga2|〈u1〉|2 −(

K0Rg +σKs0Rs

)S2

k

, (4.67)

where

a1 := 1 −βT0

fk − f ∗ν(1 + Pr)(1 + εs)(1 − f ∗ν )

, (4.68)

a2 :=ρ0Cp|〈u1〉|2

(1 − Pr)|1 − fν|2Im

[f ∗ν +

(1 + εs fν/ fk)( fk − f ∗ν )(1 + Pr)(1 +εs)

]. (4.69)

Equation (4.67) together with equations (4.46a) and (4.46b) form a complete coupled sys-tem of differential equations for T0, 〈u1〉, and p1. This system gives a generalization ofthe Swift equations [135], extended to include the case of slowly varying cross-sections.

It is possible to go one step further and improve the expression for the mean tem-perature by determining the correction term T2,0. If we integrate (4.48) and (4.49) twice

with respect to y, we can determine T2,0 and Ts2,0up to X-dependent functions T±

b2,0. In

the same way as the differential equation (4.67) for T0 followed as a solvability conditionfor T2,0 and Ts2,0

, we can derive an ordinary differential equation for Tb2,0as a solvability

condition for the fourth-order mean temperatures T4,0 and Ts4,0. Performing this analysis

it is possible to include transverse variations into the mean temperature profile, sinceT2,0 does depend on y in contrast to T0. In this thesis we will not go this far, as thiswould require a lengthy derivation of both the second and third harmonics. We will,however, compute the second harmonics in Section 4.6 below.


4.4 Integration of the generalized Swift equations

It has been shown in Section 4.2 and 4.3 that all acoustic variables can be determined, ifthe following system of ordinary differential equations is solved for T0, p1, and 〈u1〉:

dT0

dX= κ

2H2 − 2M2Cp

(T0 − Tre f

)−RgRe [a1 p1〈u1

∗〉]

Rga2|〈u1〉|2 −(

K0Rg +σKs0Rs

)S2

k

, (4.70a)

d(Rg〈u1〉)dX

= κa3Rg p1 + a4Rg

dT0

dX〈u1〉, (4.70b)

dp1

dX= κa5〈u1〉, (4.70c)

where

a1 := 1 −βT0

fk − f ∗ν(1 + Pr)(1 + εs)(1 − f ∗ν )

, (4.71a)

a2 :=ρ0Cp|〈u1〉|2

(1 − Pr)|1 − fν|2Im

[f ∗ν +

(1 + εs fν/ fk)( fk − f ∗ν )

(1 + Pr)(1 +εs)

], (4.71b)

a3 := − i

ρ0c2

[1 +

(γ − 1) fk

1 +εs

], (4.71c)

a4 := − ( fν − fk)β

(1 − Pr)(1 + εs)(1 − fν), (4.71d)

a5 := − iρ0

1 − fν. (4.71e)

With suitable boundary conditions these equations can be integrated numerically. Forexample, if a stack is positioned in a resonator with the left stack end at distance xL fromthe closed end, then one can impose

p1(0) = pL cos(kxL), 〈u1〉(0) =ipL

ρLcL

sin(kxL), T0(0) = TL, (4.72)

where k is the wave number and pL is the pressure amplitude at the closed end. Inaddition we still need to impose H2, which is nonzero in general and it may also dependon X. In a steady-state situation, without heat exchange to the surroundings and zeromass flow, it follows that H2 = 0. In this case the thermoacoustic heat flow is balancedby a return diffusive heat flow in the stack and in the gas, so that the net heat flow iszero. Alternatively, one can also impose a temperature TR on the right stack end andlook for the corresponding H2 that gives the desired temperature difference.

In Sections 6 and 7 we will show some practical examples of thermoacoustic systemsfor which the equations above have been implemented. Under certain conditions, onecan even find an exact solution as well. In the next sections we will treat a few of thesecases in full detail.

We see that the total energy flux through the stack, the cross-sectional variations, thesmall-amplitude acoustical oscillations and the resulting streaming may cause O(1)-

52 4.4 Integration of the generalized Swift equations

variations in the mean temperature. The energy flux H2 can be changed by externalinput or extraction of heat at the stack ends and consequently change the mean temper-ature profile. The acoustic oscillations affect the mean temperature via the well-knownshuttling effect (see e.g. [131]). With a nonzero mass flux M2 heat will be carried to thehot or cold heat exchanger and thus affect the mean temperature also. This can eitherbe a loss or a contribution to the heat transfer.

Note that the temperature gradient scales with κ and 1/S2k . Thus in the limits κ → 0

(short-stack limit) or Sk → ∞ (small velocities; heat conduction is dominating), the tem-perature difference across the stack will tend to zero, unless sufficient heat is suppliedor extracted (|H2| ≫ 1). Furthermore, the velocity and pressure gradients also scalewith κ, which justifies the assumption of constant-stack pressure and velocity that iscommonly applied in the short-stack approximation (see e.g. [131], [151]).

Finally, remember that these equations result from a linearization in both Ma andε. As a result there is still a dimensionless number Sk appearing in the equations thatcontains both ε and Ma. In (3.15) we showed that

Sk =κε

NLMa

,

so that the perturbation variables are weakly affected by the linearization parametersMa and ε. Hence the theory derived here is not exactly linear as it is often described,which is why we prefer to use the term weakly non-linear to indicate that there is stilla weak non-linearity involved. If the amplitude Ma becomes large enough such thatSk ≪ 1 then the temperature gradient will arise solely due to the thermoacoustic heatflow; if the amplitude Ma becomes very small, so that Sk ≫ 1, then the temperaturegradient will arise solely due to heat conduction through solid and gas.

4.4.1 Exact solution at constant temperature

If we apply equations (4.70) to a problem involving a channel supporting a constanttemperature, then the equations will simplify greatly. This case is particularly inter-esting for the analysis of insulated resonators, with or without variable cross-sections.Baks et al. [18] showed both theoretically and experimentally what kind of cooling pow-ers can be expected (in a pulse tube). In this section we will focus on deriving analyticexpressions for the pressure and velocity profiles in arbitrarily shaped channels.

Setting dT0/dX = 0, we find we have to solve

d(Rg〈u1〉)dX

= κa3Rg p1, (4.73a)

dp1

dX= κa5〈u1〉. (4.73b)

We can rewrite these equations into a reduced wave equation for the pressure

d2 p1

dX2+ C

dp1

dX+ k2 p1 = 0, C =

a5

Rg

d

dX

(Rg

a5

), (4.74)


where the complex wave number k is defined as

k = κ√−a3a5 =

κ

c

√1

1 − fν

(1 +

γ − 1

1 + εs

fk

).

The viscous dissipation due to interaction with the wall gives rise to its imaginary part.For wide tubes that have large Lautrec numbers it follows that 0 < −Im[k] ≪ 1 andthere will be little dissipation. When C and k are constant, equation (4.74) coincides withWebster’s horn equation [110, 149].

Both C and k depend on Rg in a rather complicated manner. However, for straightpores with constant Rg we find that k is constant and C = 0. It follows that the pressureand velocity can be written as

p1(X) = Ae−ikX + BeikX, (4.75a)

〈u1〉(X) =χ

ρ0c

(Ae−ikX − BeikX

), (4.75b)

with A and B integration constants and

χ =1

c

√

(1 − fν)

(1 +

γ− 1

1 +εs

fk

).

For variable cross-sections we cannot find an exact solution as above because C andk depend on Rg in a complicated manner. However, in view of the asymptotic behaviorof the hyperbolic tangent, it follows that fν, fk, and εs will not vary much in X, especially

for wide tubes, so that in good approximation we can replace k by its average value kand C by 1

Rg

ddXRg. This leads to the following equation for p1

d2 p1

dX2+

1

Rg

d

dX(Rg)

dp1

dX+ k2 p1 = 0. (4.76)

Next, for specific choices of the geometry, we can solve (4.76) analytically. For mostpractical applications Rg is either constant or linearly changing. If we put

Rg = a(X − X0),

for some a > 0 and X0 ∈ R, then we need to solve

d2 p1

dX2+

1

X − X0

dp1

dX+ k2 p1 = 0. (4.77)

After a transformation ξ = k(X − X0), we find that p1 solves the Bessel’s differentialequation

ξ2 d2 p1

dξ2+ξ

dp1

dξ+ξ2 p1 = 0.

The general solution is given in terms of the zeroth order Bessel functions of the first


and second kind.

p1(X) = AJ0[k(X − X0)] + BY0[k(X − X0)]. (4.78)

For several other geometries exact solutions can be obtained as well. For example, hy-perbolic or exponential horns occur frequently in resonators used for thermoacousticapplications (e.g. [142]). Suppose we put

Rg = ae2bX,

for some a > 0 and b ∈ R, then we need to solve

d2 p1

dX2+ 2b

dp1

dX+ k2 p1 = 0. (4.79)

The general solution is given by

p1(X) = e−aX(

A cos(kX) + B sin(kX))

. (4.80)

Alternatively, putting

Rg =1

X − X0

,

for some X0 ∈ R, we find an equation

d2 p1

dX2− 1

X − X0

dp1

dX+ k2 p1 = 0, (4.81)

which only differs from (4.77) by the minus-sign in the second term. The general solu-tion is thus given by

p1(X) = AJ0[−k(X − X0)] + BY0[−k(X − X0)]. (4.82)

4.4.2 Short-stack approximation

Many practical thermoacoustic devices have stacks of short length relative to the wave-length. One can take advantage of this property to find an approximate solution to (4.70)in the stack. The relevant dimensionless number for this analysis is the Helmholtz num-ber κ based on the length of the stack. We will expand the mean and acoustic variablesonce again in powers of κ and derive exact solutions for the perturbation variables. Letfi be one of the original perturbation variables, then we expand

fi = fi0 +κ fi1 + · · · , κ ≪ 1.

Furthermore, for illustration, we put the following boundary conditions:

X = 0 : T0 = TL, 〈u1〉 = uL, p1 = pL. (4.83)

We consider a steady-state situation H = 0, in which the thermoacoustic heat flow is


balanced by a return diffusive heat flow in the stack and in the gas. In addition, we as-sume there is no streaming, so that M = 0. Substituting the expansions into (4.70a), wefind to leading order that T00 is constant and (and thus also ρ00, a10, a20, a30, a40 and a50).Subsequently we find from (4.70b) and (4.70c) that p10 and Rg〈u10〉 must be constant aswell. It follows from (4.83) that

T00 = TL, 〈u10〉 =uL

Rg

, p10 = pL. (4.84)

Going one order higher in κ we find that T01, p11, and 〈u11〉 can be found from:

dT01

dX=

RgRe [a10 p10〈u10∗〉]

(K00Rg +σKs00Rs)S2

k − a20Rg|〈u10〉|2, (4.85a)

d(Rg〈u11〉)dX

= a30Rg p10 + a40Rg

dT01

dX〈u10〉, (4.85b)

dp11

dX= a50〈u10〉. (4.85c)

Integrating these equations, we find

T01(X) =∫ RgRe [a10 p10〈u10

∗〉](K00Rg +σKs00

Rs)S2k − a20Rg|〈u10〉|2

dX, (4.86a)

〈u11〉(X) =1

Rg

∫a30

(Rg p10 + a40Rg

dT01

dX〈u10〉

)dX, (4.86b)

p11(X) =∫

a50〈u10〉 dX. (4.86c)

The temperature difference, generated across the stack, is of order κ and thus the tem-perature difference will increase linearly with increasing stack length or frequency.

Continuing this way it is possible to determine the higher order terms in κ as well.In general we get for i = 0, 1, 2 . . .,

T0(i+1) =∫ [∑ j+k+l=i Re

[a1 j p1k〈u1l

∗〉]+ ∑ j+k+l=i+1

l≤i

aβ j|〈u1k〉|2 dT0l

dx

(K00Rg +σKs00Rs)S2

k − a20|〈u10〉|2

− ∑j+k=i+1

l≤i

(K0 jRg +σKs0 jRs)S2

k

(K00Rg +σKs00Rs)S2

k − a20|〈u10〉|2dT0k

dx

]dX, (4.87a)

〈u1(i+1)〉 =1

Rg

∫ [

∑j+k=i

a3 jRg p1k + ∑j+k+l=i

a4 jRg

dT0k

dX〈u1l〉

]dX, (4.87b)

p1(i+1) =∫

∑j+k=i

a5 j〈u1k〉 dX. (4.87c)

Consider a short parallel-plate stack placed at position xs in an acoustic standingwave with wave number k. For constant Rg all three expressions in (4.86) will be linear


in X. In particular we find that the temperature difference generated across the stack isgiven by

∆T0 =∫ 1

0

dT0

dXdX

=κRgRe [a10 p10〈u10

∗〉](K00Rg +σKs00

Rs)S2k − a20Rg|〈u10〉|2

+ o(κ), ifdRg

dX= 0. (4.88)

Note that a10, a20, K00 and Ks00are constant and are found from substitution of T00.

Putting p10 = P cos(kxs) and 〈u10〉 = −iP cos(kxs), we find that (4.88) transforms into

∆T0 =κRgIm [a10] |P|2 sin(2kxs)

2(K00Rg +σKs00Rs)S2

k − a20Rg|P|2|(1 − cos(2kxs))+ o(κ). (4.89)

This result can be seen as a generalization of the short-stack approximation per-formed by Wheatley et al [151], whose prediction is only valid for wide enough pores. In[151] and slightly modified in [11] a similar analysis was performed for a parallel-platestack positioned in an acoustic standing-wave. They solved the equations combininga short-stack approximation (κ ≪ 1) with a boundary-layer approximation (NL ≫ 1).Furthermore, only a leading-order solution was derived assuming constant pressureand velocity in the stack. The result is a similar, but not identical, expression for thetemperature difference developed across the stack,

∆T0 ∼1

8

p2ALsδk(1 +

√Pr)

ρre fg cre f (Kre f

s Rs + Kre fg Rs)(1 + Pr)

sin(2kxs)

×[

1 +1

8

p2Aδk(1 − Pr

√Pr)

ρre f Tre f

ω(Kre fs Rs + Kre f

g Rs)(γ− 1)(1 − P2r )

(1 − cos(2kxs))

]−1

. (4.90)

Numerical evaluations show that when NL ≥ 1 this expression agrees quite well withour approximation (4.89). Unsurprisingly, when NL ≪ 1 a discrepancy arises becauseof the underlying assumption of wide pores.

4.4.3 Approximate solution in short wide channels supporting a tem-perature gradient

In the previous sections we have found approximate solutions in the cases where ei-ther ∆T = 0 (section 4.4.1) or H = 0 (section 4.4.2). An approximate solution can alsobe found if a (large enough) nonzero temperature difference is imposed, provided thechannel is short and wide enough. One example of a tube that satisfies such require-ments is the thermal buffer tube that is commonly used in traveling-wave devices (seeChapter 7).

We start our analysis from the system of equations (4.70), with boundary conditions


X = 0 : T0 = TL, p1 = pL, 〈u1〉 = uL, (4.91a)

X = 1 : T0 = TR. (4.91b)

Without loss of generality we may assume TR > TL. Our first assumption is that a

considerable temperature difference is imposed across the tube, i.e. 0 ≪ TR−TL

TL= O(1).

Moreover, we assume the tube is short with respect to the wave length and wide withrespect to the penetration depths, so that κ ≪ 1 and NL ≫ 1. We will use the smallnessof κ and NL to derive a leading-order approximation of the temperature, pressure, andvelocity profiles.

Since NL ≫ 1, we find that to leading order fν = fk = 0 and

a1.= 1, a2

.= 0, a3

.= − i

ρ0c2, a4

.= 0, a5

.= − i

ρ0c2. (4.92)

Moreover, as TR ≫ TL, it is suggested by equation (4.70a) that H = O(κα), for someα ≤ −1. We therefore expand H2, M2, T0, p1, 〈u1〉 as

H2 = καH+ o(κα), κ ≪ 1, (4.93a)

M2 = κβM + o(κβ), κ ≪ 1, (4.93b)

p1 − pL = κγP + o(κγ), κ ≪ 1, (4.93c)

〈u1〉 − uL = κδU + o(κδ), κ ≪ 1, (4.93d)

T0 − TL = T + o(κ0), κ ≪ 1, (4.93e)

ρ0 = D + o(κ0), κ ≪ 1, (4.93f)

whereα, β, and δ are yet to be determined. Since c, Cp, K0, and Ks0depend only weakly

on T0 we can assume they are constant. Using the ideal gas law D can be expressed interms of T0. Substituting (4.92) and (4.93) into (4.70) we find to leading order

dTdX

= −2κα+1H− 2κβ+1CpMT −κRgRe

[(pL +κγP)(u∗

L +κδU ∗)]

(K0Rg +σKs0

Rs

)S2

k

, (4.94a)

d(RgU )

dX= −iκ1−δ Rg

Dc2(pL +κγP), (4.94b)

dPdX

= −iκ1−γD(uL +κδU ). (4.94c)

The exponents α, β, γ, and δ will now be determined by balancing the terms on eithersides of the equations in (4.94). It follows from (4.94b) and (4.94c) that the left and righthand side can only be balanced if γ = δ = 1. As a result equation (4.94a) becomes toleading order

dTdX

= −2κα+1H− 2κβ+1CpMT −κRgRe [pLu∗

L](K0Rg +σKs0

Rs

)S2

k

,


whose left and right hand side can only be balanced ifα = −1. Furthermore, it followsthat streaming will only affect the mean temperature if β ≤ −1. Putting α = β = −1,γ = δ = 1, we find we have to solve

dTdX

= −2H− 2MCpT(

K0Rg +σKs0Rs

)S2

k

, (4.95a)

d(RgU )

dX= −

iRg

Dc2pL, (4.95b)

dPdX

= −iDuL , (4.95c)

subject to

P(0) = 0, U (0) = 0, T (0) = 0, T (1) = TR − TL.

Integrating these equations, we find

T0(X) = TL −2H

g(X)

∫ X

0

g(ξ)(K0Rg(ξ) +σKs0

Rs(ξ))

S2k

dξ , (4.96a)

〈u1〉(X) = uL −iκpL

c2Rg(X)

∫ X

0

Rg(ξ)

D(ξ)dξ , (4.96b)

p1(X) = pL − iκuL

∫ X

0D(ξ) dξ , (4.96c)

H = (TR − TL)g(1)

2

∫ 1

0

g(ξ)(K0Rg(ξ) +σKs0

Rs(ξ))

S2k

dξ

−1

, (4.96d)

where

g(X) = exp

[−∫ X

0

2CpM(K0Rg(ξ) +σKs0

Rs(ξ))S2k

dξ

].

In particular, when we assume a straight tube, the expression for the temperatureprofile simplifies into

T0(X) = TL + (TR − TL)1 − eθX

1 − eθ, θ =

2CpM(K0Rg +σKs0

Rs)S2k

. (4.97)

We can now distinguish two limiting cases in which there is either a very large mass fluxor a very small mass flux. In the limit for M → ±∞ the temperature profile approaches


an almost discontinuous profile,

limM→−∞

T0(X; M) = limθ→−∞

TL + (TR − TL)1 − eθX

1 − eθ=

{TL, X = 0,

TR, 0 < X ≤ 1,(4.98a)

limM→∞

T0(X; M) = limθ→∞

TL + (TR − TL)1 − eθX

1 − eθ=

{TL, 0 ≤ X < 1,

TR, X = 1.(4.98b)

On the other hand, when there is little or no mass streaming we find a linear tempera-ture profile

limM→0

T0(X; M) = TL + (TR − TL)X. (4.99)

4.5 Acoustic streaming

This section discusses steady second-order mass flow in the stack driven by first-orderacoustic phenomena. The analysis is valid for arbitrarily shaped pores supporting atemperature gradient. Moreover, the temperature dependence of viscosity is taken intoaccount.

There are several types of streaming that can occur simultaneously. Three kindsof streaming are shown in Fig. 4.3. Gedeon streaming refers to a nett time-averagedmass flow through a stack pore, i.e. M 6= 0, typical for looped-tube thermoacousticdevices. Rayleigh streaming refers to a time-averaged toroidal circulation within a stackpore driven by boundary-layer effects at the pore walls that can occur even if M = 0.Inner streaming refers to a time-averaged toroidal circulation in the whole stack, so thatthe nett time-averaged mass flow can differ from pore to pore both in size and direction.Possible causes for inner streaming are inhomogeneities at the stack ends or asymmet-rical pores. Streaming effects are usually undesirable, but it was suggested in [135] thatfor some applications it can be useful as a substitute for heat exchangers.

(a) Gedeon streaming (b) Rayleigh streaming (c) Inner streaming

Figure 4.3: Three types of mass streaming in stack

We start with the continuity equation (4.1). If we time-average the equation andexpand in powers of Ma, then the zeroth and first order terms in Ma will drop out.

60 4.5 Acoustic streaming

Consequently we find to leading order

∂∂X

(ρ0u2,0

)+ ρ0

∂v2,0

∂y+

1

2Re

[∂

∂X(ρ1u∗

1) +∂

∂y(ρ1v∗

1)

]= 0. (4.100)

We can use this equation to express v2,0 in terms of u2,0 and known lower-order quanti-ties. Integrating over y, we find

v2,0 = − 1

ρ0

∫ y

−R−g

∂∂X

[ρ0u2,0 +

1

2Re (ρ1u∗

1)

]dy − 1

2ρ0

Re (ρ1v∗1) . (4.101)

We can then integrate (4.100) over a cross-section, while noting that v vanishes at theboundary y = ±Rg and also at the centerline y = 0, to obtain

d

dX

(Rgρ0〈u2,0〉 +

Rg

2Re [〈ρ1u∗

1〉])

= 0. (4.102)

The expression between the brackets is M2 the second-order time-averaged and cross-sectional-averaged mass flux in the X-direction. It follows that M2 is constant, whichis to be expected as there is no mass transport through the stack walls. We can nowexpress 〈u2,0〉 in terms of M2 and the first order acoustics as follows:

〈u2,0〉 =1

ρ0

(M2

Rg

− 1

2Re [〈ρ1u∗

1〉])

. (4.103)

Note that even when the average mass flux M = 0, there can still be a nonzero streamingvelocity as a result of first-order velocity and density variations.

Next we turn to equation (4.3). Expanding in powers of Ma and averaging in timewe find to leading order

∂p2,0

∂y= 0,

so that p2,0 = p2,0(X). Subsequently, time-averaging equation (4.2), we find to leadingorder

∂2u2,0

∂y2− W2

o

κµ0

dp2,0

dX= f , (4.104)

where f is a collection of products of lower-order terms given by

f :=1

2

W2o

κµ0

Re

[iκρ1u∗

1 + ρ0u∗1

∂u1

∂X+ ρ0v∗

1

∂u1

∂y− ρ0

Fr

bx

]− 1

2Re

[∂

∂y

(µ∗1

µ0

∂u1

∂y

)].

The first-order acoustics collected in f can be interpreted as a source term for the stream-ing on the left hand side, with the last term being characteristic for Rayleigh streaming.


Integrating (4.104) twice with respect to y, we can write

u2,0(X, y) = − W2o

2κµ0

dp2,0

dX((R+

g )2 − y2)−∫ R+

g

y

∫ y

0f (X, y) d y dy − C(R+

g − y). (4.105)

where we used that u2,0 vanishes on Γ+g . C is a constant of integration and will be

determined from the no-slip condition on Γ−g . Introducing F as the anti-derivative of f ,

F(X, y) :=∫ y

0f (X, y) d y,

we find

C = − W2o

2κµ0

dp2,0

dX

(R+

g −R−g

)− 〈F〉.

Substituting C into (4.105), we find that u2,0 is given by

u2,0(X, y) = − W2o

2κµ0

dp2,0

dX(R−

g + y)(R+g − y)−

∫ R+g

yF(X, y) dy + 〈F〉(R+

g − y). (4.106)

Computing the cross-sectional average we can relate dp2,0/dX to 〈u2,0〉 as follows:

dp2,0

dX= − 3κµ0

R2gW2

o

(〈u2,0〉 +

1

2Rg

∫ R+g

−R−g

∫ R+g

yF(X, y) dy dy +Rg〈F〉

). (4.107)

Summarizing, given the mass flux M2 and the first-order acoustics, 〈u2,0〉, dp2,0/dX,u2,0, and v2,0 can be determined consecutively from (4.103), (4.107), (4.105), and (4.101).

4.6 Second harmonics

The previous few sections have provided a recipe to determine all acoustic variables, themean temperature T0, and the streaming variables. In this section we will show how thesecond harmonics, the variables that oscillate at twice the fundamental frequency, canbe computed from the former variables.

Before we begin, we generalize the auxiliary functions from Section 4.2,

α j,2 :=√

2α j, j = ν, k, s. (4.108)

62 4.6 Second harmonics

Similarly we define (with Rg = 12 (R+

g +R−g ))

Fj,2 := 1 −sinh(α j,2R+

g ) + sinh(α j,2R−g )

sinh(2α j,2Rg)cosh(α j,2y),

+cosh(α j,2R+

g )− cosh(α j,2R−g )

sinh(2α j,2Rg)sin(α j,2y), j = ν, k, (4.109a)

F±k,2 := 1 −

sinh(αk,2R±g )

sinh(2αk,2Rg)cosh(αk,2y)±

cosh(αk,2R±g )

sinh(2αk,2Rg)sin(αk,2y), (4.109b)

F±s,2 := 1 − cosh(αs,2y±)

cosh(αs,2R±s )

, (4.109c)

and we introduce second-harmonic Rott’s functions

fν,2 := 1 − 〈Fν,2〉, fk,2 := 1 − 〈Fk,2〉, f ±k,2 := 1 − 〈F±k,2〉, f ±s,2 := 1 − 〈F±

s,2〉±s , (4.110)

and the stack heat capacity ratios

ε±s,2 :=

1

σ

K0Rgα2k,2 fk,2

Ks0R±

s α2s,2 f ±s,2

. (4.111)

Again we start from the momentum equation. Expanding the variables in powersof Ma according to (3.24), substituting the expansions into the y-component of the mo-

mentum equation (4.3), putting ε2 = ηM2a , and collecting terms of second order in Ma

we find∂p2,2

∂y= 0,

and we conclude p2,2 must be independent of y. Similarly, collecting all second-orderterms in Ma, we find that the X-component of the momentum equation (??) reduces to

2iκρ0u2,2 +1

2iκρ1u1 +

1

2ρ0u1

∂u1

∂X+

1

2ρ0v1

∂u1

∂y= −dp2,2

dX

+κµ0

W2o

[∂2u2,2

∂y2+

1

2

∂∂y

(µ1

µ0

∂u1

∂y

)], (4.112)

where we substituted (3.26) and (4.104). Rearranging terms we can rewrite (4.112) as

u2,2 −1

α2ν,2

∂2u2,2

∂y2=

i

2κρ0

dp2,2

dX+ A, (4.113)

where A, a source term arising from products of first-order terms, is known and givenby

A =i

4κρ0

{iκρ1u1 + ρ0u1

∂u1

∂X+ ρ0v1

∂u1

∂y− κµ0

W2o

∂∂y

(µ1

µ0

∂u1

∂y

)}.


Using the variation-of-constants formula [19], we can write a solution as follows:

u2,2 =iFν,2

2κρ0

dp2,2

dX+ Ψν(A), 〈u2,2〉 =

i(1 − fν,2)

2κρ0

dp2,2

dX+ψν(A), (4.114)

where Ψ j( f ) ( j = ν, k) is the variation-of-constants formula that is chosen such that

Ψ j( f )|y=±R±g

= 0,

Ψ j( f ) :=S j

α j,2

∫ y

−R−g

f (X,ζ)C j(X,ζ) dζ −C j

α j,2

∫ y

−R−g

f (X,ζ)S j(X,ζ) dζ

−2Rg

α j,2

(S+

j 〈 f C j〉 − C+j 〈 f S j〉

) S j − S−j C j/C−

j

S+j − S−

j C+j /C−

j

, (4.115a)

C j := cosh(α j,2y), C±j := cosh(α j,2R±

g ), (4.115b)

S j := sinh(α j,2y), S±j := sinh(α j,2R±

g ), (4.115c)

andψ j( f ) := 〈Ψ j( f )〉 . Furthermore, for the solid we define

Ψ±s ( f ) :=

Ss

αs,2

∫ y±

0f (X,ζ)Cs(X,ζ) dζ − Cs

αs,2

∫ y±

0f (X,ζ)Ss(X,ζ) dζ

+R±

s Cs

αs,2

⟨f

[Ss −

S±s

C±s

Cs

]⟩±

s

− η

αs,2

dR±t

dX

dT0

dX

[Ss −

S±s

C±s

Cs

], (4.116a)

Cs := cosh(αs,2y±), C±s := cosh(αs,2R±

s ), (4.116b)

Ss := sinh(αs,2y±), S±s := sinh(αs,2R±

s ), (4.116c)

satisfying Ψ±s ( f )|y±=R±

s= 0 and

∂Ψ±s ( f )

∂y±|y±=0 = −η dR±

t

dXdT0

dX . Additionally we introduce

the averaged functions ψ±s ( f ) := 〈Ψ±

s ( f )〉±s .Next we turn to the temperature equation. Substituting our expansions into (4.4) and

(4.5) and collecting the terms of second order in Ma, we find after some manipulationthat T2,2 and Ts2,2

can be found from

T2,2 −1

α2k,2

∂2T2,2

∂2 y= B − 1

4κ2ρ0

dT0

dX

dp2,2

dXFν,2 +

βT0

ρ0Cp

p2,2, (4.117a)

Ts2,2− 1

α2s,2

∂2Ts2,2

∂2 y= C, (4.117b)


where B and C are known and given by

4B = iβκT1 p1 + T0u1

dp1

dX− iκCpρ1T1 − 2ρ0CpΨν(A)

dT0

dX− ρ0u1

∂T1

∂X− ρ1u1

dT0

dX

− ρ0v1

∂T1

∂y+

κ

2N2L

∂∂y

(K1

∂T1

∂y

)+ 2η

d

dX

(K0

dT0

dX

)+κµ0

W2o

(∂u1

∂y

)2

,

4C =κφ

2N2s

[∂

∂y

(Ks1

∂Ts1

∂y

)+ 2η

d

dX

(Ks0

dT0

dX

)].

Using (4.115) and (4.116) and imposing the boundary conditions given in (4.7b) and(4.7d), we can write

T2,2 = Ψk(B) +βT0Fk,2

ρ0Cp

p2,2 −Fk,2 − PrFν,2

4κ2(1 − Pr)ρ0

dT0

dX

dp2,2

dX+ T+

b2,2(1 − F+

k,2)

+ T−b2,2

(1 − F−k,2), (4.118a)

T±s2,2

= Ψs(C) + T±b2,2

(1 − F±

s,2

), (4.118b)

where T+b2,2

= T2,2|Γ+g

and T−b2,2

= T2,2|Γ−g are yet to be determined. Collecting the second-

order terms in the remaining boundary condition (4.7c) and substituting (4.28), we canwrite

K0

∂T2,2

∂y±σKs0

∂Ts2,2

∂y± = −K0

(K1

K0

−Ks1

Ks0

)∂T1

∂y± η

∂R±g

∂X(K0 −σKs)

dT0

dX

if y = ±R±g and y± = R±

s . (4.119)

Applying these conditions, we find (cf. (4.33))

T±b2,2

=ε±s,2(1 + T ±

2,2)

1 + ε±s,2

(D± +

βT0

ρ0Cp

p2,2 −1

4κ2ρ0

1 − fν,2/ fk,2

1 − Pr

dT0

dX

dp2,2

dX

), (4.120)

where D± can be computed from the lower-order terms as

D± =1

K0Rgα2k,2 fk,2

[±η(K0 −σKs0

)dR±

g

dX

dT0

dX− 1

2K0

(K1

K0

−Ks1

Ks0

)∂T1

∂y

∣∣∣∣y=±R±

g

− K0

∂Ψk(B)

∂y

∣∣∣∣y=±R±

g

∓ σKs0

∂Ψ±s (C)

∂y±

∣∣∣∣y±=R±

s

]


and T ±2,2 is given by

T +2,2 =

−(ε+s,2 − ε−s,2) cosh(2αk,2Rg)

(1 + ε+s,2)(1 +ε−s,2 + (ε−s,2 − 1) cosh(2αk,2Rg) + (ε+

s,2 − ε−s,2) cosh(2αk,2Rg),

T −2,2 =

−(ε−s,2 − ε+s,2) cosh(2αk,2Rg)

(1 + ε−s,2)(1 +ε+s,2 + (ε+

s,2 − 1) cosh(2αk,2Rg) + (ε−s,2 − ε+s,2) cosh(2αk,2Rg)

.

Using relations (A.5) and (A.8) and substituting T2,2 we can derive the followingrelation for the second-harmonic density fluctuations:

ρ2,2 =1

c2

[1 + (γ− 1)(1 − Fk,2)

]p2,2 +

Fk,2 − PrFν,2

4κ2(1 − Pr)β

dT0

dX

dp2,2

dX

− ρ0βT+b2,2

(1 − F+k,2) − ρ0βT−

b2,2(1 − F−

k,2) − ρ0βΨk,B −1

2βρ1T1 (4.121)

=1

c2

[1 + (γ− 1)

(1 − Fk,2 −

ε+s,2(1 + T +

2,2)

1 + ε+s,2

(1 − F+k,2) −

ε−s,2(1 + T −

2,2)

1 +ε−s,2

(1 − F−k,2)

)]p2,2

+

[Fν,2 +

Fk,2 − Fν,2

(1 − Pr)+ε+

s,2(1 + T +2,2)

(1 − fν,2/ fk,2)(1 − F+k,2)

(1 − Pr)(1 + ε+s,2)

+ε−s,2(1 + T −2,2)

(1 − fν,2/ fk,2)(1 − F−k,2)

(1 − Pr)(1 +ε−s,2)

]β

4κ2

dT0

dX

dp2,2

dX

− ρ0β

(ε

+s,2D+

1 +ε+s,2

+ε−s,2D−

1 +ε−s,2

+ Ψk(B) +1

2

ρ1

ρ0

T1

). (4.122)

As a result we find

〈ρ2,2〉 =1

c2

[1 +

γ − 1

1 +εs,2

fk,2

]p2,2 +

[1 − fν,2 +

fν,2 − fk,2

(1 − Pr)(1 +εs,2)

]β

4κ2

dT0

dX

dp2,2

dX

− ρ0β

(ε

+s,2D+

1 +ε+s,2

+ε−s,2D−

1 +ε−s,2

+ψk(B) +1

2ρ0

〈ρ1T1〉)

, (4.123)

where εs,2 is given by

1

1 +εs,2

=1

2

(1 −ε+

s,2T +2,2

1 +ε+s,2

+1 −ε−s,2T −

2,2

1 +ε−s,2

)(4.124a)

= 1 − 1

2

(ε

+s,2(1 + T +

2,2)

1 +ε+s,2

+ε−s,2(1 + T −

2,2)

1 +ε−s,2

). (4.124b)

Finally, we turn to the continuity equation (4.1). Expanding the variables in powersof Ma and collecting terms of second order in Ma, we find

2iκρ2,2 +∂

∂X(ρ0u2,2) + ρ0

∂v2,2

∂y= − ∂

∂X(ρ1u1) −

∂∂y

(ρ1v1). (4.125)


We can use this equation to express v2,2 in terms of u2,2, ρ2,2, and the known lower-orderquantities,

v2,2 = − 1

ρ0

∫ y

0

(2iκρ2,2 +

∂∂X

(ρ0u2,2) +1

2

∂∂X

(ρ1u1)

)d y − ρ1

2ρ0

v1, (4.126)

where we applied the boundary condition that v2,2 vanishes at Γ−g .

Note that v also vanishes at the boundary Γ+g . Therefore, substituting (4.114) for u2,2,

multiplying with −2iκ, averaging over a cross-section, and using the Leibniz rule, weobtain the following equation as a consistency relation for v2,2:

4κ2〈ρ2,2〉 +1

Rg

d

dX

(Rg(1 − fν,2)

dp2,2

dX

)=

2iκ

Rg

∂∂X

(Rg [ρ0ψν(A) + 〈ρ1u1〉]).

After substituting (4.123), we obtain

4κ2

c2

[1 +

γ − 1

1 +εs,2

fk,2

]p2,2 +

fν,2 − fk,2

(1 − Pr)(1 + εs,2)β

dT0

dX

dp2,2

dX+ (1 − fν,2)β

dT0

dX

dp2,2

dX

+1

Rg

d

dX

(Rg(1 − fν,2)

dp2,2

dX

)= −2iκρ0E, (4.127)

where E is a source term arising from products of first-order or zeroth-order quantitiesand is given by

E := 2iκβ

(ε

+s,2D+

1 +ε+s,2

+ε−s,2D−

1 +ε−s,2

+ψk,B +1

2ρ0

〈ρ1T1〉)

− 1

ρ0Rg

∂∂X

(Rg [ρ0ψν(A) + 〈ρ1u1〉]

). (4.128)

Inserting (4.43), we obtain a wave equation for the second pressure harmonic

4κ2

c2

[1 +

γ − 1

1 +εs,2

fk,2

]p2,2 +

( fν,2 − fk,2)β

(1 − Pr)(1 + εs,2)

dT0

dX

dp2,2

dX

+ρ0

Rg

d

dX

(Rg

1 − fν,2

ρ0

dp2,2

dX

)= −2iκρ0E. (4.129)

Apart from the source term E this equation has a similar structure as the wave equationderived in (4.44) for the first pressure harmonic.

Combining (4.114) and (4.129) we derive the following two coupled ordinary differ-


ential equations for p2,2 and q2,2 := 〈u2,2〉 −ψν,A

dq2,2

dX= κa3,2 p2,2 +

(a4,2

dT0

dX− 1

Rg

dRg

dX

)q2,2 + E, (4.130a)

dp2,2

dX= κa5,2q2,2, (4.130b)

where

a3,2 := − 2i

ρ0c2

[1 +

(γ− 1) fk,2

1 + εs,2

], (4.131a)

a4,2 := − ( fν,2 − fk,2)β

(1 − Pr)(1 + εs,2)(1 − fν,2), (4.131b)

a5,2 := − 2iρ0

1 − fν,2

. (4.131c)

Since all the zeroth-order and first-order terms are known from the previous sections,we can compute subsequently A, B, C, D, and E. Next the system (4.130) can be inte-grated to determine p2,2 and 〈u2,2〉, provided appropriate boundary conditions are im-posed. Note that the gravitational effect only affects the streaming terms via the sourcefunction f in equation (4.104), and does not appear in the equations for the first andsecond harmonics.

4.7 Power

To analyze the performance of thermoacoustic systems it is important to have a clearunderstanding of the energy flows in the system and their interplay. In this section wewill elaborate further on the concept of total and acoustic power.

4.7.1 Acoustic power

The time-averaged acoustic power W2 is given by

W2 =1

2RgRe [p1〈u1

∗〉] . (4.132)

Thus it follows from (3.25) that the time-averaged acoustic power dW2 used or producedin a segment of length dX can be found from

dW2

dX=

d

dX

[Rg〈Re

[p1eiκt

]Re[〈u1〉eiκt

]〉]

. (4.133)

Using (3.25), we find to leading order

dW2

dX=

1

2

dRg

dXRe [p1〈u1

∗〉] +Rg

2Re

[p1

d〈u1∗〉

dX+ 〈u1

∗〉dp1

dX

]. (4.134)

68 4.7 Power

Substituting (4.45a) and (4.45b) into (4.134) we find

dW2

dX=

Rg

2

β

1 − Pr

dT0

dXRe

[f ∗k − f ∗ν

(1 − f ∗ν )(1 +ε∗s )p1〈u∗

1〉]−

Rg

2

κ(γ − 1)

ρ0c2Im

[− fk

1 + εs

]|p1|2

−Rg

2

κρ0Im [− fν]

|1 − fν|2|〈u1〉|2. (4.135)

The first term contains the temperature gradient dT0/dX and is called the sink or sourceterm. It will either absorb (refrigerator) or produce (prime mover) acoustic power de-pending on the magnitude of the temperature gradient along the stack. This term isthe unique contribution to thermoacoustics. The last two terms are the “viscous” and“thermal-relaxation” dissipation terms, respectively. These two terms arise due to theinteraction with the wall, and have a dissipative effect in thermoacoustics.

The sink/source term, which we define as Ws2 , is of greatest interest in thermoacous-

tic engines and refrigerators. For interpretation we will neglect viscosity and setσ = ∞,so that fν = Pr = εs = 0,

dWs2

dX=β

2

dT0

dX

(Re [ fk] Re [p1〈u1

∗〉] + Im [− fk] Im [p1〈u1∗〉])

. (4.136)

In a standing-wave system Im (p1〈u1∗〉) is large and therefore Im (− fk) is important.

Fig. 4.4 shows that the maximal value is attained for NL close to 1. In the case of atraveling-wave system Re (p1〈u1

∗〉) is large and Re ( fk) is important. Fig. 4.4 showsthat Re ( fk) reaches its maximal value for NL ≪ 1. This motivates why commonlystacks (NL ∼ 1) are used in standing-wave systems and regenerators (NL ≪ 1) intraveling-wave systems.

0 1 2 3 4 5 6−0.5

0

0.5

1

NL

fk

real partimaginary part

Figure 4.4: Real and imaginary part of fk, plotted as a function of the Lautrec number NL.

To test the significance of the dissipation terms we can investigate how (4.135) be-haves for small NL or Wo. To optimize the thermoacoustic effect one would like tomaximize the source term and minimize the dissipation terms. If we consider an ideal


parallel-plate stack with εs = 0, then

dW2

dX=

Rg

2

β

1 − Pr

dT0

dXRe

[f ∗k − f ∗ν(1 − f ∗ν )

p1〈u∗1〉]−

Rg

2

κ(γ − 1)

ρ0c2Im [− fk] |p1|2

−Rg

2

κρ0Im [− fν]

|1 − fν|2|〈u1〉|2. (4.137)

For small Wo and NL one can show that ( fk − fν)/(1 − fν) = O(1), Im( fk) = O(N2L),

Im( fν) = O(W2o ) and |1 − fν|2 = O(W4

o ). Therefore, it follows that the acoustic powerbehaves as

dW2

dX=

dWs2

dX− dWk

2

dX− dWν

2

dX

= O(1) − O(

N2L

)− O

(1

W2o

), Wo, NL ≪ 1,

(4.138)

where Wk2 and Wν

2 denote the thermal relaxation dissipation and viscous dissipation,respectively.

Unsurprisingly, equation (4.138) shows that the dissipation in a regenerator (NL ≪1, Wo ≪ 1) is dominated by viscous dissipation and in a stack (NL = O(1), Wo = O(1))by thermal relaxation dissipation. In a regenerator there is perfect thermal contact, butvery small pores and therefore viscous dissipation will be dominant. In a stack, on theother hand, there is imperfect thermal contact, but wider pores. Thus thermal relaxationdissipation is dominant here. Dissipation is usually undesirable, so NL should be chosencarefully.

4.7.2 Total power

In equation (4.66) we derived an expression for the total power H2,

H2 = M2Cp

(T0 − Tre f

)+

1

2RgRe

[p1u∗

1

(1 −βT0

fk − f ∗ν(1 + Pr)(1 + εs)(1 − f ∗ν )

)]

+Rg

ρ0Cp|〈u1〉|2

2κ(1 − Pr)|1 − fν|2dT0

dXIm

[f ∗ν +

(1 +εs fν/ fk)( fk − f ∗ν )

(1 + Pr)(1 + εs)

]

−(

K0Rg +σKs0Rs

) S2k

2κ

dT0

dX. (4.139)

Combining (4.53) and (4.54), we can write the total power as a sum of the acoustic powerW, the hydronamic entropy flux Q, the heat flow Qm due to a nett mass flux, and the

70 4.7 Power

heat flow Qloss due to conduction down a temperature gradient,

H2 =Rg

2Re [p1〈u∗

1〉] +1

2ρ0T0RgRe [〈s1u∗

1〉] + M2Cp(T0 − Tre f )

−(

K0Rg +σKs0Rs

) S2k

2κ

dT0

dX

=: W2 + Q2 + Qm,2 − Qloss,2. (4.140)

To illustrate the behavior of the thermoacoustic devices, idealized energy flows aredepicted in figure 4.5. The situation is ideal in the sense that heat conduction and massstreaming are neglected, so that Qm,2 = Qloss,2 = 0.

Figure 4.5(a) shows a refrigerator that is thermally insulated from the surroundingsexcept at the heat exchangers where heat is exchanged with the environment. On theleft acoustic power is supplied, possibly by means of a speaker or some other sourceof sound. Part of the acoustic power is used to sustain the standing or traveling waveagainst thermal and viscous dissipation, and part is used for the thermoacoustic effectto transport heat from the cold to the hot heat exchanger. This can be used for coolingat the the cold heat exchanger or heating at the hot heat exchanger.

Figure 4.5(b) tells a similar story for the prime mover. A large temperature differ-ence is imposed across the stack, by supplying heat at the hot heat exchanger, or byextracting heat at the cold heat exchanger. As a result acoustic power is generated, partof which is dissipated due to viscous interaction with the resonator wall. The nett resultcan then be used as a sound source for some external device, possibly even to drive athermoacoustic refrigerator [78].

If we apply the conservation of energy to a control volume surrounding each heatexchanger, then we can relate the cooling and heating power to the total power and theacoustic power as follows:

refrigerator:

{QC = −H,

QH = W − H,(4.141)

prime mover:

{QC = W + H,

QH = −H.(4.142)


TH

TC

W H

QH QC

TH TC

QH

QC

(a) Refrigerator

TH

TC

WH

QH QC

TH TC

QH

QC

(b) Prime mover

Figure 4.5: Schematic and idealized illustration of the total power H (solid line), the acousticpower W (dashed line) and the hydronamic energy flux Q (dotted line) in and around a parallel-plate stack, insulated everywhere except at the heat exchangers, positioned in (a) a thermoacous-tic refrigerator and (b) a thermoacoustic prime mover. In (a) a small temperature difference ismaintained across the stack and acoustic power is absorbed by the stack; in (b) a large temper-ature difference is maintained and acoustic power is produced. The discontinuities in H2 arisedue to heat transfer at the heat exchangers.

72 4.7 Power

Chapter 5

Thermoacoustics inthree-dimensional pores withslowly varying cross-section

Although a two-dimensional model can give a lot of information, there are many practi-cal devices that require a three-dimensional analysis because its resonator tube or stackpores have non-trivial three-dimensional shapes. These can be cylindrical, rectangular,or triangular pores, pin arrays, wired mesh, or some random porous material. In thischapter we consider shapes of the type shown in figure 5.1: narrow three-dimensionalpores with cross-sections that may vary slowly in longitudinal direction [61].

solid

gasrX

θ

Rg(X,θ)

Rs(X,θ)

Γg

Γt

Figure 5.1: Longitudinal cut of a three-dimensional pore with slowly varying cross-section. Atposition X and angle θ, the radius of the gas and solid is given by Rg(X,θ) and Rs(X,θ),respectively.

The analysis in this chapter differs from the analysis in chapter 4 in the sense thatthere is a nontrivial dependence on the third coordinate, which complicates the compu-tation of the transverse variations. It will be necessary to determine Green’s functionsfor the Poisson’s and modified Helmholtz equation. Moreover, two additional integralequations need to be solved, unless the boundary condition (3.23c) is replaced by thecondition of constant wall temperature, which is quite accurate for most purposes.

74 5.1 Acoustics

5.1 Acoustics

We follow an approach similar to that in the previous chapters. First we introduce someauxiliary functions. As in the two-dimensional case, we defineαν,αk andαs as follows:

αν = (1 + i)

√ρ0

2µ0

Wo, (5.1)

αk = (1 + i)

√ρ0

K0Cp

NL, (5.2)

αs = (1 + i)φ

√ρs0

Ks0Cs

Ns. (5.3)

Furthermore, we determine Fj ( j = ν, k) from

Fj −1

α2j

∇2τ Fj = 1 in Ag, (5.4a)

Fj = 0 on Γg, (5.4b)

and define f j := 1 − 〈Fj〉 ( j = ν, k). Note that for two-dimensional pores these defini-tions match the expressions given in the previous chapter.

Arnott et al. [5] follow a similar approach, although a slightly different notation isadopted; F(x;α j) in stead of Fj(x). Also there is an additional minus-sign in (5.4a) be-

cause they assume a time-dependence of the form eiκt, whereas we follow the conven-tional notation with a positive sign as used by Rott [115] and Swift [131].

We start from the momentum equation and expand the fluid variables in powers ofMa as shown in (3.24). Substituting the expansions into the transverse components of

the momentum equation (3.18), putting ε2 = ηM2a , and keeping terms up to first order

in Ma we find0 = −∇τ p0 − Ma∇τ p1 + o(Ma).

Collecting the leading-order terms in Ma we find we find that ∇τ p0 = 0, so that p0 is afunction of X only. Furthermore, collecting the first-order terms in Ma, we additionallyfind that ∇τ p1 = 0, so that p1 is also a function of X only.

Next we turn to the X-component of the momentum equation (3.17), which neglect-ing higher-order terms in Ma changes into

iMaκρ0u1 = −dp0

dX− Ma

dp1

dX+ Ma

κ

W2o

∇τ · (µ0∇τu1) + o(Ma).

To leading order we find that dp0/dX = 0 and therefore p0 is constant. Next assumethat the mean temperature T0 is a function of X only. Below we will show that thisis indeed the case. As a result we also find that µ0 = µ0(X) and K = K0(X). Then,collecting the terms of first order in Ma, we find that u1 satisfies

u1 =i

κρ0

dp1

dX+

1

α2ν

∇2τ u1. (5.5)

Thermoacoustics in three-dimensional pores with variable cross-section 75

With the help of equations (5.4), we can integrate (5.5) subject to v|Γg= 0 and write u1

and dp1/dX as

u1 =iFνκρ0

dp1

dXand 〈u1〉 =

i(1 − fν)

κρ0

dp1

dX. (5.6)

Next we turn to the temperature equation. Substituting our expansions into (3.20)and (3.21) and keeping terms up to first order in Ma, we find

Maρ0Cp (iκT1 + v1 ·∇T0) = iMaκβT0 p1 +κ

2N2L

∇τ ·

[K0∇τ (T0 + MaT1)

],

iMaρs0CsTs1

=φ

2

2N2s

∇τ ·

[Ks0

∇τ(Ts0

+ MaTs1

) ].

To leading order this reduces into ∇2τ T0 = ∇2

τ Ts0= 0. An obvious solution, satisfying

the boundary conditions given in (3.23), is that Ts0and T0 are equal and independent

of xτ . In view of the thermodynamic relation (A.8) it also holds that ρ0 and ρs0are

independent of xτ .Next, collecting the terms of first order in Ma, we find that T1 and Ts1

can be obtainedfrom

T1 +1

κ2ρ0

dT0

dX

dp1

dXFν −

βT0

ρ0Cp

p1 =1

α2k

∇2τ T1, (5.7a)

Ts1=

1

α2s

∇2τ Ts1

, (5.7b)

where we substituted expression (5.6) for u1.To solve the temperature from (5.7) we first need to introduce some additional aux-

iliary functions. In [5] a solution is obtained to (5.7) by assuming that the wall temper-ature Ts does not depend on time, which allows a solution as a combination of Fν andFk-functions. However, with the boundary conditions given in (3.23), this approach willnot work here. Assume for now the boundary function g := T1|Γg

is known and choose

gp and gu such that

g = gp

βT0

ρ0c2p1 −

gu

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX.

We can now write for the temperature

T1(x) =βT0Fkp

ρ0Cp

p1 −Fku − PrFν

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX, (5.8)

Ts1(x) =

βT0

ρ0Cp

(1 − Fsp)p1 −1 − Fsu

κ2(1 − Pr)ρ0

dT0

dX

dp1

dX, (5.9)

76 5.1 Acoustics

where Fk j ( j = p, u) satisfies

Fk j −1

α2k

∇2τ Fk j = 1 in Ag, (5.10a)

Fk j = g j on Γg, (5.10b)

and Fs j ( j = p, u) is found from

Fs j −1

α2s

∇2τ Fs j = 1 in As , (5.11a)

Fs j = 1 − g j on Γg, (5.11b)

∇τFs j · n′τ = 0 on Γt. (5.11c)

The standard way of solving such boundary value problems is making use of the Green’sfunctions for the given Helmholtz equations on a cross-section with appropriate bound-ary conditions. In Appendix C we will show how the g j and Fi j functions can be deter-mined using Green’s functions.

Using relations (A.5) and (A.8) and substituting T1 we can derive the following rela-tion for the acoustic density fluctuations:

ρ1 =1

c2

[γ− (γ− 1)Fkp

]p1 +

β(Fku − PrFν)

κ2(1 − Pr)

dT0

dX

dp1

dX. (5.12)

Finally, we turn to the continuity equation (3.16). Expanding the variables in powersof Ma and keeping terms up to first order in Ma, we find

iMaκρ1 + Ma

∂∂X

(ρ0u1) + Maρ0∇τ · vτ1= 0. (5.13)

Next we substitute (5.6). First note that because of the divergence theorem

∫

Ag

∇τ · vτ dS =∫

Γg

vτ · n dℓ = 0,

since v|Γg= 0. Therefore, averaging (5.13) over a cross-section and multiplying with

−iκ, we obtain the following equation as a consistency relation for v1:

κ2〈ρ1〉 = − 1

Ag

d

dX

(Ag(1 − fν)

dp1

dX

)− iκ

Ag

dAg

dXρ0u1

∣∣∣Γg

= − 1

Ag

d

dX

(Ag(1 − fν)

dp1

dX

). (5.14)


After substituting (5.12) and putting f j = 1 − 〈Fj〉 ( j = ν, kp, ku), we obtain

κ2

c2

[1 + (γ− 1) fkp

]p1 +

β( fν − fku)

1 − Pr

dT0

dX

dp1

dX+β(1 − fν)

dT0

dX

dp1

dX

+1

Ag

d

dX

(Ag(1 − fν)

dp1

dX

)= 0.

Eventually using

ρ0

d

dX

(1

ρ0

)= − 1

ρ0

dρ0

dX= − 1

ρ0

∂ρ0

∂T0

dT0

dX= β

dT0

dX, (5.15)

we obtain the dimensionless equivalent of Rott’s wave equation for general porous me-dia

κ2

c2

[1 + (γ− 1) fkp

]p1 +

β( fν − fku)

1 − Pr

dT0

dX

dp1

dX+ρ0

Ag

d

dX

(Ag

1 − fνρ0

dp1

dX

)= 0. (5.16)

We can now combine (5.6) and (5.16) to find a coupled system of first order differen-tial equations for p1 and 〈u1〉. From (5.6) we find

ρ0

Ag

d

dX

(Ag

1 − fνρ0

dp1

dX

)= −iκ

ρ0

Ag

d

dX(Ag〈u1〉). (5.17)

Substituting this result into (5.16) and repeating equation (5.6), we find that 〈u1〉 and p1

are found from

d〈u1〉dX

= κa3 p1 +

(a4

dT0

dX− 1

Ag

dAg

dX

)〈〈u1〉〉, (5.18a)

dp1

dX= κa5〈u1〉, (5.18b)

where

a3 := − i

ρ0c2

[1 + (γ − 1) fkp

],

a4 := − β( fν − fku)

(1 − Pr)(1 − fν),

a5 := − iρ0

1 − fν.

To complete the system of equations, it still remains to find an equation for the meantemperature T0. The next section explains how this can be done.


5.2 Mean temperature

As in Section 4.3 we will use the conservation of energy to determine T0 as a consistencycondition for the second-order temperature T2,0.

Putting ε2 = ηM2a , we can rewrite (3.22) as follows:

κ∂∂t

(1

2ρ(|u|2 + ηM2

a |vτ |2) + ρǫ

)

= − ∂∂X

(1

2ρu(|u|2 + ηM2

a |vτ |2) + ρuh − M2a

κηK

2N2L

∂T

∂X

)

−∇τ ·

(1

2ρvτ (|u|2 + ηM2

a |vτ |2) + ρvτh − κK

2N2L

∇τT − κµ

W2o

u∇τu

)

+∇ · T +M2

a

Fr

ρ (ubx +√ηMavτ · bτ) , (5.19)

with

T = M2aηκ

µ

W2o

(u

∂u

∂X+ ηM2

a vτ ·∂vτ∂X

+ v · ∇u

)+

ζ

W2ζ

u∇ · v

µ

W2o

(vτ · ∇τvτ + v · ∇vτ

)+

ζ

W2ζ

vτ∇ · v

.

If there is an average heat flux F to the boundary, then the left hand side of this equationwill reduce to F after averaging in time. Consequently, on expanding in powers of Ma

and keeping terms up to second order, we can neglect gravitational terms and the T -term and find

∂∂X

[M2

aρ0u1h1 + M2a h0m2 − M2

a

κηK0

2N2L

dT0

dX

]+∇τ ·

[M2

aρ0vτ1h1 + M2

a h0mτ2

−M2a

κK0

2N2L

∇τT2,0 − M2a

κ

2N2L

K1∇τT1 − M2a

κµ0

W2o

u1∇τu1

]= M2

a F2 + o(M2a),

where m2 and mτ2are the components in longitudinal and transverse directions of the

second-order time-averaged mass flux m2,

m2 := ρ0(v2,0 + Re[

v2,2e2iκt]) + Re

[ρ1eiκt

]Re[v1eiκt

]= ρ0v2,0 +

1

2Re [ρ1v∗

1] .

Plugging in relation (3.25) and rearranging terms we find

K0∇2τ T2,0 +

1

2∇τ · Re [K∗

1∇τT1] + ηd

dX

(K0

dT0

dX

)=

N2L

κ

∂∂X

(ρ0Re [u∗1h1] + 2h0m2)

+N2

L

κ∇τ ·

(ρ0Re

[v∗τ1

h1

]+ 2h0mτ2

− κ

W2o

Re [u∗1∇τu1]

)− 2N2

L

κF2. (5.20)


Similarly we can show in the solid that (3.21) reduces to

Ks0∇2τ Ts2,0

+1

2∇τ · Re

[K∗

s1∇τTs1

]+ η

d

dX

(Ks0

dT0

dX

)= 0. (5.21)

We will now use the flux condition (3.23c) to derive a differential equation for T0.Time-averaging and expanding (3.23c) in powers of Ma and collecting the second orderterms in Ma we find

(K0∇τT2,0 +

1

2Re [K∗

1∇τT1]−σKs0∇τTs2,0

−σ 1

2Re[K∗

s1∇τTs1

])· ∇τSg

= η(σKs0

− K0

) dT0

dX

∂Sg

∂X, Sg = 0.

This condition can be rewritten as

(K0∇τT2,0 +

1

2Re [K∗


−σ 1

2Re[K∗

s1∇τTs1

])· nτ

= ηK0 −σKs0

|∇τSg|∂Rg

∂X

dT0

dX= η

(K0 −σKs0)Rg√

R2g + (∂Rg/∂θ)2

∂Rg

∂X

dT0

dX, Sg = 0, (5.22)

where nτ := ∇τSg/(∇τSg) is the outward unit normal vector to Γg. Similarly, since

n′τ := ∇τSt/(∇τSt) is the outward unit normal vector to Γt, we find from (3.23d)

Ks0∇Ts2,0

· n′τ +

1

2∇τ · Re

[K∗

s1∇τTs1

]=

ηKs0Rt√

R2t + (∂Rt/∂θ)2

∂Rt

∂X

dT0

dX, St = 0. (5.23)

Now on the one hand, by applying the divergence theorem, substituting the flux condi-

tions (5.22) and (5.23), and noting that Aα=12

∫ 2π0 R2 dθ and dℓ = (R2 +(∂R/∂θ)2)1/2dθ,


we find

∫

Ag

(K0∇2

τ T2,0 +1

2∇τ · Re [K∗

1∇τT1] + ηd

dX

(K0

dT0

dX

))dS

+σ∫

As

(Ks0

∇2τ Ts2,0

+1

2∇τ · Re

[K∗

s1∇τTs1

]+ η

d

dX

(Ks0

dT0

dX

))dS

=∫

Γg

(K0∇τT2,0 +

1

2Re [K∗


− σ2

Re[K∗

s1∇τTs1

])· nτ dℓ

+σ∫

Γt

(Ks0

∇Ts2,0+

1

2Re[K∗

s1∇τTs1

])· n′

τ dℓ

+ ηAg

d

dX

(K0

dT0

dX

)+ ησAs

d

dX

(Ks0

dT0

dX

)

= η(K0 −σKs0)

dT0

dX

∫ 2π

0Rg

∂Rg

∂Xdθ+ ησKs0

dT0

dX

∫ 2π

0Rt

∂Rt

∂Xdθ

+ ηAg

d

dX

(K0

dT0

dX

)+ ησAs

d

dX

(Ks0

dT0

dX

)

= ηd

dX

[(K0 Ag +σKs0

As

) dT0

dX

]. (5.24)

On the other hand, combining (5.20) and (5.21), applying the divergence theoremand using v|Γg

= 0, we also have

∫

Ag

(K0∇2

τ T2,0 +1

2∇τ · Re [K∗

1∇τT1] + ηd

dX

(K0

dT0

dX

))dS

+σ∫

As

(Ks0

∇2τ Ts2,0

+1

2∇τ · Re

[K∗

s1∇τTs1

]+ η

d

dX

(Ks0

dT0

dX

))dS

=N2

L

κ

∫

Ag

d

dX

(ρ0Re [u∗

1h1] + 2h0m2

)dS

+N2

L

κ

∫

Ag

[∇τ ·

(ρ0Re [v∗

1h1] + 2h0mτ2− κµ0

W2o

Re [u∗1∇τu1]

)− 2N2

L

κF2

]dS

=N2

L

κ

d

dX

(Agρ0Re [〈u∗

1h1〉] + 2h0 M2

)− 2

dH2

dX, (5.25)

where we put M2 := Ag〈m2〉 and F =: 1Ag

dHdX , . Finally, equating the right hand sides of

equations (5.24) and (5.25), we get

d

dX

(2h0 M2 + Agρ0Re [〈h1u∗

1〉]− (K0 Ag +σKs0As)

S2k

κ

dT0

dX

)=

dH2

dX. (5.26)

The quantity within the big brackets is H2, the time-averaged total power (or energyflux) along X. In steady state, for a cyclic refrigerator or prime mover without heatflows to the surroundings (F = 0), H will be constant.


Substituting the thermodynamic expressions (A.7) and (A.9),

dh = Tds +1

ρdp = CpdT +

1

ρ(1 −βT)dp,

into (5.26), we find

H2

Ag

= h0

M2

Ag

+1

2ρ0CpRe [〈T1u∗

1〉] +1

2(1 −βT0)Re [p1〈u∗

1〉]

−(

K0 +σKs0

As

Ag

)S2

k

2κ

dT0

dX. (5.27)

Next we will show that

〈|Fν|2〉 = Re [〈Fν〉] , 〈Fk jF∗ν 〉 =

〈Fk j〉 + Pr〈F∗ν 〉 − fb j

1 + Pr

, j = u, p, (5.28)

where

fb j :=1

Agα2ν

∫

Γg

Fk j∇τF∗ν · nτ dℓ, j = u, p.

DefineI1 := 〈Fk jF

∗ν 〉, I2 := 〈∇τFk j · ∇τF∗

ν 〉. (5.29)

It follows from (5.10) and (5.4) that

Fν = 1 +1

α2ν

∇2τ Fν , (5.30)

Fk j = 1 +1

α2k

∇2τ Fk j, j = u, p. (5.31)

Using (5.31), the divergence theorem, and the boundary condition Fν = 0 on Γg, we canrewrite I1 as follows:

I1 =1

Ag

∫

Ag

F∗ν

(1 +

1

α2k

∇2τ Fk j

)dS

= 〈F∗ν 〉 +

1

Agα2k

∫

Ag

∇τ ·

(F∗ν∇τFk j

)dS − 1

Agα2k

∫

Ag

∇τF∗ν · ∇τFk j dS

= 〈F∗ν 〉 +

1

Agα2k

∫

Γg

(F∗ν∇τFk j

)· nτ dℓ− I2

α2k

= 〈F∗ν 〉 −

I2

α2k

. (5.32)


In the same way, substituting (5.30) rather than (5.31) and noting (α∗ν)

2 = −α2ν, we find

I1 = 〈Fk j〉 − fb j +I2

α2ν

, (5.33)

where the extra term fb j arises because Fk j does not necessarily vanish on the interfaceΓg. Eliminating the common term I2 from equations (5.32) and (5.33), we find

I1 =〈Fk j〉+ Pr〈F∗

ν 〉 − fb j

1 + Pr

, (5.34)

as claimed in (5.28). Replacing Fk j by Fν, we can repeat the same analysis to prove thefirst claim of (5.28).

Integrating (A.9), we also find that

h0 = Cp(T0 − Tre f ),

where Tre f is some reference temperature. Substituting (5.8) into (5.27) and using (5.28)we find after some manipulation

H2

Ag

=1

Ag

M2Cp

(T0 − Tre f

)+

1

2Re

[p1u∗

1

(1 −βT0

fkp − f ∗ν + fbp

(1 + Pr)(1 − f ∗ν )

)]

+ρ0Cp|〈u1〉|2

2κ(1 − Pr)|1 − fν|2dT0

dXIm

[f ∗ν +

fku − f ∗ν + fbu

1 + Pr

]

−(

K0 +σKs0

As

Ag

)S2

k

2κ

dT0

dX. (5.35)

This expression represents the total power along the X-direction (wave direction) interms of T0, p1, 〈u1〉, M2, material properties and geometry. Given H2, independent ofX, we can solve (5.35) for dT0/dX,

dT0

dX= κ

2H2 − 2M2Cp

(T0 − Tre f

)− AgRe [a1 p1u∗

1]

Aga2|〈u1〉|2 −(

K0 Ag +σKs0As

)S2

k

, (5.36)

where

a1 := 1 −βT0


(1 + Pr)(1 − f ∗ν ), (5.37)

a2 :=ρ0Cp

(1 − Pr)|1 − fν|2Im

[f ∗ν +


1 + Pr

]. (5.38)

Equation (5.36) together with equations (5.18b) and (5.18a) form a complete coupled sys-tem of differential equations for T0, 〈u1〉, and p1. This system gives a generalization ofthe Swift equations [135], for arbitrary three-dimensional slowly varying cross-sections.

It is possible to go a step further and improve the expression for the mean temper-


ature by determining the correction term T2,0. Integrating (5.20) and (5.21) using anappropriate Green’s function, we can determine T2,0 and Ts2,0

up to some X-dependent

function Tb2,0(= Tb2,0

(X)). However, before we can determine Tb2,0, we need to com-

pute the second and third harmonics, as Tb2,0follows as a solvability condition for the

fourth-order mean temperatures T4,0 and Ts4,0.

5.3 Integration of the generalized Swift equations

In equations (5.18a), (5.18b) and (5.36) we have derived a coupled system of differentialequations for the mean temperature T0, the acoustic velocity 〈u1〉, and the acoustic pres-sure p1. It follows that that all acoustic variables can be determined after integration of

dT0

dX= κ

2H2 − 2M2Cp

(T0 − Tre f

)− AgRe [a1 p1〈u1〉∗]

Aga2|〈u1〉|2 −(

K0 Ag +σKs0As

)S2

k

, (5.39a)

d〈u1〉dX

= κa3 p1 +

(a4

dT0

dX− 1

Ag

dAg

dX

)〈u1〉, (5.39b)

dp1

dX= κa5〈u1〉, (5.39c)

with

a1 := 1 −βT0


(1 + Pr)(1 − f ∗ν ), (5.40a)

a2 :=ρ0Cp

(1 − Pr)|1 − fν|2Im

[f ∗ν +


1 + Pr

], (5.40b)

a3 := − i

ρ0c2

[1 + (γ − 1) fkp

], (5.40c)

a4 := − β( fν − fku)

(1 − Pr)(1 − fν), (5.40d)

a5 := − iρ0

1 − fν. (5.40e)

Equations (5.39)form a system of five coupled equations, determining the five real vari-ables: Re(p1), Im(p1), Re(〈u1〉), Im(〈u1〉), and T0. Given the total energy flux H2, themass flux M2, the geometry, and appropriate boundary conditions in X, these equationscan be integrated numerically.

This system of equations has a similar structure as the one we obtained for two-dimensional pores in (4.70). The difference lies in the a j functions that are defined ina slightly different way. The analytic solutions obtained in Sections 4.4.1 and 4.4.2 cantherefore be generalized to three-dimensional pores straightforwardly, by replacing thea j functions by those given above. It is true that (5.40) is more difficult to compute than(4.71), but below we will discuss two cases for which the a j functions simplify greatly.


5.3.1 Ideal stack

A stack is considered ideal if the pore walls are of sufficiently high heat capacity andthermal conductivity, so that the wall temperature is locally unaffected by the acoustictemperature variations in the gas. In short, an ideal stack has Ts1

= 0. Although inreality stacks are almost never ideal, they can get close under certain conditions. Forexample in figure 4.2 we showed that a parallel-plate stack can approach an ideal stack(with εs = 0) provided the stack plates are thick enough.

If we put the assumption of Ts1= 0 into our boundary conditions, then it follows

that gu = gp = 0, so that Fsu = Fsp = 1, Fkp = Fku = Fk and fbu = fbp = 0. As a result thea j functions can be expressed as

a1 := 1 −βT0

fk − f ∗ν(1 + Pr)(1 − f ∗ν )

,

a2 :=ρ0Cp

(1 − Pr)|1 − fν|2Im

[f ∗ν +

fk − f ∗ν1 + Pr

],

a3 := − i

ρ0c2[1 + (γ− 1) fk] ,

a4 := − β( fν − fk)

(1 − Pr)(1 − fν),

a5 := − iρ0

1 − fν.

These expressions match the a j functions obtained for two-dimensional pores in (4.71)in case εs = 0.

5.3.2 Rotationally symmetric pores

Another simplification arises if we consider pores that are rotationally symmetric, suchas cylindrical or conical pores. Then the acoustic temperature fluctuations at the solid/gasinterface will only depend on X, so that the gi functions can be computed as indicatedin (C.11).

Putting

εs :=1

σ

K0 Agαk tanh(αk)

Ks0Asαs tanh(αs)

,


we can show that the a j functions can be expressed in the familiar form

a1 := 1 −βT0

fk − f ∗ν(1 + Pr)(1 + εs)(1 − f ∗ν )

,

a2 :=ρ0Cp|〈u1〉|2

(1 − Pr)|1 − fν|2Im

[f ∗ν +

(1 + εs fν/ fk)( fk − f ∗ν )

(1 + Pr)(1 +εs)

],

a3 := − i

ρ0c2

[1 +

(γ − 1) fk

1 +εs

],

a4 := − ( fν − fk)β

(1 − Pr)(1 + εs)(1 − fν),

a5 := − iρ0

1 − fν.

5.4 Acoustic streaming

In this section we generalize the analysis, derived first in Section 4.5 for two-dimensionalpores, to three-dimensional pores. We discuss steady second-order mass flow in thestack driven by first-order acoustic phenomena. The analysis is valid for arbitraryslowly varying pores supporting a temperature gradient. Moreover, the temperaturedependence of viscosity is taken into account.

We start with the continuity equation (3.16). If we time-average the equation andexpand in powers of Ma, then the zeroth and first order terms in Ma will drop out.Consequently we find to leading order

∂∂X

(ρ0u2,0

)+ ρ0∇τ · vτ2,0

+1

2Re

[∂

∂X(ρ1u∗

1) +∇τ ·

(ρ1v∗

τ1

)]= 0.

Again applying the divergence theorem and noting that vτ |Γg= 0, we can average over

a cross-section to find

d

dX

(Agρ0〈u2,0〉 +

Ag

2Re [〈ρ1u∗

1〉])

= 0.

The expression between the brackets is M2 the time-averaged and cross-sectional-averagedmass flux in the X-direction. It follows that M2 is constant, which is to be expected asthere is no mass transport through the stack walls. We can now express 〈u2,0〉 in terms

of M2 and the first order acoustics as follows:

〈u2,0〉 =1

ρ0

(M2

Ag

− 1

2Re [〈ρ1u∗

1〉])

. (5.41)

Next we turn to equation (3.18). Expanding in powers of Ma and ε and averaging intime we find to leading order in Ma and ε that

∇τ p2,0 = 0,


so that p2,0 = p2,0(X). Subsequently, time-averaging equation (3.17), we find to leadingorder

∇2τ u2,0 −

W2o

κµ0

dp2,0

dX= f , (5.42)

where f is given by

f :=1

2

W2o

κµ0

Re

[iκρ1u∗

1 + ρ0v∗1 · ∇u1 −

ρ0

Fr

bx

]− 1

2Re

[∇τ ·

(µ∗1

µ0

∇τu1

)].

The first-order acoustics collected in f can be interpreted as a source term for the stream-ing on the left hand side, with the last term being characteristic for Rayleigh streaming.We can also see this as a Poisson’s equation for the streaming velocity u2,0, which maybe solved using a Green’s function. Introducing the Green’s function Gm that, for fixedxτ ∈ Ag(X), satisfies

∇τGm(x; x) = δ(xτ − xτ), xτ ∈ Ag(X), (5.43)

Gm(x; x) = 0, Sg(x) = 0, (5.44)

we can write

u2,0(x) =W2

o

κµ0

dp2,0

dX

∫

Ag(X)Gm(x; x) dS +

∫

Ag(X)Gm(x; x) f (x) dS. (5.45)

Computing the cross-sectional average we can relate dpm20/dX to 〈um,20〉 as follows:

dp2,0

dX=κµ0

W2o

〈u2,0〉 − 〈∫AgGm(x; x) f (x) dS〉

〈∫AgGm(x; x) dS〉

. (5.46)

Summarizing, given the mass flux M2 and the first-order acoustics, it only remainsto compute the Green’s function Gm for the desired geometry. Then 〈u2,0〉, dp2,0/dXand u2,0 can be determined consecutively from (5.41), (5.46) and (5.45).

5.5 Second harmonics

We start with generalizing the auxiliary functions given in (4.9), (4.17), and (4.18)

α j,2 :=√

2α j, j = ν, k, s. (5.47)

Furthermore, for given source function f , we introduce functions Ψ j( f ) that satisfy

Ψ j,2 −1

α2j,2

∇2τ Ψ j,2 = f in Ag, (5.48a)

Ψ j,2 = 0 on Γg. (5.48b)


We expand all variables in powers of Ma according to (3.24), and substitute theexpansions into the transverse component of the momentum equation (3.18). Putting

ε2 = ηM2

a , and collecting terms of second order in Ma we find

∇τ p2,2 = 0,

and we conclude p2,2 must be independent of xτ . Similarly, collecting all second-orderterms in Ma, we find that the X-component of the momentum equation (3.17) reducesinto

2iκρ0u2,2 +1

2iκρ1u1 +

1

2ρ0u1

∂u1

∂X+

1

2ρ0vτ1

· ∇τu1 = −dp2,2

dX

+κµ0

W2o

[∇2τ u2,2 +

1

2∇τ ·

(µ1

µ0

∇τu1

)], (5.49)

where we substituted (3.26). Note that the steady component from this equation waschosen such that it vanishes. Rearranging terms we can rewrite (5.49) as

u2,2 −1

α2ν,2

∇2τ u2,2 =

i

2κρ0

dp2,2

dX+ A, (5.50)

where A, a source term arising from products of first-order terms, is known and givenby

A =i

4κρ0

{iκρ1u1 + ρ0u1

∂u1

∂X+ ρ0vτ1

· ∇τu1 −κµ0

W2o

∇τ ·

(µ1

µ0

∇τu1

)}.

The solution, satisfying the no-slip condition, can be written as

u2,2 =iFν,2

2κρ0

dp2,2

dX+ Ψν,2(A), 〈u2,2〉 =

i(1 − fν,2)

2κρ0

dp2,2

dX+ψν,2(A). (5.51)

Next we turn to the temperature equation. Substituting our expansions into (3.21)and (3.22) and collecting the terms of second order in Ma, we find after some manipula-tion that T2,2 and Ts2,2

can be found from

T2,2 −1

α2k,2

∇2τ T2,2 = B − 1

4κ2ρ0

dT0

dX

dp2,2

dXFν,2 +

βT0

ρ0Cp

p2,2, (5.52a)

Ts2,2− 1

α2s,2

∇2τ Ts2,2

= C, (5.52b)


where B and C are known and given by

4B = iβκT1 p1 + T0u1

dp1

dX− iκCpρ1T1 − 2ρ0CpΨν,2(A)

dT0

dX− ρ0u1

∂T1

∂X− ρ1u1

dT0

dX

− ρ0vτ1· ∇τT1 +

κ

2N2L

∇τ · (K1∇τT1) + 2ηd

dX

(K0

dT0

dX

)+κµ0

W2o

|∇τu1|2 ,

4C =κφ

2N2s

[∇τ ·

(Ks1

∇τTs1

)+ 2η

d

dX

(Ks0

dT0

dX

)].

As in Section 5.1 we denote the temperature on the boundary by g, in particular g2,2 :=T2,2|Γg

, and we write

g2,2 = gp,2

βT0

ρ0c2p2,2 −

gu,2

4κ2(1 − Pr)ρ0

dT0

dX

dp2,2

dX.

Using (5.48) and imposing the boundary conditions given in (3.23), we find

T2,2 = Ψk,2(B) +βT0

ρ0Cp

Fkp,2p2,2 −Fku,2 − PrFν,2

4κ2(1 − Pr)ρ0

dT0

dX

dp2,2

dX, (5.53a)

Ts2,2= Ψs,2(C) +

βT0

ρ0Cp

(1 − Fsp,2)p2,2 −1 − Fsu,2

4κ2(1 − Pr)ρ0

dT0

dX

dp2,2

dX, (5.53b)

where Fk j,2 ( j = p, u) satisfies

Fk j,2 −1

α2j,2

∇2τ Fk j,2 = 1 in Ag, (5.54a)

Fk j,2 = g j,2 on Γg, (5.54b)

and Fs j,2 ( j = p, u) is found from

Fs j,2 −1

α2s,2

∇2τ Fs j,2 = 1 in As, (5.55a)

Fs j,2 = 1 − g j,2 on Γg, (5.55b)

∇τFs j,2 · n′τ = 0 on Γt. (5.55c)

Using relations (A.5) and (A.8) and substituting T2,2 we can derive the following relationfor the second-harmonic density fluctuations:

ρ2,2 =1

c2

[1 + (γ − 1)(1 − Fkp,2)

]p2,2 − ρ0β

(Ψk,B +

1

2

ρ1

ρ0

T1

)

+

[Fν,2 +

Fku,2 − Fν,2

1 − Pr

]β

4κ2

dT0

dX

dp2,2

dX. (5.56)


As a result we find

〈ρ2,2〉 =1

c2

[1 + (γ− 1) fkp,2

]p2,2 +

[1 − fν,2 +

fν,2 − fku,2

1 − Pr

]β

4κ2

dT0

dX

dp2,2

dX

− ρ0β

(ψk,B +

1

2ρ0

〈ρ1T1〉)

. (5.57)

Finally, we turn to the continuity equation (3.16). Expanding the variables in powersof Ma and collecting terms of second order in Ma we find

2iκρ2,2 +∂

∂X(ρ0u2,2) + ρ0∇τ · vτ2,2

= − ∂∂X

(ρ1u1) −∇τ · (ρ1vτ1). (5.58)

We know that v vanishes at Γg. Therefore, substituting (5.51) for u2,2, multiplying with−iκ, and averaging over a cross-section, we obtain the following equation as a consis-tency relation for v2,2:

4κ2〈ρ2,2〉 +1

Ag

d

dX

(Ag(1 − fν,2)

dp2,2

dX

)=

2iκ

Ag

∂∂X

(Ag

[〈ρ1u1〉 + ρ0ψν,2(A)

]).

After substituting (5.57), we obtain

4κ2

c2

[1 + (γ− 1) fkp,2

]p2,2 +

fν,2 − fku,2

1 − Pr

βdT0

dX

dp2,2

dX+ (1 − fν,2)β

dT0

dX

dp2,2

dX

+1

Ag

d

dX

(Ag(1 − fν,2)

dp2,2

dX

)= −2iκρ0E, (5.59)

where E is a source term arising from products of first-order or zeroth-order quantitiesand is given by

E := 2iκβ

(ψk,B +

1

2ρ0

〈ρ1T1〉)− 1

ρ0 Ag

∂∂X

(Ag

[ρ0ψν,2(A) + 〈ρ1u1〉

]).

Inserting (4.43), we obtain a wave equation for the second pressure harmonic

4κ2

c2

[1 + (γ− 1) fkp,2

]p2,2 +

( fν,2 − fku,2)β

1 − Pr

dT0

dX

dp2,2

dX

+ρ0

Ag

d

dX

(Ag

1 − fν,2

ρ0

dp2,2

dX

)= −2iκρ0E. (5.60)

Apart from the source term E this equation has a similar structure as the wave equationderived in (4.44) for the first pressure harmonic.

Combining (5.51) and (5.60) we derive the following two coupled ordinary differen-

90 5.6 Power

tial equations for p2,2 and q2,2 := 〈u2,2〉 −ψν,2(A)

dq2,2

dX= κa3,2 p2,2 +

(a4,2

dT0

dX− 1

Rg

dRg

dX

)q2,2 + E, (5.61a)

dp2,2

dX= κa5,2q2,2, (5.61b)

where

a3,2 := − 2i

ρ0c2

[1 + (γ − 1) fkp,2

], (5.62a)

a4,2 := − ( fν,2 − fku,2)β

(1 − Pr)(1 − fν,2), (5.62b)

a5,2 := − 2iρ0

1 − fν,2

. (5.62c)

Since all the zeroth-order and first-order terms are given by the equations in the pre-vious sections, we can compute subsequently A, B, C, and E. Moreover, Appendix Cshows how the Fj and Ψ j functions can be computed for a given cross-sectional geom-

etry. Having done this, we can integrate the system (5.61) to determine p2,2 and 〈u2,2〉,provided appropriate boundary conditions are imposed.

5.6 Power

In Section 5.2 we derived the following expression for the total power H:

H2

Ag

=1

Ag

M2Cp

(T0 − Tre f

)+

1

2Re

[p1u∗

1

(1 −βT0


(1 + Pr)(1 − f ∗ν )

)]

+ρ0Cp|〈u1〉|2

2κ(1 − Pr)|1 − fν|2dT0

dXIm

[f ∗ν +


1 + Pr

]

−(

K0 +σKs0

As

Ag

)S2

k

2κ

dT0

dX. (5.63)

As mentioned in Section 4.7.2, the total power can be written as a sum of the acousticpower W, the hydrodynamic entropy flux Q, the heat flow Qm due to a nett mass flux,and the heat flow Qloss due to conduction down a temperature gradient. For the time-averaged acoustic power dW2 used or produced in a segment of length dX, secondorder in Ma, we can write

dW2

dX=

d

dX

[Ag〈Re

[p1eiκt

]Re[〈u1〉eiκt

]〉]

. (5.64)


Using (3.25), we find to leading order

dW2

dX=

1

2

dAg

dXRe [p1〈u1

∗〉] +Ag

2Re

[p1

d〈u1∗〉

dX+ 〈u1

∗〉dp1

dX

]. (5.65)

Substituting (5.18b) and (5.18a) into (5.65) we find

dW2

dX=

Ag

2

β

1 − Pr

dT0

dXRe

[f ∗ku − f ∗ν(1 − f ∗ν )

p1〈u∗1〉]−

Ag

2

κ(γ − 1)

ρ0c2Im[− fkp

]|p1|2

−Ag

2

κρ0Im [− fν]

|1 − fν|2|〈u1〉|2. (5.66)

92 5.6 Power

Chapter 6

Standing-wave devices

In this chapter we present the results from our computations on standing-wave devices.We will simulate both refrigerators and prime movers and where possible we will com-pare our numerical results to analytic approximations and experimental data. The in-fluence of geometry parameters, material and other parameters such as drive ratio andmean pressure is investigated. The performance of the machines is tested in terms ofthe temperature difference, power-output, and efficiencies.

6.1 Design

We consider two kinds of standing-wave devices: the thermoacoustic refrigerator andthe thermoacoustic prime mover. As depicted in figure 6.1, they are modeled in threeparts: an acoustically resonant tube, containing a gas, a parallel-plate stack, and twoheat exchangers. In case of a thermoacoustic refrigerator a driver is attached to oneend, and the other end is closed. In case of a prime mover, one end will be closedand one will be open. When designing a thermoacoustic system several parametersare important, related to the choice of material, working gas, geometry, and operatingconditions. Below we will discuss a few of these parameters that are important in ourcomputations.

Stack position

Perhaps the most important design parameter is the position of the stack. Without astack, a standing-wave would be maintained in the resonator, with velocity nodes at theclosed ends and pressure nodes at the open ends. With a stack, the standing-wave willbe altered, but if the stack is short enough, then the standing wave will not be perturbedappreciably, as we saw in our derivation of the short stack approximation in Section4.4.2. The distance between the stack center and the closed end is denoted by xs. Usingthe wave number k = 2π/λ, we can introduce a dimensionless number kxs that relatesthe stack to its position in the acoustic wave. Since the power output is proportional tothe product p1〈u1〉, it follows that the stack has to be positioned somewhere between apressure and a velocity node.

94 6.1 Design

xs

sound

QH QC

TH TC

(a) Standing-wave refrigerator

xs

sound

QH QC

TH TC

(b) Standing-wave prime mover

Figure 6.1: Schematic model of (a) thermoacoustic refrigerator and (b) standing-wave ther-moacoustic prime mover, illustrated with the standing-wave pressure and velocity profiles. Themodel consists of an acoustically resonant tube filled with gas, a stack of parallel plates positionedat distance xs from the closed end, and two heat exchangers. In (a) we have a half-wave-lengthresonator attached to a speaker and in (b) we have a quarter-wave-length resonator that suppliessound to the exhaust.

Stack length

As mentioned previously the stack length is vital in the design of a thermoacoustic stack.First of all, the stack should be kept short with respect to the wave length, so that theacoustic field is not significantly altered. Furthermore, as the stack length is increased,the viscous and thermal dissipation will increase as well, reducing the efficiency sig-nificantly. On the other hand, if the stack length is increased then more heat can bepumped by the stack, so that higher temperature differences and larger power-outputcan be obtained. The optimal stack length is obtained by balancing these effects.

Plate separation

In standing-wave systems it is beneficial to have pores with a radius of one or morethermal penetration depths, i.e. NL ≥ 1. This is necessary to create the optimal phasingbetween pressure and velocity and create an optimal heat-shuttling effect. It was shownin equations (4.136) and (4.138) that (in an idealized situation) the highest acoustic pow-ers are obtained if NL ∼ 1.

Standing-wave devices 95

Plate material

The term Qloss in the expression for the total power (4.140) represents the heat conduc-tion through gas and plate material and it is a loss term; it has a negative effect on theperformance. To reduce the effect of this term a material must be chosen with a lowthermal conductivity. On the other hand, we must take a heat capacity Cs larger thanthe heat capacity Cp of the working gas, so that the plate temperature can be consideredsteady (εs = 0).

Drive ratio

The drive ratio Dr is defined as the ratio between the dynamic pressure amplitude pA

(at the closed end) and the mean pressure p0,

Dr =pA

p0

,

and is a measure for the amplitude of the sound field. In general one is interested inhigh drive ratios because this leads to larger power outputs and larger temperature dif-ferences. However, at high drive ratios nonlinear effects will start to become importantthat may degrade the performance. Turbulence may occur, but also shock waves due tointeraction of the higher harmonics.

6.2 Computations

We have implemented the system (4.70) for the configuration shown in figure 6.1(a).There are three regions for which a solution has to be computed: the resonator, the heatexchangers, and the stack. Each region requires a different approach.

Resonator

We consider an insulated resonator, that is kept at a constant temperature, without heatleaks to the environment. Although in reality the resonator will be cylindrical we willmodel it as straight two-dimensional channel to simplify the calculations. In Section4.4.1 it was shown that for straight two-dimensional pores the exact solution looks asfollows:

p1(X) = Ae−ikrX + BeikrX , (6.1a)

〈u1〉(X) =χr

ρ0c

(Ae−ikrX − BeikrX

), (6.1b)

with A and B integration constants and

χr =1

c

√

(1 − fν)

(1 +

γ − 1

1 + εs

fk

), (6.2)

kr = κr

√−a3a5 =κ

c

√1

1 − fν

(1 +

γ − 1

1 +εs

fk

), (6.3)

96 6.3 A thermoacoustic couple

where χr and kr are computed from the resonator properties. The viscous dissipation inthe resonator is accounted for as the wave number kr will have a small imaginary part.

Heat exchangers

The heat exchangers are used to exchange heat with the environment. In our compu-tations they are modeled as infinitely short with temperature TH at the hot end andtemperature TC at the cold end.

Stack

Inside the stack there will be a nontrivial temperature difference and the full system ofequations (4.70) needs to be solved. For this standard ode-solvers from MatLab can beused. The result can be compared to the short-stack approximation derived in Section4.4.2, provided the stack is short enough and H = 0.

Boundary and interface conditions

To fix all the integration constants, boundary conditions have to be imposed. We startour computations at the closed end, where we give the pressure amplitude and imposezero velocity. At the interface with the stack we impose continuity of mass and momen-tum, which gives continuity of pressure p1 and volumetric velocity Ag〈u1〉.

We still need boundary conditions for the temperature. There are two constants ofintegration and so we can choose to fix the temperature at the heat exchangers on eitherside of the stack. Alternatively, one can also impose the temperature on one side and acooling (or heating) power, and the temperature on the other side will follow.

Lastly, we mention that typically there is no time-averaged mass flux M in the sys-tem. With a nonzero M there would be an accumulation or depletion of mass at theclosed end, which is not physically realistic. In the looped geometries that we considerin Chapter 7, nonzero mass fluxes may be present.

6.3 A thermoacoustic couple

The simplest type of thermoacoustic devices is the so-called thermoacoustic couple(TAC), a stack of short parallel plates without heat exchangers positioned in a half-wave-length resonator, with a driver providing a standing-wave. It is of the type shownin figure 6.1(a), but without heat exchangers.

Due to its simplicity a thermoacoustic couple is perfect for a quantitative under-standing of the basic thermoacoustic effect. There are no heat exchangers, but a steady-state temperature profile is developed across the stack from a balance of the thermoa-coustic heat flow (W + Q) by a return diffusive heat flow (Qloss) in the stack and in thegas. The resulting energy flux through a stack pore will be equal to zero (H = 0).

The first measurements of thermoacoustic couples were performed by Wheatleyet al. [151], who measured the temperature difference developed across the stack asa function of its position kxs for drive ratios of approximately 0.3%. Additionally,they derived a theoretical prediction (see (4.90)) for the temperature difference, usinga boundary-layer and short-stack approximation, which was later slightly modified by


Atchley et al. [11]. Atchley et al also extended the measurements to higher driver ratiosof up to 2%. We will validate our numerical code by comparison against these measure-ments.

6.3.1 Acoustically generated temperature differences

We consider the thermoacoustic couple TAC #3 as described in [11]. It comprises aparallel-plate stack placed in helium-filled resonator with plates that are a laminationof 302 stainless steel and G-10 fiberglass, epoxied together. All relevant parameters aregiven in Table 6.1.

Parameter Symbol Value Unit

total power H 0 Wmass flux M 0 kg m/sspeed of sound c 1020 m/sisobaric specific heat Cp 5190 J/(kg K)

isochoric specific heat Cv 3110 J/(kg K)

plate separation dg 1.52·10−3 m

thickness fiberglass d f g 1.02·10−4 m

thickness stainless steel dss 8.89·10−5 m

thickness plate ds = d f g + dss 1.91·10−4 m

frequency f 696 Hzthermal conductivity Helium Kg 0.16 W/(m K)

thermal conductivity fiberglass K f g 0.48 W/(m K)

thermal conductivity stainless steel Kss 11.8 W/(m K)

thermal conductivity plate Ks =d f g K f g+dss Kss

ds5.76 W/(m K)

stack length Ls 6.85·10−3 m

average pressure p0 1.14·105 PaPrandtl number Pr 0.68 -ambient temperature TL 298 K

Table 6.1: Specifications for TAC #3

Using the procedure given in Section 6.2, we have calculated numerically the ex-pected temperature differences developed across the stack in case H = 0 for variousdrive ratios. In figure 6.2 we plot the temperature difference across the stack as a func-tion of kxs, the relative position of the stack center for drive ratios of 0.28 and 1.99%.The numerical outcome is compared with the measurements of [11], the short-stack ap-proximation (4.89) and the short-stack/boundary-layer approximation (4.90).

For this configuration we have that κ ∼ 0.015 ≪ 1 and NL ∼ 2.75. It is therefore nosurprise that the short-stack approximation and the numerics agree very well. Further-more, the match with (4.90) is remarkably good, even though NL is not much greaterthan 1. The match with the measurements is not bad either, but for high drive ratiosthe fit seems to become worse; both the numerics and the theoretical predictions over-predict the measured values. In [11] this is attributed to uncertainties in the thermalconductivity of the stack material or possible measurement errors.

98 6.3 A thermoacoustic couple

0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.3

−0.2

−0.1

0

0.1

0.2

0.3

tem

pera

ture

diff

eren

ce (

K)

numericsshort stack approx.Wheatley et al.Atchley et al.

kxs

(a) Dr = 0.28%

0.5 1 1.5 2 2.5 3 3.5 4 4.5−6

−4

−2

0

2

4

6

tem

pera

ture

diff

eren

ce (

K)

numericsshort stack approx.Wheatley et al.Atchley et al.

kxs

(b) Dr = 1.99%

Figure 6.2: The temperature difference across the stack as a function of the relative position ofthe stack center. The solid line shows the results from the numerics, the dashed line shows theshort-stack approximation (4.89), the squares shows the values predicted by Wheatley et al. , andthe triangles show the measurements of [11].

In an article by Piccolo et al. [99], a simplified numerical model was derived for de-scribing time-average transverse heat transfer near the edges of a thermally isolatedthermoacoustic stack at low Mach numbers. The difference with the approach used herelies in the inclusion of a y-dependence in the mean temperature and various simplify-ing assumptions such as constant pressure and velocity inside the stack. This modelwas compared to the experimental data of Atchley et al. [11] and to what Picollo et al.call standard thermoacoustic theory. The latter is an approximation similar to the ap-proximation of Wheatley et al. [151]. Figure 6.3 show a close-up of figure 6.2(b) near themaximal temperature difference and includes the results of [99].

1,5 2 2,5 30

2

4

6

tem

pe

ratu

re d

iffe

ren

ce

(K

)

7

kxs

Figure 6.3: Close-up of figure 6.2(b) near the maximal value for Dr = 1.99%. The results arecompared to those of Piccolo et al [99].


To show the effect of increasing drive ratio we can compute for various drive ra-tios the temperature difference numerically. In figure 6.4 the results are compared tothose obtained experimentally in [11] for various drive ratios between 0.17% and 1.99%.Comparing the results, we again observe reasonable agreement, with larger deviationsfor high drive ratios. Both in figure 6.4(a) and figure 6.4(b) the temperature differenceprogresses from a perfect sinusoid for small amplitudes to a sawtooth curve for largedrive ratios with extremes shifting towards the pressure antinodes.

0.5 1.5 2.5 3.5 4.5−6

−4

−2

0

2

4

6

tem

per

atu

red

iffe

ren

ce(K

)

kxs

(a) Numerics

tem

per

atu

red

iffe

ren

ce(K

)

kxs

(b) Measurements

Figure 6.4: The (a) computed and (b) measured temperature difference across the stack as afunction of the relative position of the stack for drive ratios ranging between 0.17 and 1.99%.

Finally we also look at the impact of the stack on the velocity and pressure field inthe tube. Fig. 6.5 shows the pressure and velocity amplitude throughout the tube withthe stack positioned at kxs = π/4 together with the pressure and velocity amplitudesin the absence of the stack. The plot shows that the pressure and velocity are hardlyeffected by the stack because it is so short. Note the discontinuities in the velocity at thestack ends which arise to maintain the same volumetric mass flux, as the cross-sectionalarea of the gas is smaller in the stack than in the resonator.

6.3.2 Acoustic power

Since a thermoacoustic couple does not have heat exchangers, the heating and coolingpower will be equal to zero and the coefficient of performance has no meaning. How-ever, the acoustic power will be nonzero and is necessary to obtain the steady-statetemperature profile. In Section 4.7.1 we showed that the acoustic power produced orabsorbed in a segment of length dX is composed of a source/sink term, a viscous dissi-pation term, and a thermal relaxation term,

dW2

dX=

dWs2

dX− dWk

2

dX− dWν

2

dX

= O(1) − O(

N2L

)− O

(1

W2o

), Wo, NL ≪ 1.

(6.4)

100 6.4 A standing-wave refrigerator

0 0

500

1000

1500

2000

2500

p 1 (P

a)

with stackwithout stack

ππ/4 π/2 3π/4

kxs

(a) pressure amplitude

0 0

2

4

6

8

10

12

14

p 1 (P

a)

with stackwithout stack

ππ/4 π/2 3π/4

kxs

(b) velocity amplitude

Figure 6.5: (a) The absolute value of the acoustic pressure and (b) the absolute value of thecrosswise-averaged velocity throughout the tube both without a stack and with a stack positionedat kxs = π

4 . The drive ratio is 1.99 %.

To test this equation for a real application, we consider the thermoacoustic couple asintroduced in the previous section. We put the stack at 5 cm from the closed end, applya drive ratio of 2%, and vary NL. The remaining parameters are chosen as in Table 6.1.In figure 6.6 we have plotted the acoustic power ∆W2 absorbed by the stack togetherwith its source term and dissipation components.

Looking at the graph of the thermal relaxation dissipation ∆Wk2 , we indeed observe

that ∆Wk2 tends to zero for decreasing NL, but only until NL ∼ 0.1. Below this value∆Wk

2

starts to grow rapidly again because ∆Wk2 scales with |p1|2. As the pore size becomes

smaller and smaller, the pressure drop (and also the velocity) in the stack will becomelarger and larger, canceling the effect of the prefactor Im( fk). For the viscous dissipation

∆Wν2 the situation is simpler. Both |u1|2 and its prefactor will explode for small NL, and

therefore ∆Wν2 too, as the graph clearly shows. The graph also shows that the source

term is maximal for NL close to 1. Below this value the viscous dissipation increasesdramatically and therefore, for the case considered here, NL should not be taken smallerthan 1. This confirms the choice of NL ≥ 1 that is commonly applied in standing-wavedevices.

6.4 A standing-wave refrigerator

The aim of this section is to study how a standard thermoacoustic refrigerator as de-picted in 6.1(a) behaves in terms of parameters such as cooling power, drive ratio, tem-perature difference, mean pressure, stack position, stack length, and coefficients of per-formance. In particular we are interested in what happens at large amplitudes, goingup to drive ratios of 40%. In all likelihood drive ratios of 40% are too high to obtain veryaccurate predictions of the performance of the refrigerator, since nonlinearities will se-riously start to affect the performance. Furthermore, there are all kinds of practicalconsiderations and limitations that prevent us from achieving these high amplitudes.Still these computations can give some quantitative insight into the behavior of the re-frigerator and it shows how the refrigerator would work under ideal circumstances.


0 0.5 1 1.5 2−100

0

100

200

300

400

acou

stic

pow

er (

W/m

2 )

NL

acoustic powersourcetherm. dissipationvisc. dissipation

Figure 6.6: The acoustic power per unit area absorbed by the thermoacoustic couple as a func-tion of the Lautrec number for H = 0 and a drive ratio of 2%. The stack is positioned at 5cmfrom the closed end.

The cooling and heating power depend on the position of the stack inside the res-onator. In figure 6.7 we show a detailed description of all the energy flows in and aroundthe stack depending on whether the stack is placed near the closed end or near thedriver. In both cases the hot end will face the nearest end of the resonator and the totalpower H through the stack will be constant and directed from the cold to the hot end.

When the stack is positioned near the driver, we assume that the driver suppliesacoustic power Wdriver which is reduced to WH due to viscous interaction with the res-onator wall. Most of WH is then absorbed by the stack, and the remaining amount WC

will be dissipated by the resonator wall left of the cold end. Alternatively, when thestack is positioned near the closed end, we again assume that the driver supplies acous-tic power Wdriver which is reduced to WC by dissipation in the resonator. Most of WC

is then absorbed by the stack, and the remaining amount WH will be dissipated by theresonator left of the hot end. In either case, we have the following expressions for thecooling power extracted at the cold end and the heating power supplied to the hot end,

QC = H − WC , QH = H − WH . (6.5)

The performance of the refrigerator will be measured by the (relative) coefficient of per-formance

COP =QC

W, COPR =

COP

COPC≤ 1, COPC =

TC

TH − TC

. (6.6)

To compute the stack efficiency we can substitute W = WC − WH and to compute therefrigerator efficiency we can put W = Wdriver. Due to wall attenuation and the secondlaw of thermodynamics we have that 0 ≤ COPstack ≤ COPdriver ≤ COPC.

Our numerical calculations are motivated by experiments performed at Shell byAraujo [4], where a simple air-filled standing-wave refrigerator with a ceramic stack


xs

WH HH WC

QH QC

TH TC

(a) Stack near closed end

xs

WC HH WH

QHQC

THTC

(b) Stack near driver

Figure 6.7: Schematic model of the energy flows in a thermoacoustic refrigerator with a stackpositioned (a) near the closed end or (b) near the driver. The hot end always faces the nearest endof the resonator. Heating power QH is supplied to the hot end and cooling power is extractedfrom the cold end. Acoustic power WC − WH is absorbed by the stack.

was tested. To simplify calculations we assume a parallel-plate structure for the stackas opposed to the honey-comb stack tested by Araujo, but with the same porosity. Formost of our computations we have kept the stack center fixed at 2.5 cm from the closedend, to maximize the COPR. All other relevant parameters are given in Table 6.2.


mass flux M 0 kg m/s

speed of sound c 375 m/s

isobaric specific heat Cp 1000 J/(kg K)

isochoric specific heat Cv 714 J/(kg K)

tube diameter db 0.073 m

pore diameter dg 3.7·10−4 m

plate diameter ds 6·10−5 m

frequency f 393 Hz

thermal conductivity gas Kg 0.0262 W/(m K)

thermal conductivity plate Ks 2 W/(m K)

stack length Ls 0.03 m

average pressure p0 1·105 Pa

temperature hot heat exchanger TH 370 K

distance between stack centre and closed end xs 0.025 m

Table 6.2: Parameters for the standing-wave refrigerator tested in [4].

In all figures shown below the left column shows the result for a mean pressure of1 bar and the right column shows the result when the mean pressure is increased to 20bar, while keeping NL and Br constant.


0 0.5 1 1.5 2

x 104

0

0.2

0.4

0.6

0.8

1

1.2

CO

P

cooling power (W/m2)

D=5%D=10%D=20%D=30%D=40%

(a) p0 = 1 bar

0 1 2 3 4

x 105

0

0.2

0.4

0.6

0.8

1

1.2

CO

P


D=5%D=10%D=20%D=30%D=40%

(b) p0 = 20 bar

0 25 50 75 100 1250

0.5

1

1.5

CO

P

temperature difference (K)

D=5%D=10%D=20%D=30%D=40%

(c) p0 = 1 bar

0 25 50 75 100 1250

0.5

1

1.5

CO

P


D=5%D=10%D=20%D=30%D=40%

(d) p0 = 20 bar

Figure 6.8: The coefficient of power COP as a function of the temperature difference TH − TC orcooling power QC. The calculations are repeated for various drive ratios and for mean pressuresequal to 1 or 20 bar.

In figure 6.8 we have plotted the COP as a function of both the cooling power andthe temperature difference for various drive ratios. The plots show that for increasingcooling power, the COP becomes larger. With the cooling power increasing, the temper-ature difference across the stack naturally becomes smaller, and a peak COP is attainedfor a temperature difference equal to zero. Furthermore, it is noticed that for increasingdrive ratio the COP increases as well. However, plots 6.8(c) and 6.8(d) show that beyond30% for p0 = 1 bar and beyond 10% for p0 = 20 bar, increasing the drive ratio has littleeffect, because the temperature drop approaches the critical temperature difference. Itfollows that the critical temperature difference is approximately 115 K for both p0 = 1bar and p0 = 20 bar. Swift [135] predicts for the critical temperature gradient,

∇Tcrit =ω

ρ0cp

p1

〈u1〉, (6.7)

which gives a temperature difference of approximately 247 K. We conclude that thisprediction, based on an inviscid approach, overpredicts the numerical result. Finally,we note that going from 1 bar to 20 bar, the cooling power needs to be increased byroughly a factor 20 as well, to obtain the same temperature difference.

In figure 6.9 we have repeated the previous calculations for the relative coefficientof performance COPR and the same conclusions can be drawn. However, in contrast


to the COP, the COPR has a peak value, which appears to converge to a fixed valuefor increasing drive ratio. The peak for 20 bar is only slightly higher than for 1 bar.However, at a mean pressure of 1 bar a higher drive ratio are required to approach thepeak. Note also that the two zeros of the COPR occur where the cooling power andtemperature difference equal zero.

0 2 4 6 8 10

x 104

0

0.05

0.1

0.15

0.2

0.25

0.3

CO

PR


D=5%D=10%D=20%D=30%D=40%

(a) p0 = 1 bar

0 0.5 1 1.5 2

x 106

0

0.05

0.1

0.15

0.2

0.25

0.3

CO

PR


D=5%D=10%D=20%D=30%D=40%

(b) p0 = 20 bar

0 25 50 75 100 1250

0.05

0.1

0.15

0.2

0.25

0.3

CO

PR


D=5%D=10%D=20%D=30%D=40%

(c) p0 = 1 bar

0 25 50 75 100 1250

0.05

0.1

0.15

0.2

0.25

0.3

CO

PR


D=5%D=10%D=20%D=30%D=40%

(d) p0 = 20 bar

Figure 6.9: The relative coefficient of power COPR as a function of the temperature differenceTH − TC or cooling power QC. The calculations are repeated for various drive ratios and formean pressures equal to 1 or 20 bar.

In figure 6.9 we observed that the peak COPR-value appears to converge to a fixedvalue. In figure 6.10 we examine this by plotting for each drive ratio the optimal valueof the coefficients of performance, i.e. the values of the COP and COPR for which theCOPR attains its peak value. Both graphs seem to converge to the same fixed value forhigh drive ratios. Particularly for 20 bar the convergence is quite fast.

In the previous calculations the stack was fixed at 2.5 cm from the closed end. Thelocation of the stack can be optimized to maximize the efficiency. In figure 6.11 we varythe position of the stack for a drive ratio of 30% and mean pressures of 1 and 20 bar.We choose this drive ratio to ensure that the COPRopt approaches its maximal value (seefigure 6.10). The resulting graphs for 1 and 20 bar are almost overlapping. Furthermoreboth reach their peak value near kxs = 0.16, which corresponds with a stack centerpositioned at 2.5 cm from the closed end. This is in fact the case considered in all otherplots in this section.


0 0.1 0.2 0.3 0.40.86

0.88

0.9

0.92

0.94

0.96

CO

Pop

t

drive ratio

p0 = 1 bar

p0 = 20 bar

(a)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

0.3

CO

PR

opt

drive ratio

p0 = 1 bar

p0 = 20 bar

(b)

Figure 6.10: The optimal coefficients of performance COPopt and COPRopt, corresponding tothe peak COPR-values, as a function of the drive ratio. The calculations are performed for meanpressures equal to 1 or 20 bar.

0 0.5 1 1.50

0.5

1

1.5

2

CO

Pop

t

kXs

p0 = 1 bar

p0 = 20 bar

(a)

0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

0.3

CO

PR

opt

kXs

p0 = 1 bar

p0 = 20 bar

(b)

Figure 6.11: The optimal coefficients of performance COPopt and COPRopt, corresponding tothe peak COPR-values, as a function of the scaled stack position. The calculations are performedfor a drive ratio of 30% and mean pressures equal to 1 and 20 bar.

Figure 6.12 shows how the cooling power, temperature difference and drive ratio inthe stack depend on each other. We again observe the same critical temperature differ-ences of approximately 115 K, where the effect of the drive ratio is minimal, which isreflected in the nodes where all the lines cross. In both cases the corresponding cooling

power is approximately -1800 W/m2, which is caused almost completely by the conduc-tion through gas and stack material. For high drive ratios, the temperature differencewill approach the critical temperature difference more and more. Left of the node, thedevice will act as a prime mover, while right of the node, for positive cooling powers,the device will act as a refrigerator.

Shown in figure 6.13, are the acoustic and cooling power as a function of the (scaled)position of the stack center, while keeping the temperature difference over the stackequal to zero. The peak acoustic powers are obtained near the tube center, which cor-responds to 1/4 of the wave length. The peak cooling powers on the other hand areobtained at 1/4 and 3/4 of the tube length, which is at about 1/8 and 3/8 of the wave


−5000 0 5000 100000

25

50

75

100

125

150

tem

pera

ture

diff

eren

ce (

K)


D = 5%D = 10%D = 20%D = 30%D = 40%

(a) p0 = 1 bar

−1 0 1 2

x 105

0

25

50

75

100

125

150

tem

pera

ture

diff

eren

ce (

K)


D = 5%D = 10%D = 20%D = 30%D = 40%

(b) p0 = 20 bar

Figure 6.12: Mutual dependence of cooling power, temperature difference and drive ratio. Thecalculations are performed for mean pressures equal to 1 or 20 bar.

length. In the right part of the graph, the cooling power turns negative and the devicestarts to work as a heater. Note also that the power is amplified by a factor 20, as themean pressure is increased from 1 to 20 bar.

0 0.5 1 1.5 2 2.5 30

500

1000

1500

2000

2500

acou

stic

pow

er (

W/m

2 )

kXs

D=1%D=2%D=3%

(a) p0 = 1 bar

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5x 10

4

acou

stic

pow

er (

W/m

2 )

kXs

D=1%D=2%D=3%

(b) p0 = 20 bar

0 0.5 1 1.5 2 2.5 30

1000

2000

3000

4000

cool

ing

pow

er (

W/m

2 )

kXs

D=1%D=2%D=3%

(c) p0 = 1 bar

0 0.5 1 1.5 2 2.5 30

2

4

6

8x 10

4

cool

ing

pow

er (

W/m

2 )

kXs

D=1%D=2%D=3%

(d) p0 = 20 bar

Figure 6.13: Supplied cooling power and absorbed acoustic power as a function of the scaledstack position. The temperature difference across the stack is kept at 0 K, and the drive ratio isvaried between 1 and 3%. The calculations are repeated for mean pressures equal to 1 and 20bar.

So far, we have kept the stack length fixed at 3 cm in all our computations. The stack


length is, however, an important parameter in the design of a stack that should be takenneither too small nor too large for optimal performance. In figure 6.14 we show theCOPR as a function of both the cooling power and the temperature difference across thestack for various stack lengths and a drive ratio of 30%.

0 1 2 3 4 5 6

x 104

0

0.1

0.2

0.3

0.4

0.5

CO

PR


Ls = 0.1 cm

Ls = 0.5 cm

Ls = 1 cm

Ls = 2 cm

Ls = 3 cm

Ls = 4 cm

(a) p0 = 1 bar

0 2 4 6 8 10 12

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CO

PR


Ls = 0.1 cm

Ls = 0.5 cm

Ls = 1 cm

Ls = 2 cm

Ls = 3 cm

Ls = 4 cm

(b) p0 = 20 bar

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

CO

PR


Ls = 0.1 cm

Ls = 0.5 cm

Ls = 1 cm

Ls = 2 cm

Ls = 3 cm

Ls = 4 cm

(c) p0 = 1 bar

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CO

PR


Ls = 0.1 cm

Ls = 0.5 cm

Ls = 1 cm

Ls = 2 cm

Ls = 3 cm

Ls = 4 cm

(d) p0 = 20 bar

Figure 6.14: The relative coefficient of power COPR as a function of the temperature differenceTH − TC or cooling power QC. The calculations are repeated for various stack lengths, a driveratio of 30% and for mean pressures equal to 1 and 20 bar.

On the one hand, we see that in order to obtain a large COPR we need to take thestack as short as possible. On the other hand, we see that these large COPR are onlyachieved at very small temperature differences and cooling powers. Therefore, if one issatisfied with a small temperature difference across the stack, then very large COPR canbe obtained. In figure 6.15 we have shown this more clearly by plotting the peak COPRvalue as a function of the stack length. For mean pressures of both 1 and 20 bar wesee that the maximal COPR is obtained when the stack length Ls approaches zero. Theother extreme is when the stack length approaches 0.5 cm and the refrigerator becomeshighly inefficient. For longer stacks the refrigerator even stops working altogether.

If one is more interested in large cooling powers and temperature differences andless in a large COPR, then one can try to maximize the achievable cooling power andtemperature differences that is obtained from figure 6.15 at zero COPR. In figure 6.16we plot these values as a function of the stack length. It turns out that the maximalcooling power is obtained for stack lengths close to 2 cm and the maximal temperaturedifference is obtained for stack lengths close to 4 cm. Therefore, the choice of Ls = 3 cm,

108 6.5 A standing-wave prime mover

that was made in all previous calculations, seems to be a reasonable choice.

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CO

PR

opt

Ls (cm)

p0 = 1 bar

p0 = 20 bar

(a)

0 2 4 6 80

1

2

3

4

5

CO

Pop

t

Ls (cm)

p0 = 1 bar

p0 = 20 bar

(b)

Figure 6.15: (a) COPRopt and (b) COPopt as a function of the stack length for a 30% drive ratioand mean pressures of 1 and 20 bar.

0 2 4 6 80

2

4

6

8

10

12x 10

5

Qc m

ax

Ls (cm)

p0 = 1 bar

p0 = 20 bar

(a)

0 2 4 6 80

20

40

60

80

100

∆ T m

ax

Ls (cm)

p0 = 1 bar

p0 = 20 bar

(b)

Figure 6.16: The maximal achievable cooling power and temperature difference as a function ofthe stack length for a 30% drive ratio and mean pressures of 1 and 20 bar.

6.5 A standing-wave prime mover

In this section we aim to analyze a thermoacoustic prime mover of the type depictedin 6.1(b). A higher goal is a comparison with the traveling-wave prime mover that willbe discussed in Chapter 7, which is based on the “Swift”-type looped geometry [14].To ensure an accurate and meaningful comparison we will choose a configuration thatresembles this traveling-wave configuration as close as possible. In [14] a stainless steelregenerator is positioned in a helium-filled looped resonator with parameters as givenin 6.5. Instead of the regenerator we will model a parallel-plate stack placed in a straighthelium-filled resonator with the same operating conditions. The stack position, porosity,and Lautrec number will be optimized to maximize the efficiency. In the end we willalso examine how the choice of plate material affects the performance.

Given the acoustic power WH and WC at the hot and cold end, and the total power


parameter symbol value unit

mass flux M 0 kg m/sspeed of sound c 1020 m/sisobaric specific heat Cp 5193.2 J/(kg K)

isochoric specific heat Cv 3115.9 J/(kg K)thermal conductivity gas Kg 0.16 W/(m K)

stack length Ls 7.3 cmgas radius Rg 4.2 cm

solid radius Rs 3.0 cmtube radius Rb 7.3 cmfrequency f 84 Hz

mean pressure p0 3.1 · 106 Padrive ratio Dr 5 %low temperature TC 300 Khigh temperature TH 600 K

Table 6.3: Parameters: a helium-filled parallel-plate stack

through the stack H, we can compute

QC = H − WC , QH = H − WH , (6.8)

with the sign of H, WC, and WH chosen such that they are positive. The performance ofthe prime mover will be measured by the (relative) efficiency

η =WC − WH

QH

, ηR =TH

TH − TC

COP. (6.9)

Note that that 0 ≤ ηR ≤ 1 due to the second law of thermodynamics.We start with the parameter values given in Table 6.5. We vary the position of the

stack in the resonator and for each position we compute the relative efficiency ηR. Figure6.17 shows the results of these computations for different temperature differences acrossthe stack. It turns out that for all temperature differences the optimal stack position liesnear the closed end with kxs = 0.045, and as the temperature difference increases theoptimal stack position shifts more and more to the closed end. The plots also show thatnot every stack position is suitable for a prime mover operating at a given temperaturedifference. The temperature difference has to be larger than the critical temperaturegradient which is maximal near the closed end. The larger the imposed temperaturedifference, the bigger the region where the stack produces sound.

Next we fix the temperature difference at 300 K and kxs at 0.045 and we optimizethe Lautrec number NL. Figure 6.18 shows the relative efficiency as a function of theLautrec number. The optimal value is obtained for NL ≈ 0.795.

The last parameter we wish to optimize is the porosity of the stack. We vary theblockage ratio Br between 0 and 1 and compute the relative efficiency. It should be notedthat for these computations εs cannot be neglected. When Br approaches 1, εs starts togrow and the wall temperature is no longer a steady variable. Figure 6.19 shows ηR asa function of Br and it turns out that ηR is optimal when Br ≈ 0.887.

110 6.5 A standing-wave prime mover

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

ηR

kxs

TH

−TC

= 200 K

TH

−TC

= 250 K

TH

−TC

= 300 K

TH

−TC

= 350 K

TH

−TC

= 400 K

Figure 6.17: The relative stack efficiency ηR as a function of the relative stack position kxs fordifferent temperature differences across the stack. The optimal stack position is kxs ≈ 0.03 if theblockage ratio is equal to 0.72 and the Lautrec number is equal to 0.275.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

ηR

NL

0.795

Figure 6.18: The relative stack efficiency ηR as a function of the Lautrec number NL. Theoptimal Lautrec number is NL ≈ 0.795 when TH − TC = 300 K, kxs = 0.045, and Br = 0.72.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

ηR

Br

0.887

Figure 6.19: The relative stack efficiency ηR as a function of the blockage ratio Br. The optimalefficiency is obtained for Br ≈ 0.887, when TH − TC = 300 K, kxs = 0.045, and NL = 0.795.


So far, following [14], we used stainless steel for the stack plates in all our calcu-lations. However, there is a wide range of materials possible, ranging from metal-likematerials to polystyrene that is used to manufacture plastic straws. In figure 6.20 wetest several options by repeating the calculations of figure 6.19 for different plate materi-als. Furthermore in figure 6.21 we show the corresponding acoustic and cooling power,again as a function of the blockage ratio Br. It turns out that for all materials, similarefficiencies and acoustic powers are obtained , although mylar seems to work best. Fur-thermore all materials seem to achieve their maximal efficiency close to Br = 0.83. Thehighest acoustic powers are obtained for small porosities, but at a very low efficiency,which requires an enormous input of heating power,. We will show in Chapter 7 thatfor a traveling-wave geometry, the choice of material has a much greater impact on boththe efficiency and the heating and acoustic power.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

ηR

Br


Figure 6.20: The relative stack efficiency ηR as a function of the blockage ratio Br for severalmaterials, when TH − TC = 300 K, kxs = 0.045, and NL = 0.795.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2x 10

6

φ

heat

ing

pow

er (

W)


(a)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2x 10

6

acou

stic

pow

er (

W)

Br


(b)

Figure 6.21: The (a) supplied heating power and (b) produced acoustic power at the stack as afunction of the blockage ratio Br for several materials, when TH − TC = 300 K, kxs = 0.045,and NL = 0.795.

112 6.6 Streaming effects in a thermoacoustic stack

6.6 Streaming effects in a thermoacoustic stack

In this section we will try to investigate how the streaming velocity u2,0 behaves as a

function of NL when M = 0. Waxler [148] examined the streaming velocity withinstraight two-dimensional pores using a comparable formal perturbation expansion.

Waxler considered a thermoacoustic stack filled with dry air at room temperaturepositioned inside a resonator at λ/8 from a closed end. Across the stack a temperaturedifference of 20K was imposed with the highest temperature at the side of the closedend. The exact value for the drive ratio and mean pressure were not given and neitherwere the stack plate material and thickness. As a result his results cannot be reproducedexactly . Qualitatively, however, we can compare our results. The parameter values forour computations are shown in Table 6.4.


drive ratio Dr 2 %mass flux M 0 kg m/sspeed of sound c 347 m/sisobaric specific heat Cp 1000 J/(kg K)isochoric specific heat Cv 714 J/(kg K)isobaric specific heat solid Cs 1300 J/(kg K)gas radius Rg NLδk m

plate radius Rs 1.5·10−5 mthermal conductivity air Kg 0.0264 W/(m K)

thermal conductivity plate Ks 0.08 W/(m K)stack length Ls 0.05λ m

mean pressure p0 1.14·105 PaPrandtl number Pr 0.68 -temperature hot heat exchanger TH 290 Ktemperature cold heat exchanger TC 270 K

Table 6.4: Parameters: an air-filled parallel-plate stack

Figure 6.22 shows the streaming velocities u2,0 together with the mass flux m as afunction of X and y in the stack, neglecting gravitational effects (Fr → ∞). In figure 6.23we zoom in at the center of the stack and show the streaming velocity as a function of yonly. The computations are repeated for various Lautrec numbers ranging between 0.1and 8. Note that the area in the graph below m is equal to zero so that for every X theaverage mass flux is zero.

Just like Waxler we observe several transitions as NL increases. For NL ≪ Pr, thevelocity is mostly towards the right, where the acoustic wave originates. This is theregime where the acoustic Bernouilli effect is unimportant. Due to the acoustic sourcemass is driven in the direction that acoustic intensity must flow. To maintain the zeromass flux, the mean velocity must drive the mass back to the left. As NL increases,the Bernoulli effect becomes more important. For NL ∼ Pr there is a transition and asNL increases more, boundary layer flow starts to develop. Finally for sufficiently largeNL, in addition to the boundary-layer flow, the parabolic profile starts to appear that istypical for wide channels.


(a) NL = 0.1 (b) NL = 0.1

(c) NL = 0.7 (d) NL = 0.7

(e) NL = 2 (f) NL = 2

(g) NL = 8 (h) NL = 8

Figure 6.22: The streaming velocity u2,0 (left column) and the mass flux m (right column) as afunction of X and y inside the stack. The profiles are computed for NL varying between 0.1 and8.

114 6.6 Streaming effects in a thermoacoustic stack

−1 −0.5 0 0.5 1−3

−2

−1

0

1

2x 10

−6

y/Rg

u 2,0 (

m/s

)

NL = 0.1

NL = 0.7

NL = 2

NL = 8

Figure 6.23: The streaming velocity profile u2,0 as a function of y in the center of the stack. Theprofiles are computed for NL varying between 0.1 and 8.

Chapter 7

Traveling-wave devices

Traveling-wave devices come in all sorts and sizes. These can be refrigerators [84, 142],prime movers [14, 78, 141, 155], or combinations of the two [34, 134, 154]. Ceperley [23]was the first to realize that a traveling acoustic wave propagating through a regeneratorundergoes a thermodynamic cycle similar to the Stirling cycle, so that potentially higherefficiencies could be obtained. These concepts were first transformed into a workingprime mover by Yazaki et al. [155], who managed to design a machine consisting of aregenerator placed in a looped tube, generating spontaneous traveling-wave gas oscil-lations around the loop. However, due to the high acoustic velocities there were largeviscous losses, reducing the efficiency significantly. Backhaus and Swift [14] improvedon these ideas by combining the loop with a resonance tube, so that in the loop thetraveling-wave phasing could be obtained, but at low acoustic velocities, thereby ob-taining efficiencies up to 30%.

In this chapter we will present the results from our simulations of a traveling-waveprime mover similar to that of Backhaus and Swift [14], consisting of a looped tube witha regenerator and thermal buffer tube, connected to a resonator tube with an acous-tic load. The usual approach in simulating such devices [14, 34, 76, 78, 141] consists ofproviding a geometry and looking for the right values of system parameters such asfrequency, power input and imposed temperature difference that give a stable system.Here we will employ the opposite approach. We only fix the geometry of the regeneratorand thermal buffer tube, and look for an appropriate configuration of the loop and res-onator that gives us the desired system parameters. Such an approach can be used as auseful guide for the design of practical devices. Given certain specifications, like power-input, temperature impedance, frequency, drive ratio, and regenerator impedance, theoptimal geometry can be computed. Moreover, the machine can be designed to preventthe occurrence of mass streaming, which is known to have a negative impact on theperformance.

We will derive an optimization procedure that finds the traveling-wave geometrythat fits given system parameters. In addition we will compute the efficiency of theprime mover, and its sensitivity to the choice of parameters related to the regenerator. Itwill follow that this kind of traveling-wave prime mover works potentially much betterthan an equivalent standing-wave prime mover.

116 7.1 A traveling-wave prime mover

7.1 A traveling-wave prime mover

We consider a traveling-wave prime mover of the type shown in figure 7.1 (cf. [14]): ashort looped tube attached to a half-wave-length resonance tube with a variable acousticload. Within the loop several segments can be distinguished: the regenerator, thermalbuffer tube, T-junction, inertance tube, and compliance. Below we will discuss each ofthese segments.

Wload

TC

TH

TC

WC

WH

12

3

4

5 6

B

A

(b) Detailed drawing of torus section

12λ

(a) Schematic scale drawing of traveling-wave prime mover

Figure 7.1: (a) Schematic scale drawing of a traveling-wave prime mover consisting of a shortlooped tube attached to a half-wave-length resonator with variable acoustic load. (b) Detailedview of torus section that provides sound to the resonator. The system is divided into severaldifferent segments: regenerator (A), thermal buffer tube (B), tube with T-junction (1,2), iner-tance tube (3), compliance (4), and the resonator (5,6) with acoustic load. The (parallel-plate)regenerator amplifies the sound from WC to WH and Wload is supplied to the acoustic load.

Regenerator (A)

Standing-wave devices require stacks that have pores of several penetration depthswide. Due to the resulting imperfect thermal contact between gas and solid, entropyis created, so that the ideal Carnot efficiency can never be achieved. In traveling-wavedevices regenerators can be used that have pores that are very small compared to thepenetration depth, so that we can make use of the reversible Stirling cycles [23], whichrequires the pressure and velocity to be in phase at the regenerator. Furthermore, smallgas velocities are preferred to prevent large amounts of viscous dissipation in the nar-row regenerator pores. Additionally, we want to maximize the product of pressure and

Traveling-wave devices 117

velocity to maximize the produced acoustic power. It is therefore advantageous to re-duce the velocity oscillations and increase the pressure oscillations at the regenerator.For this purpose a resonator is attached in such a way that the loop lies at a velocitynode (and pressure antinode) of the resonator.

We will simulate a helium-filled parallel-plate regenerator made of stainless steelwith heat exchangers on either side. The heat exchangers are used to supply heatingpower and amplify the acoustic power from WC at the cold end to WH at the hot end.The difference between WH and WC is the acoustic power produced by the regeneratorand will be denoted by WA. As in [14] we will start with a porosity Br = 0.72 andLautrec number NL = 0.275. Later we will optimize the geometry and material of theregenerator to maximize the efficiency.

Thermal buffer tube (B)

The thermal buffer tube is used as a thermal buffer between the hot heat exchanger andthe room temperature. Usually its tapered to reduce Rayleigh streaming, but in ouranalysis we consider a straight tube for simplicity.

T-junction (1,2,5)

The T-junction connects the loop to the resonator. Part of the sound produced by theregenerator will be supplied to the resonator and part will be fed back to the loop tocompensate for the wall attenuation. Due to the presence of sharp edges flow sepa-ration may occur, resulting in vortex shedding and subsequent turbulent dissipation.Disselhorst and Van Wijngaarden [37] showed that vortex shedding will only take placeif the Strouhal number, pertaining to the radius of curvature of the edges and the acous-tic velocity amplitude, is smaller than one. In our modeling and simulations we willignore such effects.

Inertance tube and compliance (3,4)

Clockwise from the T-junction we have the inertance tube and the compliance that givea contraction and expansion of the tube. Their lengths and diameters have to be cho-sen carefully so that the pressure and velocity match periodically across the loop. Inlumped-elements models the inertance tube is modeled as a piston with a certain massand the compliance as a large volume at constant pressure.

Resonance tube with acoustic load (5,6)

For the resonance tube we use a straight tube with a variable acoustic load to whichacoustic power Wload is supplied. The difference between Wload and WA is caused bywall attenuation in the loop and resonator. In [14] a quarter-wave length resonator isused with a conical shape to reduce the acoustic velocities, but here we have used astraight half-wave length resonator to simplify calculations.

The acoustic load is modeled as a narrow channel connected to a large reservoir thatis kept at constant pressure. The highest load efficiencies will be obtained when the loadis positioned near the T-junction, but in practice it is necessary to keep some distancedue to the complicated flow patterns around the T-junction. In practical applications the

118 7.2 Computations

T

x

WC

HAWH

HB

QHQC

1 2 3 4

5

BA

Figure 7.2: Torus section unrolled. We distinguish the regenerator (A), thermal buffer tube (B),T-junction (1,2,5), inertance tube (3), and the compliance (4). The top graph shows a schematictemperature profile. In the regenerator the temperature increases and in the thermal buffer tubethe temperature drops back to its original value. Everywhere else the temperature is assumedconstant. Heating power QH is supplied to the hot end and cooling power QC is extracted fromthe cold end of the regenerator. As a result the acoustic power is amplified by the regeneratorfrom WC to WH . The total power in the regenerator and thermal buffer tube is denoted by HA

and HB, respectively.

load may be replaced by a traveling-wave refrigerator that is driven thermoacousticallyby the prime mover (see e.g. [34]).

7.2 Computations

If we cut the torus in figure 7.1 at the regenerator and unroll the geometry we get aconfiguration as shown in figure 7.2. In our computations we will work on the unrolledgeometry and model the ends by periodicity; influence of edge effects is not taken intoaccount.

Figure 7.2 also shows the energy flows in and around the regenerator. Heatingpower QH is supplied to the hot end and cooling power QC is extracted from the coldend of the regenerator. This leads to a total power HA in the regenerator directed to itscold heat exchanger and a total power HB inside the thermal buffer directed to its coldheat exchanger. Moreover, the acoustic power is amplified from WC at the cold end toWH at the hot end. Putting control volumes around the hot and cold heat exchangersand applying conservation of energy, we find

QC = WC − HA, QH = HB − HA, WA = WH − WC , (7.1)


where WA denotes the acoustic power produced by the regenerator. We can now intro-duce efficiencies for the regenerator and acoustic load as

ηA =WA

QH

, ηload =Wload

QH

. (7.2)

The relative efficiencies are obtained by dividing by the Carnot efficiency ηC,

ηR =η

ηC

, ηC =TH − TC

TH

. (7.3)

The Carnot efficiency gives an upper bound for the efficiency, so that 0 ≤ ηR ≤ 1. AsWA will be larger than Wload because of wall attenuation, we obtain ηload ≤ ηA ≤ ηC.

The computations consist of two parts. First we give the geometry for the regen-erator and the thermal buffer tube, we impose a impedance and pressure amplitudeat the regenerator, and we impose a temperature difference across the regenerator andthermal buffer tube. Integration of the system (4.70) will then give the complete veloc-ity, pressure and temperature profile in the regenerator and thermal buffer tube. Fromthis we get boundary conditions for the pressure and velocity in the loop and resonator.As a result of this approach we impose more conditions than available unknowns. Wecircumvent this problem, by adapting the geometry parameters of the loop and res-onator until all boundary conditions are satisfied. In particular we will vary the radiiand lengths of the various segments.

7.2.1 Regenerator and thermal buffer tube

The first part of the geometry consists of the regenerator and thermal buffer tube asdepicted in figure 7.2.1. The variables in the regenerator will be indicated by an indexA and in the thermal buffer tube by an an index B.

pA

uA

pB

uB

TC TH TC

A B

Figure 7.3: Regenerator (A) and thermal buffer tube (B).

Left of the regenerator a pressure pA and an impedance ZA is imposed, which leadsto the following initial conditions:

p1A(0) = pA, (7.4a)

〈u1A〉(0) = uA = pA/ZA. (7.4b)

At the interface between the regenerator and thermal buffer tube we impose continuityof mass and momentum, which results in continuity of pressure and volumetric velocity,


so that

p1A(1) = p1B(0), (7.5a)

〈u1A〉(1) =1

Br

〈u1B〉(0), (7.5b)

where Br is the blockage ratio as given in Table 3.1. Additionally, we impose a temper-ature difference TH − TC across the regenerator and thermal buffer tube,

T0A(0) = TC, T0A(1) = TH , (7.6a)

T0B(0) = TH , T0B(1) = TC. (7.6b)

Integrating the system of ODE’s given in (4.70), subject to these boundary conditions,we can compute numerically T0, p1 and 〈u1〉 throughout the regenerator and thermalbuffer tube. This will yield a pressure pB and velocity uB at the end of the thermalbuffer tube. Note that only pA and uA are imposed; pB and uB are an outcome of thecomputations.

7.2.2 Optimization procedure

Having solved for the regenerator and thermal buffer tube, we still need to model andsimulate the remaining part of the system that is shown in figure 7.4. The lengths (L j)and radii (R j) of each segment will be adapted so that all boundary conditions are satis-fied. The regenerator and thermal buffer tube are kept fixed throughout this procedure.

Wload

5 6

1

2

3

4

(pA, uA)

(pB, uB)

Figure 7.4: Tube with T-junction, inertance tube, compliance and reso-nance tube with acoustic load.

Since each segment is a straight tube at constant temperature, we can use the expres-sions derived in equations (4.75), which express the pressure and velocity in segment j( j = 1, . . . 6) as

p1 j(X) = A je−ik jX + B je

ik jX , 0 ≤ X ≤ 1, (7.7a)

〈u1 j〉(X) =χ j

ρ0c

(A je

−ik jX − B jeik jX)

, 0 ≤ X ≤ 1, (7.7b)


with A j and B j integration constants and

χ j =1

c

√√√√(1 − fν j)

(1 +

γ − 1

1 +εs j

fk j

), (7.8)

k j = κ j

√−a3 ja5 j =

κ j

c

√√√√ 1

1 − fν j

(1 +

γ − 1

1 + εs j

fk j

), κ j =

ωL j

cre f. (7.9)

The unknowns A j and B j differ in each section and have to be determined from theboundary and interface conditions. The left and right end of each segment correspondwith X = 0 and X = 1, respectively.

It still remains to model the side branch with acoustic load. The acoustic load isdissipated via a thin channel leading to a reservoir which is kept at constant pressure.We can apply Darcy’s law to relate the pressure pload and the velocity uload at the load,

uload = Zload pload, (7.10)

where Zload is an impedance depending on the area, viscosity, and length of the channel.We will use pA, uA, pB and uB as boundary conditions for the ends of the loop as

indicated in figure 7.4. In addition we will impose continuity of mass and momentumacross every interface. Moreover, at the end of the resonator we have a closed ending,which results in a zero velocity. This leads to the following set of conditions:

uB = 〈u11〉(0), pB = p11(0), (7.11a)

〈u11〉(1) = b2〈u12〉(0) + b5〈u15〉(0), p11(1) = p12(0) = p15(0), (7.11b)

〈u12〉(1) = b3〈u13〉(0), p12(1) = p13(0), (7.11c)

b3〈u13〉(1) = b4〈u14〉(0), p13(1) = p14(0), (7.11d)

b4〈u14〉(1) = BruA, p14(1) = pA, (7.11e)

b5〈u15〉(1) = bloaduload + b6〈u16〉(0), p15(1) = p16(0) = pload, (7.11f)

〈u16〉(1) = 0. (7.11g)

where b j = R2j /R2

A gives the ratio between the cross-sectional area of segments j and A.Substituting (7.7), we find we have to solve the following equation for the constants A j

and B j: (M− 0

0 M+

)(AB

)= F (7.12)

with vectors

A =[

A1 A2 A3 A4 A5 A6

],

B =[

B1 B2 B3 B4 B5 B6

],

F =[

pB ρ0cuB 0 0 0 0 0 0 0 pA ρ0cBruA 0 pload ρ0cbloaduload 0]

,


and matrices

M− =

1

χ1

e−ik1 −1

e−ik1 −1

χ1e−ik1 −b2χ2 −b5χ5

e−ik2 −1

χ2e−ik2 −b3χ3

e−ik3 −1

χ3e−ik3 −b4χ4

e−ik4

χ4e−ik4

e−ik5 −1

e−ik5

χ5e−ik5 −b6χ6

e−ik6

,

and

M+ =

1

−χ1

eik1 −1

eik1 −1

−χ1eik1 b2χ2 b5χ5

eik2 −1

−χ2eik2 b3χ3

eik3 −1 −1

−χ3eik3 −b4χ4 b4χ4

eik4

−χ4eik4

eik5 −1

eik5

−χ5eik5 b6χ6

−eik6

.

Note that M−, M+ ∈ C15 × C

6, A, B ∈ C6, F ∈ C

15. As there are more equations(15) than unknowns (12) this is an overdetermined system of equations. By variation ofthe geometry parameters we will try to minimize the distance between the least-squaressolution and the real solution. In particular we will do this by minimizing the relativeerror

RELTOL :=‖(M(M∗M)−1 M∗ − I)F‖

‖F‖ , with M :=

(M− 0

0 M+

),

subject to 8 (real) geometry parameters L1, L2, L3, L4, b3, b4, b5, Zload.


7.3 Results

We have implemented the optimization procedure above using the parameter valuesgiven in Table 7.1. Following Backhaus and Swift [14] we choose 30-bar helium as theworking gas and a stainless-steel regenerator that we model with a parallel-plate ge-ometry. The regenerator parameters are chosen such that the regenerator has the sameporosity, length, hydraulic radius as in [14]. Additionally the thermal buffer tube hasalso been given the same length as in [14]. The geometry parameters of the remainingsegments will be determined from the optimization procedure.

variable value unitDr 5 %

ZA 30ρAcA N s m−3

LA 7.3 cmLB 24 cmRg 0.42 mm

Rs 0.30 mmRA 4.45 cmRB 4.45 cmf 84 Hz

p0 3.1 · 106 PaTC 300 KTH 600 K

M 0 kg s−1

Table 7.1: Input parameters for numerical simulation of regenerator and thermal buffer tube

In the sections below we will first show what kinds of temperature profiles one canexpect in the regenerator and thermal buffer tube, and how they are affected by thelengths of the components and the presence of mass streaming. Next, we will focuson the regenerator and try to look for its optimal design that maximizes its efficiency.Finally, we will give the machine geometry that follows from the minimization routinefor the optimal regenerator and thermal-buffer-tube design, and we show the pressure,velocity, and acoustic-power profiles that can be expected throughout the prime mover.

7.3.1 Temperature

In figure 7.5 we show the temperature profiles in the regenerator and thermal buffertube computed using the parameter values given in Table 7.1. The temperature profileincreases and decreases almost linearly as is commonly assumed.

If the length of the regenerator or thermal buffer tube is changed then a deviationfrom the linear temperature profile can occur. This is shown in figure 7.6 where thetemperature is plotted for different lengths of the regenerator or thermal buffer tube.For very long thermal buffer tubes the temperature even increases first before droppingto the low temperature TC.

So far we have only considered the case with no streaming, i.e. M = 0. To show theeffect of streaming on the temperature profile we have plotted for some choices of M the

124 7.3 Results

0 0.05 0.1 0.15 0.2 0.25 0.3300

350

400

450

500

550

600

x(m)

mea

n te

mpe

ratu

re (

K)

A B

Figure 7.5: Mean temperature profile in regenerator (A) and thermal buffer tube (B).

0 0.05 0.1 0.15300

350

400

450

500

550

600

x(m)

mea

n te

mpe

ratu

re (

K)

Ls = 2 cm

Ls = 4 cm

Ls = 6 cm

Ls = 8 cm

Ls = 10 cm

Ls = 12 cm

Ls = 14 cm

(a) regenerator

0.1 0.2 0.3 0.4 0.5 0.6 0.7300

350

400

450

500

550

600

650

x(m)

mea

n te

mpe

ratu

re (

K)

Ls = 10 cm

Ls = 20 cm

Ls = 30 cm

Ls = 40 cm

Ls = 50 cm

Ls = 60 cm

(b) thermal buffer tube

Figure 7.6: Mean temperature profiles in (a) regenerator and (b) thermal buffer tube. In (a) LB

was fixed at 24 cm and LA was varied and in (b) LA was fixed at 7.3 cm and LB was varied.

corresponding temperature distribution in 7.7. As |M| increases the temperature profilein the thermal buffer tube changes from linear to exponential to almost boundary-layer-like behavior, whereas the regenerator-temperature profile is not affected so much. Thisshows that the presence of a mass flux can distort the mean-temperature profile signif-icantly. This behavior was in fact predicted in Section 4.4.3. It was shown in equations(4.97)-(4.99) that for a relatively short and wide tube (as we have here) with little or nomass streaming a linear temperature profile is expected and as the mass flux increasesthe temperature distribution will start to look more and more like an exponential profile


until eventually we get

limM2→−∞

T0(X; M2) =

{TH , X = 0,

TC, 0 < X ≤ 1,

limM2→∞

T0(X; M2) =

{TH , 0 ≤ X < 1,

TC, X = 1.

The same limiting behavior is observed in figure 7.7. In addition we see that for positivemass fluxes the temperature first increases before dropping to the low temperature. Itis possible that due to the high positive mass flux, high-temperature gas parcels aretransported away from the hot end whose heat cannot be pumped fast enough to thecold end, and heat is accumulated near the hot heat exchanger which gives rise to atemperature increase.

0 0.02 0.04 0.06300

350

400

450

500

550

600

x(m)

mea

n te

mpe

ratu

re (

K)

(a) regenerator

M/Ag = −0.025

M/Ag = −0.005

� M/Ag = −0.001

� M/Ag = 0

N M/Ag = 0.001

× M/Ag = 0.005

+ M/Ag = 0.025

0.1 0.15 0.2 0.25 0.3300

350

400

450

500

550

600

650

x(m)

mea

n te

mpe

ratu

re (

K)

(b) thermal buffer tube

Figure 7.7: Temperature distributions in (a) regenerator and (b) thermal buffer tube, for different

average mass fluxes M/Ag [kg s−1m−2] through the regenerator.

7.3.2 Regenerator efficiency

So far we have used the design parameters for the regenerator and thermal buffer tubethat were also used in [14]. In this section we will look a bit more closely on whatthe optimal design is for the regenerator in terms of power-output and efficiency. Theefficiency in this section will therefore be defined as the relative regenerator efficiency

ηR = ηA/ηC .

Several parameters will affect the performance of the prime mover. The most im-portant geometry parameters are the Lautrec number NL, the blockage ratio Br and theregenerator length LA. Important operating conditions are the impedance ZA, the driveratio Dr and the temperature difference TH − TC. In figure 7.8 we show how variationof these parameters degrades or improves the performance. In all these computationswe have used the parameter values as given in Table 7.1, unless otherwise specified.

126 7.3 Results

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ηR

NL

0.275

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

Br

ηR

0.917

(b)

0 3 6 9 12 15 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ηR

LA

6.8

(c)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

ηR

ZA/(ρ c)

16.5

(d)

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

ηR

drive ratio

(e)

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

ηR


233

(f)

Figure 7.8: The relative regenerator efficiency as a function of (a) the Lautrec number NL (com-puted at room temperature), (b) the blockage ratio Br, (c) the regenerator length LA, (d) theimpedance ZA, (e) the drive ratio Dr, and (f) the temperature difference across the stack.

It is well-known that for the optimal working of a traveling-wave device the Lautrecnumber should be small, so that there is a perfect thermal contact between the gas andthe wall. In figure 7.8(a) we see that reality is a bit more subtle. Indeed, NL shouldbe smaller than 1, but the optimal value is reached for NL ∼ 0.275 (computed at roomtemperature) which corresponds with a hydraulic radius of 42 µm, exactly the value asused in [14]. When NL becomes too small the viscous dissipation becomes importantand it starts affecting the performance.

For a given hydraulic radius we can also change the porosity Br, which is effectively


the same as changing the thickness of the solid. We see in figure 7.8(b) that the more gasthe better the efficiency. However, for high porosities, as the plates become infinitelythin, the efficiencies drop back to zero because there is not enough regenerator materialto transport the heat.

The effect of the regenerator length is shown in figure 7.8(c). When the stack is tooshort, only a small temperature difference can be achieved and therefore the efficiencywill drop also. On the other hand, when the stack is too long, there will be lots ofdissipation, which also reduces the efficiency. We see from the graph that the optimalvalue is obtained near LA = 6.8 cm, which is quite close to the choice of 7.3 cm usedin [14].

We mentioned earlier that for the gas in the regenerator to undergo the ideal Stirlingcycle, it is necessary that the pressure and velocity are in phase. Moreover, the velocityoscillations should be small to reduce viscous dissipation and the pressure oscillationsshould be large to maximize the acoustic power. A trade-off is therefore expected, andit is shown in figure 7.8(d) that an impedance of ZA ≈ 16.5 ρc gives the highest efficien-cies.

Next, in figure 7.8(e), we demonstrate how the efficiency is influenced by the driveratio. When the drive ratio is increased at a fixed temperature difference, the poweroutput will increase, but the losses stay the same. As a result we observe higher effi-ciencies for increasing drive ratios, but as the driver ratios increases more and more, theefficiency profiles converge due to the restriction of the Carnot efficiency.

Lastly, we mention the impact of the imposed temperature difference. We first notethat it should be large enough, otherwise the device cannot act as a prime mover; itfollows from 7.8(f) that a minimal temperature difference of approximately 24 K is nec-essary. The optimal efficiency is achieved near 233 K and for larger temperature differ-ences the efficiency starts to decrease slowly.

−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.050.2

0.3

0.4

0.5

0.6

0.7

mass flux (kg/m2/s)

ηR

0

Figure 7.9: The relative regenerator efficiency as a function of (d) the average mass flux M/Ag

[kg s−1 m−2] through the regenerator.

So far we have neglected the influence of streaming. If a nonzero mass flux is in-cluded the efficiencies ηR should decrease because when M 6= 0 a large time-averagedconvective enthalpy flux can arise, mixing gas of different temperatures and degrad-ing the performance. If we plot the relative efficiency as a function of the average

128 7.3 Results

mass flux M/Ag through the regenerator (figure 7.9), then we see that this is indeed

the case. As expected, the maximal efficiency is obtained when M = 0 and decreasesrapidly for increasing or decreasing mass flux. The graph does not show values beyond

M/Ag ≈ 0.03 kg s−1 m−2 because of numerical difficulties; a very steep temperatureprofile is necessary (see figure 7.7) to reach the desired low temperature TC, which re-quires a very precise guess of the total power HB.

In all the previous calculations we have used stainless steel for the regenerator ma-terial. If we want to make a study of what material is best we can repeat the previouscalculations for different materials. In particular we are interested in the effect of theporosity on the performance. Apart from the material and porosity, the remaining pa-rameters are chosen as in Table 7.1. In figures 7.10 and 7.11 we have plotted the relativeefficiency of the regenerator, the supplied heating power, and the produced acousticpower as a function the blockage ratio Br for stainless steel, alumina ceramics, glass,mylar, and plastic.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ηR

Br


Figure 7.10: The relative regenerator efficiency as a function of the blockage ratio Br for severalmaterials.

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

heat

ing

pow

er (

W)

Br


(a)

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

acou

stic

pow

er (

W)

Br


(b)

Figure 7.11: The (a) supplied heating power and (b) produced acoustic power at the regeneratoras a function of the blockage ratio Br for several materials.


We see that for most materials the optimal efficiency is obtained for high porosities,i.e. very thin stack plates. However, when the plates become too thin (Br → 1), theefficiency drops back to zero again because there is not enough plate material to carryall the heat. For some materials, such as mylar or plastic straws, a high efficiency isreached for almost all porosities, provided the stack plates are neither too thick (Br → 0)or too thin (Br → 1). However, although the efficiency may be high, low values of Br

are not practical, as almost no acoustic power is produced (see figure 7.11). We observethat for all materials the most acoustic power is produced for Br between 0.75 and 0.9with still a relatively high efficiency. Depending on whether one is interested more inhigh efficiencies or high power output it follows that either glass or stainless steel arethe best options for the regenerator material.

We can compare these results to stack efficiency of the standing-wave prime moversimulated in Section 6.5, whose configuration has also been optimized to obtain maxi-mal efficiencies under the same operating conditions. Comparing figures 6.20 and 7.10 itfollows that the efficiencies for the traveling-wave configuration are about twice as highas those of the standing-wave configuration. Furthermore, as opposed to the standing-wave configuration, the choice of material is much more critical for the regenerator,especially at low porosities.

7.3.3 Geometry optimization

In this section we will determine the geometry for the loop and resonator that fits theregenerator and thermal buffer tube given by the input parameters in Table 7.2. We usedthe results from figure 7.8 to choose the optimal design that for a temperature differenceof 300 K and drive ratio of 5% maximizes the regenerator efficiency, i.e. Br = 0.917,NL = 0.275, LA = 6.8cm, ZA = 16.5ρc, and M = 0. Using MATLAB’s lsqnonlin wehave minimized RELTOL by careful choice of the parameters L1, L2, L3, L4, b3, b4, b5, andZload, subject to the the constraints that all parameters be positive, b3 < 1 (contraction),and b4 > 1 (expansion).

The routine converged with a relative error of approximately 4 · 10−8 and yieldedthe geometry given in figure 7.12 and Table 7.2. This choice of loop and resonator givesa machine (load) efficiency of 0.35. Note that viscous dissipation due to interactionwith the wall is included in all segments. Although the addition of viscous dissipationsaffects the wave number only slightly (e.g. k1 = 0.5172 vs. k1 = 0.5185− 0.0007i), it hasa significant impact on the performance of the device. Most importantly the efficiencyη is affected, which decreases from 0.39 at the regenerator to 0.35 at the load. Also theacoustic power Wload provided to the load is reduced significantly from WA that wasgenerated by the regenerator.

It should be noted that the minimization procedure is quite sensitive to the initialguess. Because we have so many geometry parameters to choose (more than availableunknowns), different geometries might work. Small changes in the initial guess canlead to very different geometries and some of these can be quite unrealistic, e.g. a veryshort and wide compliance or a very thin and long inertance tube.

In figure 7.13 we show the acoustic power as a function of its position in the primemover. Within the regenerator there is a sharp amplification of the sound and there is anincrease in acoustic power of about WA = 211 W. Some of this is lost in the loop due toviscous dissipation, and 206 W is supplied to the resonator, which explains the drop in

130 7.3 Results

Input Outputvariable value unit variable value unitDr 5 % L1 14.1 cmZA/(ρAcA) 16.5 - L2 15.9 cmLA 6.8 cm L3 18.0 cmLB 24 cm L4 20.3 cmL5/λ 1/8 - R3 2.93 cmL6/λ 1/2 - R4 4.96 cmRg 42.1 µm R5 3.23 cm

Rs 3.81 µm HA 123 WRA 4.45 cm HB 661 WRB 4.45 cm QH 538 WR1 4.45 cm WA 211 WRload 9.94 mm Wload 185 Wf 84 Hz ηA 0.39 -

p0 3.1 · 106 Pa ηload 0.35 -TC 300 K ηA,R 0.79 -TH 600 K ηload,R 0.69 -

M 0 kg s−1 RELTOL 4.14 · 10−8 -

Table 7.2: Input and output parameters of minimization routine.

A

B

1

2

3

4

load

5 6

1 m

Figure 7.12: Scaled version of geometry obtained from minimization routine. From top to bot-tom and left to right we have: regenerator (A), thermal buffer tube (B), tube with T-junction(1,2), inertance tube (3), compliance (4), resonator with load (5,6). The input and output param-eters are given in Tables 7.1 and 7.2.

W at the T-junction. Inside the resonator there is dissipation as well, so that eventuallyan amount of Wload = 185 W can be provided to the acoustic load.

Next, in figures 7.14-7.17 we show the volumetric velocity Ag〈u1〉 and the pressurep1 all along the system. At the regenerator we have a small velocity, a large pressureand a real impedance. The length of the whole loop is much smaller than a wave length,and it is positioned near a pressure antinode of the resonator, so that the pressure isapproximately constant in the loop and the velocity close to zero. At the T-junctionthere is a drop in Ag〈u1〉 due to the flow into the resonator.


0 0.2 0.4 0.6 0.8 1400

450

500

550

600

650

700

x(m)

acou

stic

pow

er (

W)

A

B 1

2 3 4

(a) in loop

0 1 2 3 4 5 60

50

100

150

200

250

x(m)

acou

stic

pow

er (

W)

5

6

(b) in resonator

Figure 7.13: Acoustic power as a function of the position in the prime mover.

0 0.2 0.4 0.6 0.8 15

6

7

8

9

10x 10

−3

x(m)

Re(

Ag<

u 1>)

(m

3 /s)

A

B1

2 3 4

(a) Re(

Ag〈u1〉)

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x(m)

Im(A

g<u 1>

) (

m3 /s

)

A

B1

2 3

4

(b) Im(

Ag〈u1〉)

Figure 7.14: Real and imaginary part of volumetric velocity as a function of the position in theloop.

0 2 4 6 8−4

−2

0

2

4

6x 10

−3

x(m)

Re(

Ag U

1) (

m3 /s

) 5

6

(a) Re(

Ag〈u1〉)

0 1 2 3 4 5 6−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

x(m)

Im(A

g U1)

(m

3 /s)

5

6

(b) Im(

Ag〈u1〉)

Figure 7.15: Real and imaginary part of volumetric velocity as a function of the position in theresonance tube.

132 7.3 Results

0 0.2 0.4 0.6 0.8 11.46

1.48

1.5

1.52

1.54

1.56x 10

5

x(m)

Re(

p 1) (

Pa)

A

B

1

2

3

4

(a) Re(

p1

)0 0.2 0.4 0.6 0.8 1

−500

0

500

1000

1500

2000

2500

3000

x(m)

Im(p

1) (

Pa)

A

B

1 2

3

4

(b) Im(

p1

)

Figure 7.16: Real and imaginary part of acoustic pressure as a function of the position in theloop.

0 1 2 3 4 5 6

0

5

10

15x 10

4

x(m)

Re(

p 1) (

Pa)

5

6

(a) Re(

p1

)0 1 2 3 4 5 6

−3000

−2000

−1000

0

1000

2000

3000

4000

x(m)

Im(p

1) (

Pa)

5

6

(b) Im(

p1

)

Figure 7.17: Real and imaginary part of acoustic pressure as a function of the position in theresonance tube.

Chapter 8

Nonlinear standing waves

It is well-known [28, 32, 33, 39] that disturbances can arise in a closed tube, when a pis-ton supplies oscillations at or near a resonance frequency. In particular shock wavesmay occur, or deformed sinusoidal profiles. Nonlinear standing waves may also occurin practical thermoacoustic devices, where high amplitudes are quite common. Highlynonlinear acoustic-wave forms have been observed experimentally [10], arising due togeneration of higher harmonics and generating losses that degrade the performance.Dequand et al. [36] also observed shock formation in side-branch systems of the typeshown in Figure 1.9. Replacing the thermoacoustic stack by a piston, Gaitan and Atch-ley [43] analyzed the energy transfer into the higher harmonics for a prime-mover con-figuration and showed that it can be partly suppressed by varying the cross-section ofthe prime mover. Hofler’s refrigerator (Figure 1.4) also uses a variable cross-sectionwith a horn and reservoir to avoid the formation of shock waves.

An elaborate theoretical model for calculating nonlinear disturbances (both shockand pre-shock) in closed tubes was derived by Chester [28] and improved upon by var-ious others [62,65,93,94,144]. Chester’s model includes effects of compressive viscosityand of shear viscosity in the boundary layers near the walls, and starts from the fun-damental one-dimensional hydrodynamic equations with a correction for wall frictionusing boundary-layer equations.

Coppens and Sanders [32] derive a one-dimensional nonlinear acoustic-wave equa-tion describing finite-amplitude (pre-shock) standing waves in closed tubes. Using aperturbation expansion they arrive at a set of linear equations that need to be solved it-eratively. This was later extended [33] to a three-dimensional model for acoustic wavesin “lossy” cavities.

Menguy and Gilbert [87] have also analyzed nonlinear oscillations in gas-filled tubes,both theoretically and experimentally. In their approach several small parameters areidentified and a multiple-scales perturbation solution is attempted. Starting from theNavier-Stokes equations, two generalized Burgers equations are derived that describethe propagation of a weakly nonlinear acoustic wave propagating in a cylindrical tubebefore shock formation. These equations are solved numerically to second order usingtruncated Fourier series, while allowing for arbitrary boundary conditions. Finally, theresults are successfully validated against experimental results for standing-wave andtraveling-wave configurations.


Much of the later work on nonlinear standing waves [1, 31, 39, 51, 60, 94, 122] startswith the Kuznetsov’s equation [70], a single second-order nonlinear wave equation thatdescribes the nonlinear propagation of sound in a dissipative and irrotational medium.Neglecting wall friction, Enflo and Hedberg [40] used Kuznetsov’s equation to derive aone-dimensional evolution equation for the velocity potential and found a steady-statesolution also using a multiple-scales approach. The main difference with the analysisof Menguy and Gilbert [87] is the assumption of irrotational flow, which allows for thecomputation of an analytical solution.

Makarov and Ochmann [82, 83, 96] have written an extensive three-part review onall modern investigations of nonlinear and thermoviscous phenomena in acoustic fieldsin fluids. They treat [83] various nonlinear evolution equations, most of which startfrom the simple wave equation with nonlinear source terms. In particular they also gointo the Kuznetsov equation and the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equa-tion which is a generalization of the Kuznetsov equation and includes the effect of weakdiffraction. Ochmann and Makarov [96] also address the topic of thermoacoustic vibra-tions in a closed pipe with an infinitesimally thin heater located in the center of the pipe.By relating the heat release of the heater to the pressure fluctuations, shock waves canbe predicted [29, 30].

In this chapter we will try to predict the wave forms that may arise when a ther-moacoustic refrigerator or prime mover is excited near its resonance frequency. Onlywide tubes are considered with uniform cross-sections in which the viscous interactionwith the wall can be neglected. We generalize the approach of Enflo and Hedberg [40]to predict the nonlinear wave profiles when a thermoacoustic stack is placed inside thetube. As in [40] we will use Kuznetsov’s equation to determine the sound field in thetube, but with a reflection condition to simulate the presence of a thermoacoustic stack.Gusev et al. [49] also use a reflection condition to model a prime mover, but assume thatthe sound field inside the stack is not affected by the nonlinearities.


To simplify analysis we assume irrotational flow, i.e. ∇× v = 0. As a result the flowvelocity can be described as the gradient of a velocity potential,

v = −∇Φ.

The assumption of irrotational flow is correct, provided the viscous interaction with thewall can be neglected. This is indeed the case, since we consider wide and straight tubeswith negligible wall dissipation. Moreover, we consider such cases in which the walldissipation can be neglected with respect to the dissipation in the shock.

8.1.1 Kuznetsov’s equation

We start from the Kuznetsov equation [70], which is given by

∂2Φ

∂t2− c2

0∇2Φ =

∂∂t

[(∇Φ)2 +

1

2c20

(γ − 1)

(∂Φ∂t

)2

+b

ρ0

∇2Φ

], (8.1)

Nonlinear standing waves 135

where c0 and ρ0 are the equilibrium values of the speed of sound and density of thefluid, independent of x, and b is a constant that represents the total effect of viscosityand heat conduction,

b = K

(1

Cv

− 1

Cp

)+

4

3µ +ζ .

As before γ is defined as the ratio of the specific heats

γ =Cp

Cv

.

Let the ends of the tube be denoted by x = 0 and x = L. From here on we willrestrict ourselves to one space dimension with the velocity u = −∂Φ/∂x as dependentvariable,

∂2Φ

∂t2− c2

0

∂2Φ

∂x2=

∂∂t

[(∂Φ∂x

)2

+1

2c20

(γ − 1)

(∂Φ∂t

)2

+b

ρ0

∂2Φ

∂x2

]. (8.2)

If the left end of the tube is closed then a zero velocity is imposed

u(0, t) = 0.

It follows that the sound reflects at x = 0 with reflection coefficient R = 1. We will nowmodel the presence of a stack at x = 0 by requiring the wave to reflect effectively at apoint −τL < 0 with reflection coefficient R0,

R(−τL, t) = R0.

Depending on how much sound is dissipated in the stack R0 will vary between 0 (verydense stack) and 1 (no stack). Assuming a short stack with relatively wide pores onecan expect that R0 will be close to 1. At x = L we impose an excitation h with angularfrequencyω and amplitude U.

u(L, t) = Uh(ωt) = ℓωh(ωt).

where ℓ is the maximal displacement of the excitation. The sound source will be mod-eled by putting h(t) = sin(t).

We next rescale

x = Lx, t =L

c0

t, u = c0u, Φ = Lc0Φ, p = ρ0c20 p.

The relevant dimensionless numbers for this problem are

ε =U

c0

, µ =ω

2Lb

2ρ0c30

, κ =ωL

c0

, β =γ + 1

2, γ =

Cp

Cv

, (8.3a)

δ = 1 − R0, µ0 = µ(1 + τ), κ0 = κ(1 + τ), ∆ = κ0 − π . (8.3b)


Dropping the tildes we arrive at the following boundary value problem:

∂2Φ

∂t2− ∂2

Φ

∂x2=

∂∂t

[(∂Φ∂x

)2

+γ − 1

2

(∂Φ∂t

)2

+2µ

κ2

∂2Φ

∂x2

], (8.4)

subject to

R(−τ , t) = R0, (8.5a)

u(1, t) = −∂Φ∂x

(1, t) = εh(κt). (8.5b)

8.1.2 Bernoulli’s equation

If the density is assumed constant and gravity is neglected, the momentum balanceas part of the Euler equations can be integrated to Bernoulli’s equation for unsteadypotential flow

∂Φ∂t

+1

2u2 +

p

ρ0

= C, (8.5c)

where C only depends on time. Therefore the pressure p can be expressed inΦ and u as

p = ρ0

(C − 1

2u2 − ∂Φ

∂t

). (8.5d)

With the rescaling given above, we find that the dimensionless pressure is given by

p(x, t) = C(t)− ∂Φ∂t

(x, t)− 1

2u2(x, t). (8.6)

8.1.3 Perturbation expansion

We will use the smallness ofΦ, ε, µ, δ, and ∆ to find an approximate solution correct upto second order in these small parameters. To make a small-parameter analysis possiblewe will use ε as the small parameter and relate µ, δ and ∆ to ε. First, assuming Φ issmall, we expand

Φ(x, t;ε) = εΨ(x, t) +ε2ψ(x, t) +O(ε3), ε≪ 1. (8.7)

Substituting the expansion into (8.4) we find that to leading order

∂2Ψ

∂t2− ∂2

Ψ

∂x2= 0, ε≪ 1,µ ≪ 1.

The general solution is given as a sum of left and right-running waves

Ψ(x, t) = F1(t − x) + F2(t + x), (8.8)


for still arbitrary functions F1 and F2. Next assuming µ = O(ε) and collecting the first-order terms in ε and µ, we find

∂2ψ

∂t2− ∂2

ψ

∂x2= (γ+ 1)

[f1(t − x) f ′1(t − x) + f2(t + x) f ′2(t + x)

]

+ (γ − 3)[

f1(t − x) f ′2(t + x) + f ′1(t − x) f2(t + x)]+

2µ

εκ2

[f ′′1 (t − x) + f ′′2 (t + x)

].

where f1,2 ≡ F′1,2. If we change to the variables

ξ = t − x, η = t + x, (8.9)

then we find

4∂2ψ

∂ξ∂η= (γ+ 1)

[f1(ξ) f ′1(ξ) + f2(η) f ′2(η)

]+ (γ − 3)

[f1(ξ) f ′2(η) + f ′1(ξ) f2(η)

]

+2µ

εκ2

[f ′′1 (ξ) + f ′′2 (η)

].

Integrating with respect to ξ and η we find

4ψ(ξ , η) =γ + 1

2

[η f 2

1 (ξ) +ξ f 22 (η)

]+ (γ− 3)

[F1(ξ) f2(η) + f1(ξ)F2(η)

]

+ 2µ

εκ2

[η f ′1(ξ) +ξ f ′2(η)

]+ A(ξ) + B(η).

A suitable redefinition of the integration constants

A(ξ) := −γ + 1

2ξ f 2

1 (ξ)− 2µ

εκ2ξ f ′1(ξ) + G1(ξ),

B(η) := −γ + 1

2η f 2

2 (η) − 2µ

εκ2ξ f ′2(η) + G2(η),

leads to

ψ(x, t) = G1(t − x) + G2(t + x) +γ + 1

4x[

f 21 (t − x) + f 2

2 (t + x)]

+γ − 3

4

[F1(t − x) f2(t + x) + f1(t − x)F2(t + x)

]

+µ

εκ2

x[

f ′1(t − x) + f ′2(t + x)]. (8.10)


and

Φ(x, t) = ε[

F1(t − x) + F2(t + x)]+ ε2

[G1(t − x) + G2(t + x)

]

+ ε2γ + 1

4x[

f 21 (t − x) − f 2

2 (t + x)]

+ ε2γ − 3

4

[F1(t − x) f2(t + x) + f1(t − x)F2(t + x)

]

+εµ

κ2

x[

f ′1(t − x) − f ′2(t + x)]+ O(ε3), (8.11)

so that

u(x, t) = ε{

f1(t − x) − f2(t + x)}

+ ε2{

g1(t − x) − g2(t + x)}

−ε2γ + 1

4

{[f 21 (t − x) − f 2

2 (t + x)]

−2x[

f1(t − x) f ′1(t − x) + f2(t + x) f ′2(t + x)]}

−ε2γ − 3

4

{F1(t − x) f ′2(t + x)− f ′1(t − x)F2(t + x)

}

− εµκ

2

{f ′1(t − x) − f ′2(t + x) − x

[f ′′1 (t − x) + f ′′2 (t + x)

]}+O(ε3). (8.12)

Note that some of the second-order terms on the right hand side are of the same form.We can therefore redefine gi as

gi(t) := gi(t)−{γ + 1

4f 2i (t) +

µ

εf ′i (t)

}, i = 1, 2,

and find

u(x, t) = ε{

f1(t − x) − f2(t + x)}

+ ε2{

g1(t − x) − g2(t + x)}

+ε2γ + 1

2x[

f1(t − x) f ′1(t − x) + f2(t + x) f ′2(t + x)]

+ε2γ − 3

4

{f ′1(t − x)F2(t + x)− F1(t − x) f ′2(t + x)

}

+εµ

κ2

x[

f ′′1 (t − x) + f ′′2 (t + x)]+ O(ε3). (8.13)

In [39] a zero velocity was imposed at x = 0, so that f1 = f2 and g1 = g2. Herethe situation is a bit different due to condition (8.5a). It assumes that because of thepresence of the stack the left-running wave is effectively reflected at some point −τ ≤ 0with reduced amplitude R0. As a result we can put

f1(t − x) = f (t − x), f2(t + x) = f (t + x), x = x + τ , (8.14)

g1(t − x) = g(t − x), g2(t + x) = g(t + x), (8.15)


where f and g are yet to be determined. Thus we can write for u

u(x, t) = ε{

R0 f (t − x)− f (t + x)}

+ε2βx[

R20 f (t − x) f ′(t − x) + f (t + x) f ′(t + x)

]

+ε2γ − 3

4R0

{f ′(t − x)F(t + x) − F(t − x) f ′(t + x)

}(8.16)

+εµ

κ2

x[

R0 f ′′(t − x) + f ′′(t + x)]+ε2

{R0g(t − x) − g(t + x)

}+O(ε3).

The remaining boundary condition (8.5b) can be used to derive an equation for theunknown functions f and g,

h(κt) ={

R0 f (t − t0) − f (t + t0)}

+εβt0

[R2

0 f (t − t0) f ′(t − t0) + f (t + t0) f ′(t + t0)]

+εγ − 3

4

{R0 f ′(t − t0)F(t + t0)− F(t − t0) f ′(t + t0)

}

+εµ0

κ2

[R0 f ′′(t − t0) + f ′′(t + t0)

]+ ε{

R0g(t − t0)− g(t + t0)}

+O(ε2), (8.17)

where t0 = 1 + τ .In the sections below we will solve this equation for two cases; we first look for a

solution away from resonance for an arbitrary excitation h and R0 = t0 = 1, and nextwe look for a solution near resonance with h(t) = sin(t) and arbitrary R0 and t0. Havingdone that we can use the second solution to determine the sound field between the rightstack end and the piston when the tube is near resonance and the first solution can beused to determine the sound field between the closed end (with R0 = t0 = 1) and theleft stack end.

8.2 Solution away from resonance

Depending on the order of magnitude of ε, µ, and δ, different approximations to (8.17)are possible. We assume µ, δ = O(ε) and introduce

µ =µ

ε, δ =

δ

ε. (8.18)

Moreover, for the subsequent analysis we also need

κ0 6= nπ , n ∈ Z. (8.19)

The last condition implies that we have to stay away from resonance (∆ 6= 0) and isnecessary to avoid division by zero. Below we will consider two cases; first for arbitraryexcitation h and then for harmonic excitation h(t) = sin(t).

140 8.2 Solution away from resonance

8.2.1 Arbitrary excitation

Substituting (8.18), we find that (8.17) transforms into

h(κt) ={

f (t − t0) − f (t + t0)}

+εβt0

[f (t − t0) f ′(t − t0) + f (t + t0) f ′(t + t0)

]

− 2εδ f (t − t0) +εγ − 3

4

{f ′(t − t0)F(t + t0) − F(t − t0) f ′(t + t0)

}

+εµ0

κ2

[f ′′(t − t0) + f ′′(t + t0)

]+ ε{

g(t − t0) − g(t + t0)}

+ O(ε2). (8.20)

Setting ε = 0 we find to leading order

f (t − t0) − f (t + t0) = h(κt).

We will look for a solution to this equation in the frequency domain. Taking the Fouriertransform F we find

− 2i sin(ωt0) [F f ] (ω) =1

κ[Fh]

(ωκ

). (8.21)

Rearranging terms and taking the inverse Fourier transform F−1, we obtain

f (t) =[F−1 A

](t) +

∞

∑n=−∞

aneinπt/t0 , A(ω) =i [Fh] (ω)

2 sin(ωt0), (8.22)

where the second term is a solution of the homogeneous problem. From causality argu-ments we conclude that an = 0 for all n.

Next, collecting the second-order terms, we find

g(t − t0)− g(t + t0) = 2δ f (t − t0) −βt0

[f (t − t0) f ′(t − t0) + f (t + t0) f ′(t + t0)

]

− µ0

κ2

[f ′′(t − t0) + f ′′(t + t0)

]− β− 2

2

{f ′(t − t0)F(t + t0) − F(t − t0) f ′(t + t0)

}.

Taking the Fourier transform on either side, we find

− 2i sin(ωt0) [F g] (ω) = [Fk] (ω), (8.23)

with

k(t) = 2δ f (t − t0) −βt0

[f (t − t0) f ′(t − t0) + f (t + t0) f ′(t + t0)

]

− µ0

κ2

[f ′′(t − t0) + f ′′(t + t0)

]− β− 2

2

{f ′(t − t0)F(t + t0) − F(t − t0) f ′(t + t0)

}.

Finally, applying the inverse Fourier transform we obtain

g(t) =[F−1B

](t), B(ω) =

i [Fk] (ω)

2 sin(ωt0). (8.24)


8.2.2 Harmonic excitation

Substituting h(t) = sin(t) into (8.21), we find

[F f ] (ω) = −√

2π

4 sin(ωt0)[δ(ω+κ) − δ(ω−κ)] =

√2π

4 sin(κ0)[δ(ω+κ) + δ(ω−κ)] .

Here and below δ(·) is used to denote the δ-function and is not to be confused with theconstant δ defined in (8.3). We can take the inverse Fourier transform and find

f (t) =cos(κt)

2 sin(κ0). (8.25)

Next, substituting f into (8.23), we obtain

[F g] (ω) =βκ0

8 sin(2κ0) tan(κ0)

√2π

2[δ(ω+ 2κ) + δ(ω− 2κ)]

− (µ0 + δ)

2 sin(κ0) tan(κ0)

√2π i

2[δ(ω+κ) − δ(ω−κ)]

+δ

2 sin(κ0)

√2π

2[δ(ω+κ) + δ(ω−κ)] .

Finally, applying the inverse Fourier transform we obtain

g(t) =βκ0

4

cos(2κt)

sin2(κ0)− µ0 + δ

2

cos(κ0) sin(κt)

sin2(κ0)+δ

2

cos(κt)

sin(κ0). (8.26)

The full solution, correct up to second order in ε, can be found by substituting f and ginto (8.16). For example, if we use only the leading-order terms, we find

u(x, t) = ε [ f (κt −κx)− f (κt +κx)] + o(ε)

= εsin(κx)

sin(κ0)sin(κt) + o(ε). (8.27)

For t0 = 0 this coincides with the standard solution for the sound field in closed tubeswithout wall friction.

8.3 Solution near resonance

In the previous section we showed how equation (8.4) can be solved using Fourier trans-formations, provided one stays away from resonance. In this section we will show thatunder certain conditions we can find a solution near resonance using matched asymp-totic expansions.

With the substitution

f (t) = v(κt), g(t) = w(κt), v′ = v, ζ1 = κt −κ0, ζ2 = κt +κ0,

142 8.3 Solution near resonance

we arrive at the following equation

h(κt) ={

R0v(ζ1) − v(ζ2)}

+εβκ0

[R2

0v(ζ1)v′(ζ1) + v(ζ2)v′(ζ2)]

+µ0

[R0v′′(ζ1) + v′′(ζ2)

]+εR0

β− 2

2

{v′(ζ1)v(ζ2) − v(ζ1)v′(ζ2)

}

+ε{

R0w(ζ1)− w(ζ2)}

+ o(ε). (8.28)

Whenω approaches the first resonance frequency, in the sense that ∆≪ 1, we can findan approximate solution to (8.28). Since ∆ = κ0 − π , we have

ζ1 = κt − π −∆, ζ2 = κt + π +∆.

Moreover, we can substitute for v(ζ1) − v(ζ2) the difference of their series expansion

v(ζ1) − v(ζ2) = v(κt − π) − v(κt + π) −∆[v′(κt − π) + v′(κt + π)

], ∆≪ 1.

If h is a periodic excitation near resonance, we can expect the function v to be almostperiodic. We model this by introducing two time scales,

ζ = κt + π , T = εκt, v(ζ , T) = v(ζ),

where ζ is the “fast” time and T is the “slow” time. We then assume that the deviationfrom periodicity occurs in the slow time scale and not in the fast time scale, so thatv(ζ , T) = v(ζ − 2π , T). Dropping the tildes, we can now write

v(ζ − 2π , T − 2πε) = v(ζ − 2π , T)− 2πε∂v

∂T(ζ − 2π , T)

= v(ζ , T)− 2πε∂v

∂T(ζ , T), ε≪ 1.

As a result we have

v(κt + π)− v(κt − π) = v(ζ , T)− v(ζ − 2π , T − 2πε) = 2πε∂v

∂T(ζ , T), ε≪ 1,

and

v(ζ1)− v(ζ2) = −2πε∂v

∂T(ζ , T)− 2∆

∂v

∂ζ(ζ , T), ∆,ε≪ 1. (8.29)

Inserting (8.29) into (8.28) and putting R0 = 1 − δ, we find

h(ζ − π) = −2πε∂v

∂T− 2∆

∂v

∂ζ− δv + 2εβκ0v

∂v

∂ζ+ 2µ0

∂2v

∂ζ2+ o(ε,∆,µ0, δ). (8.30)

It follows that to match the left and right hand side of the equation, we need to rescale


v. We introduce V by

v(ζ , T) =∆

εβκ0

+

√2

εβκ0

V(z, T), z =ζ

2,

This rescaling does not violate the asymptotic expansion given in (8.7) since ∆≪ 1. Wefind that (8.30) transforms into

V∂V

∂z+ν

[1

2

∂2V

∂z2− δ

µ0

V −νδ∆µ

20

− 2πε

µ0

∂V

∂T

]=

1

2h(2z − π), (8.31)

where

ν =

√µ

20

2εβκ0

.

In Section 8.3.1 we will give an analytical (z, T)-dependent solution for the case δ =∆ = 0, derived by Rudenko et al. [122].

We conclude from (8.31) that the effects of the stack (δ), resonance (∆), viscosity (µ0),and nonlinearity (ε) are balanced with the energy inflow (h), when

δ ∼ ∆ ∼ µ0 ∼√ε.

However, with this ordering of dimensionless parameters it is not possible to derive ananalytic solution to (8.31). If instead we assume

δ ∼ ∆ ∼ µ0, ν ≪ 1,

then a steady-state solution (T → ∞) can be determined using the smallness of ν. AtT → ∞ a steady-state field is reached due to the interaction of dissipation, nonlinearlosses and energy inflow from the the source. Setting ∂

∂T = 0, putting h(ζ) = sin(π),and introducing

δ =δ

µ0

, ∆ =∆

µ0

,

we find

2VdV

∂z+ν

[d2V

∂z2− 2δV

]− 2ν2

δ∆ = − sin(2z). (8.32)

In Section 8.3.2 we will derive a solution using matched asymptotic expansions, whenδ, ∆ = O(1). Next in Section 8.3.3 we show numerically how the solution changes when

the dissipation due to the stack is increased such that δ = O(ν−1).

8.3.1 Exact solution when δ = ∆ = 0 using Mathieu functions

Consider a closed tube at resonance (∆ = 0), periodically excited (h(t) = sin(t)), andwithout a stack ( τ = δ = 0). For this case it was shown by Rudenko et al. [64, 122] thatthe solution to (8.31) with zero initial condition v(T = 0,ζ) = 0 can be written in terms


of periodic Mathieu functions:

V(z, T) =∞

∑n=0

an exp

(−λ2nµ

4πεT

)CE2n (z, q) , (8.33)

where

a2n =

∫ 2π0 CE0

(ζ2 , q)

dζ∫ 2π

0 CE22n

(ζ2 , q)

dζ, q =

επβ

2µ2=

1

4ν2,

and λ2n is the characteristic eigenvalue corresponding to the periodic Mathieu eigen-function CE2n in the notation of Abramowitz and Stegun [3]. In particular when q ≫ 1we can write

λ0 = −2q + 2√

q − 1

4− 1

32√

q+ · · · .

The solution above simplifies in a few special cases. In the limit for T → ∞ a steady-state solution is approached and

v(ζ) =2µ′

πβ

d

dζln CE0

(ζ

2, q

). (8.34)

Since we assume ν ≪ 1 we also have q ≫ 1. As a result v can be approximated by [3]

v(ζ) =

√2

επβ

[cos

(ζ

2

)− 2 exp

(−2

√qζ)

1 + 2 exp(−2

√qζ)]

, 0 ≤ ζ ≤ π , (8.35)

so that for q → ∞ (ν → 0) we find

v(ζ) =

√2

επβcos

(ζ

2

)sgn(ζ), −π ≤ ζ ≤ π . (8.36)

8.3.2 Steady-state solution for δ = O(1)

Assuming δ, ∆ = O(1), we can use the smallness of ν to find an approximate solutionusing the method of matched asymptotic expansions [57]. First we look for an outersolution V that is valid outside a narrow region around some point zi ∈ (−π/2, π/2).Then we look for an inner solution V that is valid within this small region. A combina-tion of these two solution will give the full composite expansion.

We need some boundary conditions to guarantee a unique solution. A natural con-dition would be

V(−π/2) = V(π/2), (8.37)

so that in resonance, when ∆ = 0, v wil be periodic as well. Moreover, in the transi-tion point we will assume V(zi) = 0 for reasons of symmetry. The location zi will bedetermined from the matching conditions for V and V in the intermediate region.

We expand the outer solution in powers of ν,

V = V0 + νV1 + · · · , ν ≪ 1.


Inserting the expansions into (8.32) and collecting the terms of order ν0 and ν1, we find

2V0

dV0

dz= − sin(2z), (8.38)

d2V0

dz2+ 2

dV0V1

dz− 2δV0 = 0. (8.39)

Integrating (8.38) we find

V0 = ±√

cos2(z) + C0. (8.40)

where C0 is a constant of integration. Applying condition (8.37), we find that C0 = 0and

V0 = ± cos(z). (8.41)

We see that two possible solutions for V0 exist. Now before we go to V1, we willfirst turn to the inner solution. We expect that in the narrow region around zi there willbe a transition between the “-” and the “+” solution. Therefore an inner solution V isintroduced as

V(z) = V(s), s =z − zi

ν.

Substitution into equation (8.32) yields

d2V

ds2+ 2V

dV

ds− 2ν2

δV = −ν sin(2zi + 2νs) + 2δ∆ν3, ν ≪ 1. (8.42)

Again we expand in powers of ν,

V(s) = V0(s) +νV1(s) + · · · , ν ≪ 1.

It turns out that to find a solution we need to let zi depend on ν and we expand

zi = zi0+νzi1

+ · · · , ν ≪ 1.

Next we substitute the expansion and equate the coefficients of ν0 and ν1 to find

d2V0

ds2+ 2V0

dV0

ds= 0, (8.43)

d2V1

ds2+ 2

dV0V1

ds= − sin(2zi0

). (8.44)

We integrate (8.43) subject to V(0) = 0. By separation of variables we find

V0 = a0 tanh (a0s) , (8.45)

where a0 is yet to be determined. We can determine a0 by matching the inner solution


V0 to the outer solution V0 found above. Taking the limit for s → ±∞, we observe that

lims→−∞

V0 = −|a0|, lims→+∞

V0 = |a0|,

and we see that the inner solution V0 and outer solution V0 match, provided the changeis from the “-” solution to the “+” solution and a0 = cos(zi0

) ≥ 0. Summarizing, wefind

V0(z) =

{− cos(z), ζ < zi,

cos(z), ζ > zi,(8.46)

provided zi stays away from ±π/2. Furthermore

V0(s) = a0 tanh [a0s] , a0 = cos(zi0). (8.47)

Next we will compute the first-order terms. Substituting V0 into (8.39), we find afterintegration

V1 = ±1

2

C1

cos(z)+

1

2

(1 + 2δ

)tan(z). (8.48)

Applying (8.37), we obtain C1 = −1 − 2δ and

V1(z) =

(1 + 2δ

) sin(z) + 1

2 cos(z), ζ < zi,

(1 + 2δ

) sin(z) − 1

2 cos(z), ζ > zi.

(8.49)

Note that only this choice for C1 cancels the singularity that would otherwise arise inz = ±π/2. It still remains to determine V1. We integrate (8.44) once with respect to sand apply the integrating factor

exp

[2∫

V0 ds

]= cosh2(a0s).

We can then rewrite (8.44) as

d

ds

[cosh2(a0s)V1

]= −s sin(2zi0

) cosh2(a0s) − B1 cosh2(a0s),

where B1 is a constant of integration. Integrating once again with respect to s, we get

V1 =1

cosh2(a0s)

{− sin(2zi0

)s2

4− B1

s

2+

sin(2zi0)

8a20

[cosh (2a0s) − 1]

− 1

4a0

[B1 + sin(2zi0

)s]

sinh (2a0s)

}.

However, the first-order outer and inner solution do not necessarily attain a commonlimit as is the case for the zeroth-order terms. We will show that they only match when


sin(2zi0) is close to zero, thereby fixing zi0

. For this purpose we introduce an intermedi-ate variable

σ =z − zi

η(ν)=

ν

η(ν)s, η(ν) ≪ 1,

that is positioned between the O(1) coordinate of the outer layer and the O(ν) coordi-nate of the inner layer. We therefore assume

ν ≪ η(ν) ≪ 1,

and substitute σ in the outer and inner solution in some overlapping region where bothsolutions should hold. For this we choose η1(ν) and η2(ν) such that ν ≤ η1(ν) ≪η(ν) ≪ η2(ν) ≤ 1 where the outer solution is valid for η that satisfies η1(ν) ≪ η(ν) ≤ 1and the inner solution for η that satisfies ν ≤ η(ν) ≪ η2(ν). Without loss of generalitywe may assume σ > 0, so that

V1 =(1 + 2δ

) sin(zi + ησ) − 1

2 cos(zi + ησ)=(1 + 2δ

) sin(zi0) − 1

2 cos(zi0)

+ o(1),

and since η/ν ≫ 1

V1 =1

cosh2(a0σην)

{− sin(2zi0

)(ση

2ν

)2− B1

ση

2ν+

sin(2zi0)

8a20

[cosh

(2a0σ

η

ν

)− 1]

− 1

4a0

[B1 + sin(2zi0

)ση

ν

]sinh

(2a0σ

η

ν

)}

=

(1 + 4

δ

µ0

)sin(zi0

) − 1

2 cos(zi0)

+ o(1),

providedzi0

= 0, B1 = 1 + 2δ. (8.50)

With these restrictions V1 and V1 will attain a common limit for z − zi = O(η(ν)). Thusthe matching condition fixes the position of the boundary layer. If necessary, zi can bedetermined up to higher-order accuracy by computing and matching the higher-orderinner and outer solutions. However, knowledge of zi0

is enough to determine V up tofirst-order accuracy.

Summarizing we find that the outer solution V and the inner solution V are givenby

V(z) =

− cos(z) + ν(1 + 2δ

) sin(z) + 1

2 cos(z)+ o(ν), ζ < zi,

cos(z) + ν(1 + 2δ

) sin(z) − 1

2 cos(z)+ o(ν), ζ > zi,

(8.51)

and since a0 = 1,

V(s) = tanh (s) − ν

2

(1 + 2δ

){

s

cosh2(s)+ tanh(s)

}. (8.52)


The composite solution can be found by adding both solutions and subtracting the com-mon part. We find

V(z) = sgn(z − zi) [cos(z) − 1] + tanh

(z − zi

ν

)

+ν

2

(1 + 2δ

) { sin(z) − sgn(z − zi)

cos(z)+ sgn(z − zi)

− (z − zi)/ν

cosh2 ( z−zi

ν

) − tanh

(z − zi

ν

)}+ o(ν). (8.53)

As a result we obtain for ν ≪ 1

v(ζ) =∆

εβκ0

+

√2

εβκ0

{sgn (ζ)

[cos

(ζ

2

)− 1

]+ tanh

(ζ

2ν

)}(8.54)

+µ0 + 2δ

2εβκ0

{sin(ζ2

)− sgn (ζ)

cos(ζ2

) + sgn (ζ)− ζ

2ν cosh2 ( ζ2ν

) − tanh

(ζ

2ν

)}+ · · · ,

where we put zi = 0. Since tanh(x) approaches sgn(x) if x → ±∞ and cos(x) ap-proaches 1 if x → 0, we can rewrite the solution as

v(ζ) =∆

εβκ0

+

√2

εβκ0

cos

(ζ

2

)tanh

(ζ

2ν

)

+µ0 + 2δ

2εβκ0

{tan

(ζ

2

)− tanh

(ζ2ν

)

cos(ζ2

) − ζ

2ν cosh2 ( ζ2ν

)

}+ · · · . (8.55)

Letting ∆, δ,ν → 0 this expression coincides with the one derived in equation (8.36)using Mathieu functions.

Finally note that if we compute the average v, while omitting the odd parts of theintegrand, we find to leading order

v =1

2π

∫ π

−πv(ζ) dζ =

∆

βκ0ε+ o(1), ν ≪ 1. (8.56)

We conclude that in exact resonance, when ∆ = 0, v will have zero average.

8.3.3 Steady-state solution for δ = O(ν−1)

Previously we assumed δ = O(1) with ε. In this section we will investigate what hap-pens when the dissipation by the stack is increased by assuming νδ = O(1). Moreprecisely we will define

δ := 2νδ =2νδ

µ0

. (8.57)


With this rescaling, equation (8.32) transforms into

νd2V

dz2+ 2V

dV

dz− δV = − sin(2z) +νδ∆. (8.58)

Again we will use the smallness of ν to find an approximate solution via the methodof matched asymptotic expansions using an outer solution V and an inner solution Vthat is valid inside a narrow region around some point zi ∈ (−π/2, π/2). Rememberthe boundary conditions are periodicity for the outer solution,

V(−π/2) = V(π/2), (8.59)

and symmetry for the inner solution,

V(zi) = 0. (8.60)

Before we turn to the outer solution we will first determine the inner solution. Asbefore we introduce V by

V(z) = V(s), s =z − zi

ν.

Then V satisfies

d2V

ds2+ 2V

dV

ds− δνV = −ν sin(2zi + 2νs) + δ∆ν2, ν ≪ 1. (8.61)

We expand in powers of ν ≪ 1,

V(s) = V0(s) +νV1(s) + · · · ,

zi = zi0+ νzi1

+ · · · .

Next we substitute the expansions into (8.58) and equate the coefficients of ν0 and ν1 tofind

d2V0

ds2+ 2V0

dV0

ds= 0, (8.62)

d2V1

ds2+ 2

dV0V1

ds− δV0 = − sin(2zi0

). (8.63)

Then we integrate (8.62) subject to V(0) = 0. By separation of variables, we find

V0 = a0 tanh [a0s] , (8.64)

where a0 is yet to be determined. Without loss of generality we may assume a0 ≥ 0. Itstill remains to determine V1. We integrate (8.63) once with respect to s and apply theintegrating factor

exp

[2∫

V0 ds

]= cosh2(a0s).


We can then rewrite (8.63) as

d

ds

[cosh2(a0s)V1

]= −

(sin(2zi0

)s + B1

)cosh2(a0s) + δ log [cosh(a0z)] ,

where B1 is a constant of integration. Integrating once again with respect to s, we get

V1 =1

cosh2(a0s)

{− sin(2zi0

)s2

4− B1

s

2+

sin(2zi0)

8a20

[cosh (2a0s) − 1]

− 1

4a0

[B1 + sin(2zi0

)s]

sinh (2a0s) + δL(s)

}.

where

L(s) :=∫ s

0log [cosh(a0σ)] dσ

=1

2s2 − s log 2 +

π2

24+

1

2dilog(1 + e−2s)

=

1

2s2 − s log 2 +

π2

24+

1

2

∞

∑n=1

(−1)ne−2ns

n2s ≥ 0,

−1

2s2 − s log 2 − π

2

24− 1

2

∞

∑n=1

(−1)ne2ns

n2s ≤ 0.

It follows that to avoid blow-up of the solution for large s we need to have sin(2zi0) = 0,

i.e. zi0= 0, and we obtain

V1 =1

cosh2(a0s)

{−B1

s

2− B1

4a0

sinh (2a0s) + δL(s)

}. (8.65)

The constants a0 and B1 will be determined from the matching condition between theinner and outer solution. Taking the limits for s → ±∞, we find the following matchingconditions for V:

limz↑0

V0 = −a0 ≤ 0, limz↓0

V0 = a0 ≥ 0, (8.66)

limz↑0

V1 =B1

2a0

, limz↓0

V1 = − B1

2a0

. (8.67)

Next we turn to the outer solution. Substituting the expansion

V = V0 + νV1 + · · · , ν ≪ 1,


into (8.58) and collecting the terms of order ν0 and ν1, we find

2V0

dV0

dz− δV0 = − sin(2z), (8.68)

d2V0

dz2+ 2V0

dV1

dz+

(2

dV0

dz− δ)

V1 = δ∆. (8.69)

The equation for V0 cannot be solved in closed form and will be solved numerically.Therefore we will first turn to V1, which can be expressed in V0. With the integratingfactor

exp

[∫ z

0

1

2V0(s)

(2

dV0(s)

ds− δ)

ds

]= E(z)V0(z), E(z) = exp

[∫ z

0

δ

2V0(s)ds

],

we can rewrite (8.69) as

dEV0V1

dz=

E

2

[δ∆− d2V0

dz2

].

One more integration yields

V1(z) =∫ z E(s)

2E(z)V0(z)

[δ∆− d2V0(s)

ds2

]ds. (8.70)

Therefore once V0 is known we can apply (8.70) to determine V1. However, before wecompute V0 numerically we will first examine some properties of V0.

We consider the nonlinear differential equation

2YdY

dz− δY = − sin(2z), (8.71)

subject to either Y(π/2) = α, or Y(−π/2) = β. Since 2Y′ = δ− sin(2z)/Y it is clear thatif Y(z) = 0 for any z away from 0 then the solution will break down. This is shown infigure 8.1 in which we plot Y for variousα orβ. Forα orβ too close to zero, the solutionmay cross the the horizontal axis and break down. Theorem 1 below shows that, givenα 6= 0, Y can only cross zero in the right part of the domain and for given β 6= 0 in theleft part of the domain.

Theorem 1Suppose Y satisfies (8.71), then for any ǫ with 0 < ǫ < π/2 it holds that

(i) ifα > 0, then Y(z) > 0 for all z ∈ (ǫ, π/2];

(ii) if β < 0, then Y(z) > 0 for all z ∈ [−π/2, −ǫ);

(iii) ifα < 0, then Y(z) < 0 for all z ∈ (ǫ, π/2];

(iv) if β > 0, then Y(z) > 0 for all z ∈ [−π/2, −ǫ).

Proof.


(i) We will prove by contradiction. Suppose 0 < ǫ < π/2, α > 0 and z0 ∈ [ǫ, π/2)such that Y(z0) = 0 and Y(z0) > 0 for all z ∈ [0, z0). Then Y′(z0) > 0. However,from (8.71) we also find

limz↓z0

Y′(z) = limz↓z0

δ

2− sin(2z)

2Y(z)= −∞,

and we arrive at a contradiction.

(ii-iv) The proof goes analogous to that of (i). 2

−1.5 −1 −0.5 0 0.5 1 1.5−3

−2

−1

0

1

2

3

z

Y

α = −3α = −2α = −1α = −0.1α = 0.1α = 1

(a)

−1.5 −1 −0.5 0 0.5 1 1.5−3

−2

−1

0

1

2

3

z

Y

β = 3β = 2β = 1β = 0.1β = −0.1β = −1

(b)

Figure 8.1: Y as a function of z for δ = 1 and (a) Y(−π/2) = α or (b) Y(π/2) = β. If0 ≤ α ≤ 2.5 or 0 ≥ β ≥ −2.5, then the solution will break down when it crosses the horizontalaxis.

It now follows from Theorem 1 and the matching condition (8.66) that a necessarycondition for the inner and outer solutions to match is a change from a “-” solution V−

0

(with β < 0) to a “+” solution V+0 (with α > 0). Moreover, Theorem 1 ensures that

such a negative solution always exists for z < 0 and such a positive solution for z > 0.Since α > 0 and β < 0 it is clear that (8.59) can only be satisfied when α ↓ 0 and β ↑ 0,although neither can be exactly equal to zero. Finally, there are still the two matchingconditions in (8.66) that need to be satisfied. If we take a0 := V+

0 (0), then the second ofthe two matching conditions will be satisfied. Next, if we also choose β = −α, then itfollows from Theorem 2 that V−

0 (0) = −V+0 (0) = −a0 and the first condition in (8.66)

will be satisfied as well.

Theorem 2Suppose Y satisfies (8.71) subject to Y(π/2) = α, then Y: z 7→ −Y(−z) satisfies (8.71)with Y(−π/2) = −α.Proof.The statement follows immediately from substitution of Y into (8.71). 2

As an example of what V0 looks like we have plot in figure 8.2 the composite solutionthat is obtained by combining the leading order inner and outer solutions. The plots

have been computed for various values of ν and δ, α = 10−5 and ∆ = 0. We observe


that as the dissipation in the stack is increased and δ gets bigger, the shock wave getsless steep, until it completely disappears for δ ≫ 1. Similarly, when ν increases, theviscous effects in the tube become more dominant, and the shock wave becomes lesssteep.

−1.5 −1 −0.5 0 0.5 1 1.5−0.75

−0.5

−0.25

0

0.25

0.5

0.75

z

V0

ν = 0.01ν = 0.05ν = 0.1ν = 0.2

(a)

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

z

V0

δ = 0δ = 1δ = 2δ = 5

(b)

Figure 8.2: V0 as a function of z for α = 10−5 and ∆ = 0. In (a) ν is varied for δ = 1 and in(b) δ is varied for ν = 0.05.

Once V−0 and V+

0 have been computed, we can use (8.70) to compute

V+1 (z) =

∫ π/2

z

E+(s)

2E+(z)V+0 (z)

[d2V+

0 (s)

ds2− δ∆

]ds + C1, (8.72)

V−1 (z) =

∫ z

−π/2

E−(s)

2E−(z)V−0 (z)

[δ∆− d2V−

0 (s)

ds2

]ds + C1, (8.73)

where we imposed V+1 (π/2) = V−

1 (−π/2). Substituting V−0 (z) = −V+

0 (−z), we canwrite

V−1 (z) = −V+

1 (−z) − δ∆∫ z

−π/2

E−(s)

E−(z)V−0 (z)

ds + C1. (8.74)

Furthermore the two matching conditions in (8.67) can be reduced to one condition ifwe impose V−

1 (0) = −V+1 (0). This leads to the following expression for C1:

C1 = δ∆

∫ 0

−π/2

E−(s)

E−(0)V−0 (0)

ds = −δ∆∫ π/2

0

E+(s)

E+(0)V+0 (0)

ds.

Substituting C1 back into (8.72) and (8.74) we find

V+1 (z) =

∫ π/2

z

E+(s)

2E+(z)V+0 (z)

[d2V+

0 (s)

ds2− δ∆

]ds − δ∆

∫ π/2

0

E+(s)

E+(0)V+0 (0)

ds, (8.75)

V−1 (z) = −V+

1 (−z) + δ∆∫ z

−π/2

E−(s)

E−(z)V−0 (z)

ds + δ∆∫ 0

−π/2

E−(s)

E−(0)V−0 (0)

ds. (8.76)

154 8.4 Results

Lastly we determine B1 from the remaining matching condition in (8.67). We find

B1 = −2a0 limz↓0

V+1 (z)

= −∫ π/2

0E+(s)

[d2V+

0 (s)

ds2+ δ∆

]ds,

where we used a0 = V+0 (0) and E+(0) = 1.

The full composite solution is now found by adding the inner and outer solutionsand subtracting the common part.

V(z) =

V−0 (z) + V−

0

( z

ν

)+ a0 +ν

[V−

1 (z) + V−1

( z

ν

)− B1

2a0

]z ≤ 0.

V+0 (z) + V+

0

( z

ν

)− a0 +ν

[V+

1 (z) + V+1

( z

ν

)+

B1

2a0

]z ≥ 0.

Note that since V0(z) = −V0(−z), it follows that V = 0 + O(ν), so that to leadingorder V has zero average. If in addition ∆ = 0, then it follows from (8.74) that even

V = 0 +O(ν2).Figure 8.3 below shows the V1 profiles for various ν and δ. For the calculations

we used the V0 profiles shown in figure 8.2 . Special precaution has to be taken nearz = ±π/2 where there is a removable singularity similar to what we observed in theprevious section in (8.49) ; the numerical integration converges if z = ±π/2 is excluded.

−1.5 −1 −0.5 0 0.5 1 1.5

−2

−1

0

1

2

z

V1

ν = 0.01ν = 0.05ν = 0.1ν = 0.2

(a)

−1.5 −1 −0.5 0 0.5 1 1.5−6

−4

−2

0

2

4

6

z

V1

δ = 0δ = 0.5δ = 1δ = 1.5

(b)

Figure 8.3: V1 as a function of z for α = 10−5 and ∆ = 0. In (a) ν is varied for δ = 1 and in(b) δ is varied for ν = 0.05.

8.4 Results

We have tested the equations derived in the previous sections for two kinds of configu-rations:


(a) A closed tube excited near resonance by a speaker generating velocity oscillationsat a relative amplitude ε≪ 1 with h(t) = sin(t) (figure 8.4(a)). This configurationcan be used to model a thermoacoustic prime mover, with the speaker simulatingthe presence of the sound-producing stack.

(b) A thermoacoustic refrigerator, consisting of a closed tube with a stack of parallelplates positioned near the closed end. The speaker provides sound at a relativeamplitude ε≪ 1 with h(t) = sin(t) (figure 8.4(b)).

L

sound

(a) Closed tube

sound

LA LB LC

L

(b) Closed tube with thermoacoustic stack

Figure 8.4: We consider two kinds of configurations: (a) a closed tube driven near resonanceand (b) a thermoacoustic refrigerator driven near resonance with a stack near the closed end.

8.4.1 Nonlinear standing waves in a closed tube

We have simulated a closed tube at exact resonance with the parameters as given inTable 8.1. For this configuration we can compare the results of the matched asymptoticsin Section 8.3.2 for δ = 0 with the solution computed in Section 8.3.1 using Mathieufunctions.

First in figure 8.5 we give the nonlinear pressure and velocity profiles as a functionof both time and the position in the tube. Then in figure 8.6 we show how the nonlin-earity travels inside the tube at given times and we compare the result of the matchedasymptotics (solid line) with the zero-viscosity solution (dashed line) given in (8.36).

From figures 8.5 and 8.6 we conclude that as time changes the nonlinearity movesback and forth through the tube and enters and leaves the tube at the piston (x = L).Note that the pressure is an odd function of time and velocity is an even function of time.The zero-viscosity solution shows a discontinuous shock-like profile. We expect that asν decreases the matched asymptotics will converge to the same shock profile. Indeed itfollows from figure 8.7 that as ν decreases and the effect of viscosity becomes smallerthe profile will look more and more like the zero-viscosity shock wave. Note also that

although the sound waves were excited with a Mach number of only ε ≈ 6 · 10−6, we

find peak Mach numbers close to 6 · 10−3.

156 8.4 Results

(a) velocity (b) pressure

Figure 8.5: (a) The velocity u and (b) the pressure p as a function of place and time in the closedtube. The profiles were computed at exact resonance using the parameter values from Table 8.1.

0 0.2 0.4 0.6 0.8 1−400

−300

−200

−100

0

100

200

300

400

x/L

p (P

a)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/L

u (m

/s)

ωt = 0ωt = −π/4ωt = −π/2ωt = −3π/4ωt = −π

(a)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

u (m

/s)

x/L0 0.2 0.4 0.6 0.8 1

−400

−300

−200

−100

0

100

200

300

400

x/L

p (P

a)

ωt = 0ωt = π/4ωt = π/2ωt = 3π/4ωt = π

(b)

Figure 8.6: The velocity and pressure inside the closed tube at various times. The computationswere performed at exact resonance (∆ = 0), and there is a good agreement between the matchedasymptotics (-) and the zero-viscosity solution (- -).


Physical parameters Dimensionless parameterssymbol value unit symbol value

U 2.1 · 10−3 m s−1κ π

c0 330 m s−1∆ 0

L 0.5 m δ 0

f 330 - ν 4.4 · 10−2

b 1 · 10−2 kg m−1 s−2γ 1.4

ρ0 1 kg m−3ε 6.3 · 10−6

ℓ 1 · 10−6 m τ 0

Table 8.1: Parameter values for simulation of closed tube.

0−1

−0.5

0

0.5

1

ω t

u (m

/s)

ν = 0.1,ν = 0.05ν = 0.01

ππ/2-π -π/2

(a)

0−200

−100

0

100

200

ω t

p (P

a)

ν = 0.1,ν = 0.05ν = 0.01

ππ/2-π -π/2

(b)

Figure 8.7: (a) The velocity u and (b) the pressure p as a function of time in the center of thetube. As we vary ν from 0.1 to 0.01 (and viscosity decreases) we see that the nonlinear waveprofile starts to look more and more like a shock wave.

8.4.2 Nonlinear standing waves in a thermoacoustic refrigerator

In this section we will simulate a thermoacoustic refrigerator of the type shown in figure8.4(b). In particular we will simulate the thermoacoustic couple of Atchley et al. [11] thatwas described in Section 6.3. In addition to the stack specifications given in Table 6.1,we will use the parameter values given in Table 8.2 to describe segments A and C. Thevalue of τC depends on the frequency f and the length of the third segment LC and ischosen such that

∆C =ωLC(1 + τC)

c0

− π = 0,

and a nonlinearity will arise in the right part of the tube. In the left part of the tube wehave τA = 0 and

∆A =ωLA

c0

− π 6= 0.

We have implemented this configuration in three steps:

158 8.4 Results

Physical parameters Dimensionless parameterssymbol value unit symbol value symbol value

U 4.37 · 10−3 m s−1κA 0.215 κC π

c0 1.02 · 103 m s−1∆A -2.93 ∆C 0

LA 5 cm δA 0 δC 5e-3

LB 6.85 cm τA 0 τC 8.44 · 10−2

LC 73.1 cm γ 1.67 νC 1.58 · 10−3

ℓ 1e-6 m εC 4 · 10−4

f 696 Hz

b 2.87 · 10−4 kg m−1 s−2

ρ0 0.184 kg m−3

Table 8.2: Parameter values for thermoacoustic refrigerator.

1. First we simulate the segment C, between the stack and the speaker. At x = 0we impose a reflection coefficient by fixing δ and at x = LC we assume harmonicexcitation with h(t) = sin(t). Implementing the equations given in Section 8.3.3we can compute the pressure pC and the velocity uC.

2. We then compute the discrete Fourier transform of pC and uC at x = 0 and use itas a boundary condition for the sound field in the stack by applying continuity ofmass and momentum for each mode. For each mode the pressure, velocity, andtemperature profiles are computed by implementing the system of ode’s given in(4.70). Finally the full nonlinear sound field in the stack is computed using theinverse of the discrete Fourier transform.

3. Segment A is not at resonance and thus we can compute the sound field by imple-menting the off-resonance solution given in Section 8.3.1. The nonlinearity comesin via the right velocity condition which follows from the nonlinear velocity fieldin the stack by continuity of mass across the interface.

In figure 8.8 we give the nonlinear pressure and velocity profiles as a function of bothtime and the position for a reflection coefficient R0 = 0.995 (δ ≈ 1.67). Then in fig-ure 8.9 we show how the nonlinearity travels inside the thermoacoustic refrigerator atgiven times. As for the closed tube without stack we conclude that as time changesthe nonlinearity moves back and forth through the tube. Inside the stack the nonlinear-ity is damped due to dissipation in the narrow pores. Moreover, it follows from fromfigure 8.10 that as δ is increased (and R0 becomes smaller), the sharp nonlinearity issmoothened, and the velocity and pressure profiles change from the nonlinear shockwaves into a more familiar linear sinusoid profile.


(a) velocity in stack (b) velocity in tube

(c) pressure in stack (d) pressure in tube

Figure 8.8: The velocity u and pressure p as a function of place and time in (a,c) the stack and(b,d) the tube. The computations were performed at resonance with a reflection coefficient ofR = 0.995 and at a frequency of 696 Hz.

160 8.4 Results

0 0.2 0.4 0.6 0.8−2

−1

0

1

2x 10

−3

x(m)

Ag u

(m

3 /s)

(a) volumetric velocity

ωt = 0

ωt = −π/4

· · · ωt = −π/2

ωt = −3π/4

ωt = −π

0 0.2 0.4 0.6 0.8−300

−200

−100

0

100

200

300

x(m)

p (P

a)

(b) pressure

0 0.2 0.4 0.6 0.8−2

−1

0

1

2x 10

−3

x(m)

Ag u

(m

3 /s)

(c) volumetric velocity

ωt = 0

ωt = π/4

· · · ωt = π/2

ωt = 3π/4

ωt = π

0 0.2 0.4 0.6 0.8−300

−200

−100

0

100

200

300

x(m)

p (P

a)

(d) pressure

Figure 8.9: The velocity and pressure as a function of the position in the refrigerator at varioustimes.

0−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

ω t

u (m

/s)

ππ/2-π -π/2

δ = 0δ = 1δ = 2

(a) velocity

0−600

−400

−200

0

200

400

600

ω t

p (P

a)

ππ/2-π -π/2

δ = 0δ = 1δ = 2

(b) pressure

Figure 8.10: (a) The velocity u and (b) the pressure p as a function of time at the right stackend. As we vary δ from 0 to 2 the nonlinearity is smoothened.

Chapter 9

Conclusions and discussion

A weakly non-linear theory of thermoacoustics for arbitrary and slowly varying porecross-sections, applicable for stacks, regenerators, and resonators, has been developedbased on systematic dimensional analysis and using small-parameter asymptotics. Cru-cial assumptions for the asymptotic expansions are small amplitudes (Dr, Ma ≪ 1) andslow longitudinal variation (ε≪ 1). The theory is weakly non-linear in the sense that al-though the equations are linearized, terms up to second order in Ma and ε are included;in addition to the first and second harmonics this also includes streaming, second-ordertime-averaged mass flow.

Using the smallness of ε it is possible to decouple the transverse and longitudinalvariation of the fluid variables. It follows that a coupled system of ordinary differentialequations has to be solved for the mean temperature, the acoustic pressure, and the av-erage acoustic velocity, which can easily be implemented numerically. The problem ofdetermining the transverse velocity variation is reduced to finding Green’s functions fora modified Helmholtz equation on the given cross-section and solving two additionalintegral equations. Next the streaming velocity can be determined using a Green’s func-tion for the Poisson equation. Finally it is shown that the second-harmonic pressure andvelocity can also be determined from a system of ordinary differential equations simi-lar to the one found for the first harmonics. The asymptotic theory can in principle beextended to include higher-order terms, such as the higher harmonics and higher-orderstreaming, but this is avoided as it would require a lengthy and messy derivation andwould add little more understanding.

In addition to Ma and ε we have identified several other dimensionless parame-ters (Sk,κ, NL, Br) that are crucial in thermoacoustics. For these parameters we haveindicated various parameter regimes, each signifying specific geometrical or physicalconstraints. For specific parameter regimes an approximate analytic solution can be ob-tained. For example, at constant temperature the acoustic velocity and pressure in astraight tube can be written as a linear combination of sines and cosines with a complexwave number; for linearly changing cross-sections this becomes a combination of Besselfunctions of the first and second kind. The wave number will have a small imaginarypart to account for viscous dissipation, depending on the width of the tube. Anotherapproach predicts the steady-state temperature difference across the stack using a short-stack approximation (κ ≪ 1), which shows a change from a sinusoid profile for small

162

Prime MoverHeat Pump

Resonator

Figure 9.1: Combination of traveling-wave heat pump and prime mover using a double-Helmholtz resonator. The regenerators are located near velocity nodes of the tube to reducesviscous dissipation and the gas loop surrounding the regenerator gives the traveling-wave phas-ing. The sound produced by the prime mover is used to drive the refrigerator.

amplitudes to a sawtooth profile for large amplitudes. A third approximation assumeswide tubes (NL ≫ 1) and large temperature differences. It turns out that a tube thatsatisfies these conditions will have a temperature profile that changes from a linear pro-file in the absence of streaming to an exponential profile when the average mass flux isincreased.

The thermoacoustic equations have been applied to two kinds of thermoacousticsystems: standing-wave (straight-tube) devices and traveling-wave (looped-tube) de-vices. The standing-wave devices are modeled as (closed) straight tubes with a stackand heat exchangers placed inside and, in case of a refrigerator, driven by a speaker. Tovalidate our equations a numerical simulation of a thermoacoustic couple has been per-formed and compared to experimental and analytic results with very good agreement.Subsequently it has been shown numerically how a standing-wave prime mover andrefrigerator perform under various operating conditions. In particular we have investi-gated how parameters like drive ratio, temperature difference, mass flux, stack length,stack position, hydraulic radius, porosity, plate material, mean pressure, power-input,and power-output affect the relative coefficient of performance and efficiency.

A traveling-wave prime mover has been modeled as a resonance tube with variableacoustic load connected to a short looped tube containing a regenerator and thermalbuffer tube. The resonator is there to create the optimal phasing between pressure andvelocity at the location of the loop. In addition the loop has a contraction and expansionto create the proper phasing at the regenerator. In our numerical simulations we fixedthe regenerator and thermal buffer tube and performed an optimization routine thatdetermines a loop and resonator configuration that gives a stable system for given sys-tem parameters such as drive ratio, regenerator impedance, temperature difference, andfrequency. This optimization routine can be a very useful tool in the design of practi-cal traveling-wave systems. In addition we have optimized several of the regenerator’sproperties, such as impedance, drive ratio, temperature difference, regenerator length,porosity, hydraulic radius, mass flux, and regenerator material, to maximize the regen-erator efficiency. A comparison with an equivalent standing-wave prime mover showsthat the traveling-wave prime mover can potentially yield efficiencies that are twice ashigh.

The logical next step in the analysis of traveling-wave devices would combine thetraveling-wave prime mover with a traveling-wave refrigerator. One could for exampleremove the acoustic load and attach a looped refrigerator in its place, a so-called ther-moacoustically driven refrigerator (cf. [77]). Another possible approach uses a double-

Conclusions and discussion 163

Helmholtz resonator (figure 9.1), i.e. a closed resonator tube that expands towards thetube ends, with regenerators placed near both ends. If sufficient heat is supplied toone of the regenerators it is possible to produce enough sound to generate a significanttemperature difference across the other regenerator (cf. [78]). A third configuration, asproposed by Yazaki et al. [154], assumes the looped geometry given in figure 9.2, wherethe first regenerator generates spontaneous gas oscillations which are absorbed by thesecondary regenerator that is placed further along the loop. This is a much simplergeometry which ensures the gas in the regenerator to execute the ideal Stirling cycle.However, large velocities may occur near the regenerators, causing large viscous lossesand degrading the performance. The first two approaches are therefore recommended.

Refrigerator

Prime mover

Figure 9.2: Looped tube equipped with prime mover and refrigerator. If enough heat is suppliedto the first regenerator, then spontaneous gas oscillations will arise around the loop, generatinga temperature difference across the secondary regenerator.

Lastly, we have investigated the nonlinear disturbances that can arise in a thermoa-coustic refrigerator or prime mover, when it is excited near resonance. It is shownthat nonlinear shock-like profiles may arise, due to a balance of energy inflow fromthe source with nonlinear absorption and viscous dissipation in the gas and stack. Thepresence of a stack in a closed tube is modeled by means of a reflection condition, asa result of which we can compute the nonlinear wave profiles both in the stack in thetube.

164

Appendix A

Thermodynamic constants andrelations

In dimensional form we have the following thermodynamic constants and relations(taken from [25]).

c2 =

(∂p

∂ρ

)

s

, (A.1)

Cp = T

(∂s

∂T

)

p

=

(∂h

∂T

)

p

, (A.2)

Cv = T

(∂s

∂T

)

ρ

=

(∂ǫ∂T

)

ρ

, (A.3)

β = − 1

ρ

(∂ρ∂T

)

p

, (A.4)

c2β

2T = Cp(γ− 1), (A.5)

p = ρh − ρǫ, (A.6)

ds =Cp

TdT − β

ρdp, ⇒ s1 =

Cp

T0

T1 −β

ρ0

p1, (A.7)

dρ =γ

c2dp − ρβ dT, ⇒ ρ1 =

γ

c2p1 − ρ0βT1, (A.8)

dh = Tds +1

ρdp, ⇒ h1 = T0s1 +

p1

ρ0

, (A.9)

dǫ = Tds +p

ρ2

dρ, ⇒ ǫ1 = T0s1 −p0

ρ20

ρ1. (A.10)

166

Appendix B

Derivations

B.1 Total-energy equation

From the conservation of mass (3.1), momentum (3.2), and energy (3.3), we can derivethe following expressions

∂ρ∂t

= −∇ · (ρv) , (B.1)

ρ∂v

∂t= −ρ(v · ∇)v −∇p +∇ · τ + ρb, (B.2)

ρ∂ǫ∂t

= −ρ(v · ∇)ǫ+∇ · (K∇T) − p∇ · v + τ :∇v. (B.3)

We can use these relations to derive an equation for the change in total energy per unit

mass 12ρ|v|

2 + ρǫ,

∂∂t

(12ρ|v|

2 + ρǫ)

=∂ρ∂t

(1

2|v|2 +ǫ

)+ ρv ·

∂v

∂t+ ρ

∂ǫ∂t

= − 1

2|v|2∇ · (ρv) − ρv · (v · ∇)v −ǫ∇ · (ρv) − ρv · ∇ǫ+ ρv · b

− v · ∇p − p∇ · v + v · (∇ · τ) + τ :∇v +∇ · (K∇T) . (B.4)

Since∇ · (v · τ) = v · (∇ · τ) + τ :∇v, (B.5)

we can write

∂∂t

(12ρ|v|

2 + ρǫ)

= −∇ ·

[1

2ρ|v|2v + ρǫv + pv − K∇T − v · τ

]+ ρv · b,

= −∇ ·

[ρv

(1

2|v|2 + h

)− K∇T − v · τ

]+ ρv · b, (B.6)

where we used that h = ǫ + p/ρ. The same equation appears in [71], but without thegravitational term. It describes the rate of change of the total specific energy of the fluid

168 B.2 Temperature equation

due to an energy flux arising from four phenomena: transfer of mass by the motion ofthe fluid, transfer of heat, internal friction, and gravitational effects.

B.2 Temperature equation

Starting from equation (3.3), we can derive an equation for the fluid temperature T.After substitution of (3.5) and ǫ = h − p/ρ, we obtain

ρdh

dt= ρ

d

dt

(p

ρ

)+∇ · (K∇T) − p∇ · v + τ :∇v

=dp

dt+∇ · (K∇T) − p

ρ

dρ

dt− p∇ · v + τ :∇v

=dp

dt+∇ · (K∇T) + τ :∇v, (B.7)

where we substituted the continuity equation (3.1) into the last equality. It follows fromthe thermodynamic relations (A.9) and (A.7) that the total derivative of h can be writtenas follows:

ρdh

dt= ρT

ds

dt+

dp

dt

= ρCp

dT

dt+ (1 −βT)

dp

dt. (B.8)

If we substitute this back into (B.7), then we find

ρCp

dT

dt= βT

dp

dt+∇ · (K∇T) + τ :∇v. (B.9)

Appendix C

Green’s functions

In this section we will first show how the Fj-functions given in Section 5.1 and the Fj,2-functions given in Section 5.5 can be computed using Green’s functions [38]. Then wewill show how the Green’s functions can be computed for specific geometries.

C.1 Fj-functions

First we introduce the Green’s functions Gν and Gk. For every X we fix xτ ∈ Ag(X), setx := Xex + xτ , and solve for j = ν, k

G j(x; x) − 1

α2j

∇2τ G j(x; x) = δ(xτ − xτ), xτ ∈ Ag(X), (C.1a)

G j(x; x) = 0, Sg(x) = 0. (C.1b)

Using the Green’s identities, the Fj-functions introduced in (5.4) can be expressed in G j

as follows:

Fj(x) =∫

Ag

G j(x; x) dS, j = ν, k. (C.2)

In the same way we can also introduce the Green’s functions Gs. For fixed xτ ∈ As(X)we solve

Gs(x; x) − 1

α2s

∇2τ Gs(x; x) = δ(xτ − xτ), xτ ∈ As(X), (C.3a)

Gs(x; x) = 0, Sg(x) = 0, (C.3b)

∇τGs(x; x) · n′τ = 0, St(x) = 0. (C.3c)

170 C.1 Fj-functions

Given g j it can be shown that

Fk j(x) =∫

Ag

Gk(x; x) dS −∫

Γg

g j(x)∇τGk(x; x) · nτ dℓ,

Fs j(x) =∫

As

Gs(x; x) dS +∫

Γg

(1 − g j(x))∇τGs(x; x) · nτ dℓ.

The hats in the gradients and integrals are used to indicate that the differentiation orintegration is with respect to x. It only remains to determine the unknown boundaryfunctions gu and gp for which we will use the boundary condition (3.23c). If we impose

∇τFkp · nτ = −σ∇τFsp · nτ , Sg = 0,

∇τ (Fku − PrFν) · nτ = −σ∇τFsu · nτ , Sg = 0,

then (3.23c) is satisfied to leading order. We now find that gu and gp are found from thefollowing integral equations:

∫

Γg

Kb(x; x)g j(x) dℓ = Φ j(x), Sg(x) = 0, j = u, p, (C.4)

where Φu, Φp and Kb are defined as

Φp(x) =∫

Ag

∇τGk(x; x) · nτ dS,

Φu(x) =∫

Ag

∇τ (Gk(x; x)− PrGν(x; x)) · nτ dS,

Kb(x; x) = ∇τ(∇τ(

Gk(x; x) −σφGs(x; x))

· nτ

)· nτ .

In most cases these integral equations are not trivially solved. In [5] this problemwas avoided by neglecting the acoustic fluctuations in the wall, i.e. gu = gp = 0. Inthat case Fsu = Fsp = 1 and Fkp = Fku = Fk. The second case for which the solution issimple is when Ag is rotationally symmetric, i.e. circular or conical pores. Then Kb and

the g j-functions will be constant on Γg(X) and we find

g j =Φ j

Kb

, j = u, p. (C.5)

These two cases are discussed in more detail in Sections 5.3.1 and 5.3.2. For the generalcase one has to resort to a numerical implementation or to the general theory of integralequations. For example a solution can be attempted in the form of a sum of orthogonalbasis functions.

Green’s functions 171

C.2 Fj,2-functions

The Fj,2-functions can be determined in the same way as the Fj-functions. We first de-termine Green’s functions G j,2 (ν, k) from

G j,2(x; x)− 1

α2j,2

∇2τ G j,2(x; x) = δ(xτ − xτ), xτ ∈ Ag(X), (C.6a)

G j,2(x; x) = 0, Sg(x) = 0, (C.6b)

and Gs,2 from

Gs,2(x; x) − 1

α2s,2

∇2τ Gs,2(x; x) = δ(xτ − xτ), xτ ∈ As(X), (C.7a)

Gs,2(x; x) = 0, Sg(x) = 0, (C.7b)

∇τGs,2(x; x) · n′τ = 0, St(x) = 0. (C.7c)

It follows that the Ψ j,2-functions as introduced in (5.48) are given by

[Ψ j,2( f )

](x) =

∫

Ag

f (x)G j,2(x; x) dS, j = ν, k. (C.8)

In particular Fj,2 follows from Ψ j,2 by substitution of f ≡ 1,

Fj,2(x) =∫

Ag

G j,2(x; x) dS, j = ν, k. (C.9)

Furthermore Fk j,2 (k = u, p) is given by

Fk j,2(x) =∫

Ag

Gk,2(x; x) dS −∫

Γg

g j,2(x)∇τGk,2(x; x) · nτ dℓ,

Fs j,2(x) =∫

As

Gs,2(x; x) dS +∫

Γg

(1 − g j,2(x))∇τGs,2(x; x) · nτ dℓ,

where the boundary functions gu,2 and gp,2 are found from the following integral equa-tions:

∫

Γg

Kb,2(x; x)g j,2(x) dℓ = Φ j,2(x), Sg(x) = 0, j = u, p, (C.10)

where Φu,2, Φp,2 and K are defined as

Φp,2(x) =∫

Ag

∇τGk,2(x; x) · nτ dS,

Φu,2(x) =∫

Ag

∇τ(Gk,2(x; x) − PrGν,2(x; x)

)· nτ dS,

Kb,2(x; x) = ∇τ(∇τ(

Gk,2(x; x) −σφGs,2(x; x))

· nτ

)· nτ .

172 C.3 Green’s functions for various stack geometries

n Laplace: ∇2τ Modified Helmholtz: ∇2

τ − 1/α2j

1 −1

2|xτ − xτ |

1

2α j

exp(−α j|xτ − xτ |

)

2 − 1

2πlog |xτ − xτ |

1

2πK0(−α j|xτ − xτ |)

Table C.1: Free-space Green’s functions on Rn.

As above we note that if we assume a steady wall temperature then gu,2 = gp,2 = 0, sothat Fsu = Fsp = 1 and Fkp = Fku = Fk. Moreover, for rotationally symmetric pores we

find that Kb,2 and the g j,2-functions will be constant on Γg(X), so that

g j,2 =Φ j,2

Kb,2

, j = u, p. (C.11)

C.3 Green’s functions for various stack geometries

There is more than one way to determine the Green’s functions G j. One way is usingthe method of images [38]. The method of images adds homogeneous solutions to thefree-space Green’s function in such a way that their sum satisfies the right boundaryconditions. The free-space Green’s functions are given in Table C.1 and are fundamentalsolutions of the Laplace and modified Helmholtz equations that have suitable behaviorat infinity.

As an example we consider the case n = 1 where we have a geometry as shown inFig. C.1(a), so that xτ = y. Define

Φ j(x, x) :=1

2α j

exp(−α j|y − y|

), j = ν, k, s.

We now want to add a homogeneous function such that the resulting function vanishesat Γg. Introducing sources at the reflection points 2Rs − y and −2Rs − y, we can cancelthe contribution of y on Γg. However, to eliminate the contributions of 2Rs − y and−2Rs − y we have to introduce even more sources. Continuing this way we can writethe Green’s functions G j ( j = ν, k) in the form of an infinite sum,

G j(x, x) =∞

∑k=−∞

(Φ j [x; ak]−Φ j [x; bk]

), j = m,ν, k, (C.12)

whereak = (X, y + 4kRs), bk = (X, −y + (4k − 2)Rs).

Similarly we can show that in the solid Gs is given by the sum

Gs(x, x) =∞

∑k=−∞

(−1)k(Φs [x; ask]−Φs [x; bsk]

), (C.13)

Green’s functions 173

2Rs2Rt

y

(a) Parallel plates

θr

Rs Rt

(b) Circular cross-sections

y

z

2bt

2at

2bg

2ag

(c) Rectangular cross-sections

Figure C.1: Various stack geometries.

where

ask = (X, y + 2k(Rt −Rs)), bsk = (X, −y + 2k(Rt −Rs) + 2Rs).

For n = 2 we will employ a different approach that is also used by [38] and thatsolves for the Green’s functions by expanding in eigenfunctions. For circular pores thisleads to the following expressions:

Gm(x; x) =1

πR2s

∞

∑n=−∞

∞

∑i=1

Jn(knir)Jn(knir)

k2ni J

′2n (kniRs)

cos[n(θ− θ)

], (C.14)

G j(x; x) =1

πR2s

∞

∑n=−∞

∞

∑i=1

α2j Jn(knir)Jn(knir)

(k2ni −α2

j )J ′2n (kniRs)cos

[n(θ− θ)

], j = k,ν, (C.15)

Gs(x; x) =2

π

∞

∑n=−∞

∞

∑i=1

α2jJni(lnir)Jni(lnir) cos[n(θ− θ)]

ǫn(k2ni −α2

j )[R2t J ′2

ni (lniRt)−R2sJ ′2

ni (lniRs)], (C.16)

where the prime denotes differentiation and

Jni(r) = Yn(lniRs)Jn(r)− Jn(lniRs)Yn(r), ǫn =

{2, n = 0,

1, n > 0,

and Jn and Yn are the Bessel functions of the first and second kind, respectively. Fur-thermore, the eigenvalues kni and lni are computed from

Jn(kniRs) = 0,dJn

dr(lniRt) = 0.

174 C.3 Green’s functions for various stack geometries

−1−0.5

00.5

1

−1

0

1−0.2

−0.15

−0.1

−0.05

0

yz

(a) Re(

G j

)−1

0

1

−1

0

1−0.5

0

0.5

1

yz

(b) Im(

G j

)

Figure C.2: The Greens function G j for the Helmholtz equation over a rectangular region withDirichlet boundary conditions on the sides when ag = bg = 1,α j = 1 + i, and y = z = −0.3.

For rectangular pores we obtain the following Green’s functions:

Gm =4

agbg

∞

∑i,n=1

gin

i2π

2/a2g + n2

π2/b2

g

, (C.17)

G j =4

agbg

∞

∑i,n=1

α2j gin

i2π

2/a2g + n2

π2/b2

g − 4α2j

, j = k,ν, (C.18)

Gs =4

(as)(bs)

∞

∑i,n odd

α2s sin

i2π

2/(as)2 + n2

π2/(bs)

2 − 4α2s

, (C.19)

with eigenfunctions

gin(x; x) = sin

[iπ

y + ag

2ag

]sin

[iπ

y + ag

2ag

]sin

[nπ

z + bg

2bg

]sin

[nπ

z + bg

2bg

],

sin(x; x) = sin

[iπ

2

y − ag

as

]sin

[iπ

2

y − ag

as

]sin

[nπ

2

z − bg

bs

]sin

[nπ

2

z − bg

bs

],

where as = at − ag and bs = bt − bg. As an example of how such a Green’s function maylook, we have plotted G j in Fig. C.2 as a function of y and z on the unit square.

Bibliography

[1] S.I. Aanonsen, T. Barkve, J.N. Tjøtta, and S. Tjøtta, Distortion and harmonic genera-tion in the nearfield of a finite amplitude sound beam, Journal of Acoustical Society ofAmerica (1984), 749–768.

[2] P.C.H. Aben, P.R. Bloemen, and J.C.H. Zeegers, 2-D PIV measurements of oscillatoryflow around parallel plates, Experiments in Fluids (2008).

[3] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, National Bu-reau of Standards, Washington D.C., 1964,http://www.math.sfu.ca/ ˜ cbm/aands/ .

[4] S.B. Araujo, Downhole electric power generation by aeroacoustic oscillations, Univer-siteitsdrukkerij Eindhoven, Eindhoven, 2004.

[5] W. P. Arnott, H.E. Bass, and R. Raspet, General formulation of thermoacoustics forstacks having arbitrarily shaped pore cross sections, Journal of Acoustical Society ofAmerica 90 (1991), 3228–3237.

[6] A.A. Atchley, Standing wave analysis of a thermoacoustic prime mover below onset ofself-oscillation, Journal of Acoustical Society of America 92 (1992), 2907–2914.

[7] , Study of a thermoacoustic prime mover below onset of self-oscillation, Journalof Acoustical Society of America 91 (1992), 734–743.

[8] , Analysis of a thermoacoustics prime mover above onset of self-oscillation, Jour-nal of Acoustical Society of America 94 (1993), 1773.

[9] , Analysis of the initial buildup of oscillations in a thermoacoustic prime mover,Journal of Acoustical Society of America 95 (1994), 1661–1664.

[10] A.A. Atchley, H.E. Bass, and T. Hofler, Frontiers of nonlinear acoustics, ch. Develop-ment of nonlinear waves in a thermoacoustic prime mover, pp. 603–608, ElsevierScience Publishers, London, 1990.

[11] A.A. Atchley, T. Hofler, M.L. Muzzerall, M.D. Kite, and Chianing Ao, Acousti-cally generated temperature gradients in short plates, Journal of Acoustical Society ofAmerica 88 (1990), 251.

[12] J.L. Auriault, Heterogeneous medium. is an equivalent macroscopic description possible?,International Journal of Engineering Science 29 (1983), 785–795.

http://www.math.sfu.ca/~cbm/aands/

176 Bibliography

[13] , Upscaling heterogeneous media by asymptotic expansions, Journal of Engi-neering Mechanics 128 (2002), 817–822.

[14] S. Backhaus and G.W. Swift, A thermoacoustic stirling heat engine: Detailed study,Journal of Acoustical Society of America 107 (2000), 3148–3166.

[15] , New varieties of thermoacoustic engines, Proceedings of theNinth International Congress on Sound and Vibration (2002),www.lanl.gov/thermoacoustics/Pubs/ICSV9.pdf .

[16] S. Backhaus, E. Tward, and M. Petach, Traveling-wave thermoacoustic electric gener-ator, Applied Physics Letters 85 (2004), 1085–1088.

[17] H. Bailliet, V. Gusev, R. Raspet, and R.A. Hiller, Acoustic streaming in closed ther-moacoustic devices, Journal of Acoustical Society of America 110 (2001), 1808–1821.

[18] M.J.A. Baks, A. Hirschberg, B.J. van der Ceelen, and H.M. Gijsman, Experimen-tal verification of an analytical model for orifice pulse tube refrigeration, Cryogenics 30(1990), 947–951.

[19] W.E. Boyce and R.C. DiPrima, Elementary differential equations and boundary valueproblems, Wiley Interscience, New York, 1965.

[20] E. Buckingham, On physically similar systems: Illustrations of the use of dimensionalequations, Physical Review (1914), 345–376.

[21] H.B. Callen, Thermodynamics, ch. Appendix F, Wiley, New York, 1960.

[22] R.L. Carter, M. White, and A.M. Steele, Private communication of Atomics Interna-tional Division of North American Aviation, Inc., (1962).

[23] P.H. Ceperley, A pistonless stirling engine - the traveling wave heat engine, Journal ofAcoustical Society of America 66 (1979), 1508.

[24] , Gain and efficiency of a short traveling wave heat engine, Journal of AcousticalSociety of America 77 (1985), 1239–1244.

[25] C.J. Chapman, High speed flow, University Press, Cambridge, 2000.

[26] S. Chapman and T.G. Cowling, The mathematical theory of non-uniform gases; anaccount of the kinetic hteory of viscosity, thermal conduction, and diffusion in gases,Cambridge University Press, Cambridge, 1939.

[27] R. Chen and S.L. Garrett, A large solar/heat-driven thermoacoustic cooler, Journal ofAcoustical Society of America 108 (2000), 2554.

[28] W. Chester, Resonant oscillations in closed tubes, Journal of Fluid Mechanics 18(1964), 44–66.

[29] B.T. Chu, Analysis of a self-sustained thermally driven nonlinear vibration, The Physicsof Fluids 6 (1963), 1638–1644.

[30] B.T. Chu and S.J. Ying, Thermally driven nonlinear oscillations in a pipe with travelingshock waves, The Physics of Fluids 6 (1963), 1625–1637.

www.lanl.gov/thermoacoustics/Pubs/ICSV9.pdf

Bibliography 177

[31] A.B. Coppens and A.A. Atchley, Encyclopedia of acoustics, ch. Nonlinear standingwaves in cavities, pp. 237–247, Wiley, New York, 1997.

[32] A.B. Coppens and J.V. Sanders, Finite-amplitude standing waves in rigid-walled tubes,Journal of Acoustical Society of America 43 (1968), 516–529.

[33] , Finite-amplitude standing waves within real cavities, Journal of AcousticalSociety of America 58 (1975), 1133–1140.

[34] W. Dai, E. Luo, Y. Zhang, and H. Ling, Detailed study of a traveling wave thermoacous-tic refrigerator driven by a traveling wave thermoacoustic engine, Journal of AcousticalSociety of America 119 (2006), 2686–2692.

[35] A.T.A.M. de Waele, P.P. Steijaert, and J. Gijzen, Thermodynamical aspects of pulsetubes, Cryogenics 37 (1997), 313–324.

[36] S. Dequand, S.J. Hulshoff, and A. Hirschberg, Self-sustained oscillations in a closedside branch system, Journal of Sound and Vibration 265 (2003), 359–386.

[37] J.H.M. Disselhorst and L. van Wijngaarden, Flow in the exit of open pipes duringacoustic resonance, Journal of Fluid Mechancis 99 (1980), 293–321.

[38] D.G. Duffy, Green’s functions with applications, Chapman and Hall, London, 2001.

[39] B.O. Enflo and C.M. Hedberg, Theory of nonlinear acoustics in fluids, Kluwer Aca-demic Publishers, Dordrecht, 2002.

[40] , Theory of nonlinear acoustics in fluids, ch. Nonlinear standing waves inclosed tubes, Kluwer Academic Publishers, Dordrecht, 2002.

[41] K.T. Feldman Jr., Review of the literature on rijke thermoacoustic phenomena, Journalof Sound and Vibration 7 (1968), 83–89.

[42] , Review of the literature on sondhauss thermoacoustic phenomena, Journal ofSound and Vibration 7 (1968), 71–82.

[43] D.F. Gaitan and A. Atchley, Finite amplitude standing waves in harmonic and anhar-monic tubes, Journal of Acoustical Society of America 93 (1993), 2489–2495.

[44] D.L. Gardner, J.R. Olson, and G.W. Swift, Turbulent losses in thermoacoustic res-onators, Journal of Acoustical Society of America 98 (1995), 2961.

[45] S.L. Garrett, Thermoacoustic engines and refrigerators, American Journal of Physics72 (2004), 11–17.

[46] S.L. Garrett, J.A. Adeff, and T.J. Hofler, Thermoacoustic refrigerator for space applica-tions, Journal of Thermophysics and Heat Transfer 7 (1993), 595–599.

[47] S.L. Garrett and S. Backhauss, The power of sound, American Scientist 88 (2000),516–525.

[48] W.E. Gifford and R.C. Longsworth, Surface heat pumping, Advances in CryogenicEngineering 1 (1966), 302.

178 Bibliography

[49] V. Gusev, H. Bailliet, P. Lotton, and M. Bruneau, Asymptotic theory of nonlinearacoustic waves in a thermoacoustic prime-mover, Acustica 86 (2000), 25–38.

[50] V. Gusev, S. Job, H. Bailliet, P. Lotton, and M. Bruneau, Acoustic streaming in an-nular thermoacoustic prime-movers, Journal of Acoustical Society of America 108(2000), 934–945.

[51] V.E. Gusev, Build-up of forced oscillations in acoustic oscillators, Soviet Physics -Acoustics 30 (1984), 121–125.

[52] M.F. Hamilton, Y.A. Ilinskii, and E.A. Zabolotskaya, Acoustic streaming generated bystanding waves in two-dimensional channels of arbitrary width, Journal of AcousticalSociety of America 113 (2003), 153–160.

[53] L.S. Han, Hydrodynamic entrance lengths for incompressible laminar flow in rectangularducts, Journal of Applied Mechanics 27 (1960), 403–409.

[54] B. Higgins, Journal of Natural Philosophy and Chemical Arts 129 (1802), 22.

[55] T.J. Hofler, Thermoacoustic refrigerator design and performance, PhD Thesis, Physicsdepartment, University of California, San Diego, 1986.

[56] T.J. Hofler, J.C. Wheatley, G.W. Swift, and A. Migliori, Acoustic cooling engine,(1988), US Patent No. 4,722,201.

[57] M.H. Holmes, Introduction to perturbation methods, Springer, New York, 1995.

[58] Ulrich Hornung (ed.), Homogenization and porous media, Springer-Verlag NewYork, Inc., New York, NY, USA, 1997.

[59] Y. Ilinskii, B. Lipkens, T.S. Lucas, D.K. Perkins, and T.W. van Doren, Measurementsof macrosonic standing waves in oscillating closed cavities, Journal of Acoustical Soci-ety of America 104 (1998), 623–636.

[60] Y. Ilinskii, B. Lipkens, T.S. Lucas, T.W. van Doren, and E.A. Zabolotskaya, Nonlin-ear standing waves in an acoustical resonator, Journal of Acoustical Society of Amer-ica 104 (1998), 2664–2674.

[61] P.H.M.W. in ’t panhuis, S.W. Rienstra, J. Molenaar, and J.J.M. Slot, Weakly nonlin-ear thermoacoustics for stacks with slowly varying pore cross-sections, Journal of FluidMechanics 618 (2009), 41–70.

[62] J. Jiminez, Nonlinear gas oscillations in pipes. Part 1: Theory, Journal of Fluid Me-chanics 59 (1973), 23–46.

[63] M. M. Kaminski, On probabilistic viscous incompressible flow of some composite fluids,Computational Mechanics 28 (2002), 505–517.

[64] A.A. Karabutov, E.A. Lapshin, and O.V. Rudenko, Interaction between light wavesand sound under acoustic nonlinearity conditions, Journal of Experimental and Theo-retical Physics 44 (1976), 58.

Bibliography 179

[65] J. Keller, Resonant oscillations in closed tubes: the solution of Chester’s equation, Journalof Fluid Mechanics 77 (1976), 279–304.

[66] G. Kirchhoff, Ueber den Einfluss der Warmteleitung in einem Gas auf die Schallbewe-gung, Annalen der Physik 134 (1868), 177.

[67] H.A. Kramers, Vibrations of a gas column, Physica 15 (1949), 971.

[68] P.C. Kriesels, M.C.A.M. Peters, A. Hirschberg, and A.P.J. Wijands, High amplitudevortex-induced pulsations in a gas transport system, Journal of Sound and Vibration184 (1995), 343–368.

[69] E. Kroner, Modeling small deformations of polycrystals, ch. Statistical modeling, Else-vier, London, 1986.

[70] V.P. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics - Acoustics 16 (1971),467–470.

[71] L.D. Landau and E. M. Lifshitz, Fluid mechanics, Pergamon Press, Oxford, 1959.

[72] S. Laplace, Equations of nonlinear acoustics, Annales des Chemie et des Physique 3(1816), 328.

[73] J. Liang, Thermodynamic cycles in oscillating flow regenerators, Journal of AppliedPhysics 82 (1997), 4159–4165.

[74] W. Licht Jr. and D.G. Stechert, The variation of the viscosity of gases and vapors withtemperature, Journal of Physical Chemistry 48 (1944), 23–47.

[75] J. Lighthill, Acoustic streaming, Journal of Sound and Vibration 61 (1978), 391–418.

[76] H. Ling, E. Luo, and W. Dai, A numerical simulation method and analysis of a completethermoacoustic-stirling engine, Ultrasonics 44 (2006), 1511–1514.

[77] E. Luo, W. Dai, Y. Zhang, and H. Ling, Thermoacoustically driven refrigerator withdouble thermoacoustic-stirling cycles, Applied Physics Letters 88 (2006), 1511–1514.

[78] J.A. Lycklama a Nijeholt, M.E.H. Tijani, and S. Spoelstra, Simulation of a traveling-wave thermoacoustic engine using computational fluid dynamics, Journal of AcousticalSociety of America 118 (2005), 2265–2270.

[79] I. Lyulina, Numerical simulation of pulse-tube refrigerators, PhD Thesis, Eind-hoven University Press, 2005,http://alexandria.tue.nl/extra2/200510289.pdf .

[80] I.A. Lyulina, R.M.M. Mattheij, A.S. Tijsseling, and A.T.A.M. de Waele, Numericalsimulation of pulse-tube refrigerators, International Journal of Nonlinear Sciencesand Numerical Simulation 74 (2003), 91–97.

[81] , Numerical simulation of pulse-tube refrigerators: 1d model, Proceedings of theEurotherm Seminar: Heat Transfer in Unsteady and Transitional Flows 5 (2004),79–88, 287.

http://alexandria.tue.nl/extra2/200510289.pdf

180 Bibliography

[82] S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics,Part I, Acustica 82 (1996), 579–606.

[83] , Nonlinear and thermoviscous phenomena in acoustics, Part II, Acustica 83(1997), 197–222.

[84] J.L. Martin, J.A. Corey, and C.M. Martin, A pulse-tube cryocooler for telecommunica-tions applications, Cryocoolers 10 (1999), 181–189.

[85] R.M.M. Mattheij, S.W. Rienstra, and J.H.M. ten Thije Boonkkamp, Partial differen-tial equations: Modeling, analysis, computation, SIAM, Philadelphia, 2005.

[86] D.J. McKelvey and S.C. Ballister, Shipboard electronic thermoacoustic cooler, Mastersof Science in Engineering Acoustics and Master of Science in Applied Physics(1995), 161, DTIC Report No. AD A300-514.

[87] L. Menguy and J. Gilbert, Weakly nonlinear gas oscillations in air-filled tubes; solutionsand experiments, Acustica 86 (2000), 798–810.

[88] P. Merkli and H. Thomann, Thermoacoustic effects in a resonant tube, Journal of FluidMechanics 70 (1975), 161.

[89] , Transition to turbulence in oscillating pipe-flow, Journal of Fluid Mechanics68 (1975), 567–575.

[90] G. Mozurkewich, Time-average temperature distribution in a thermoacoustic stack,Journal of Acoustical Society of America 103 (1998), 380–388.

[91] U.A. Muller and N. Rott, Thermally driven acoustic oscillations. Part VI: Excitationand power, Journal of Applied Mathematics and Physics 34 (1983), 609–626.

[92] I. Newton, Philosophicae naturalis principia mathematica, vol. II, London, 1687,http://www.archive.org/details/newtonspmathema00new trich .

[93] A.L. Ni, Non-linear resonant oscillations of a gas in a tube under the action of a periodi-cally varying pressure, PMM USSR 47 (1983), 498–506.

[94] Ch. Nyberg, Spectral analysis of a two frequency driven resonance in a closed tube,Acoustical Physics 45 (1999), 94–104.

[95] W.L.M. Nyborg, Physical Acoustics, vol. IIB, ch. Acoustic streaming, p. 265, Aca-demic, New York, 1965.

[96] M. Ochmann and S. Makarov, Nonlinear and thermoviscous phenomena in acoustics,Part III, Acustica 83 (1997), 827–846.

[97] J.R. Olson and G.W. Swift, Similitude in thermoacoustics, Journal of Acoustical So-ciety of America 95 (1994), 1405–1412.

[98] , Acoustic streaming in pulse-tube refrigerators: Tapered pulse tubes, Cryogenics(1997), 769–776.

http://www.archive.org/details/newtonspmathema00newtrich

Bibliography 181

[99] A. Piccolo and G. Pistone, Computation of the time-averaged temperature fields andenergy fluxes in a thermally isolated thermoacoustic stack at low acoustic mach numbers,International Journal of Thermal Sciences 46 (2007), 235–244.

[100] M.E. Poesse and S.L. Garrett, Performance measurements on a thermoacoustic refriger-ator driven at high amplitudes, Journal of Acoustical Society of America 107 (2000),2480–2486.

[101] M.E. Poesse, R.W. Smith, S.L. Garrett, R. van Gerwen, and P. Gosselin,Thermoacoustic refrigeration for ice cream sales, Proceedings of 6th GustavLorentzen Natural Working Fluids Conference (2004),http://www.thermoacousticscorp.com/pdf/10.pdf .

[102] A.A. Putnam and W.R. Dennis, Survey of organ-pipe oscillations in combustion sys-tems, Journal of Acoustical Society of America 28 (1956), 246.

[103] M. Quintard and S. Whitaker, Transport in ordered and disordered porous media:volume-averaged equations, closure problems, and comparison wiht experiments, Chem-ical Engineering Science 48 (1993), 2537–2564.

[104] R. Radebaugh, A review of pulse tube refrigeration, Advances in Cryogenic Engi-neering 35 (1990), 11911205.

[105] , Development of the pulse tube refrigerator as an efficient and reliable cryocooler,Proceedings of the Institute of Refrigeration 96 (1999), 1–27.

[106] R. Raspet, H.E. Bass, and J. Kordomenos, Thermoacoustics of traveling waves: Theo-retical analysis for an inviscid ideal gas, Journal of Acoustical Society of America 94(1993), 2232–2239.

[107] R. Raspet, J. Brewster, and H.E. Bass, A new approximation method for thermoacousticcalculations, Journal of Acoustical Society of America 103 (1998), 2395–2400.

[108] R.L. Raun, M.W. Beckstead, J.C. Finlinson, and K.P. Brooks, A review of rijke tubes,rijke burners and related devices, Progress in Energy and Combustion Science 19(1993), 313–364.

[109] Lord Rayleigh, Theory of sound, Vol. II, Dover, New York, 1945.

[110] S.W. Rienstra and A. Hirschberg, An introduction into acoustics, ch. Webster’s hornequation, pp. 237–239, Eindhoven University of Technology, Eindhoven, 2008,http://www.win.tue.nl/ ˜ sjoerdr/papers/boek.pdf .

[111] P.L. Rijke, Notiz uber eine neue Art, die in einer an beiden Enden offenen Rohre enthal-tene Lift in Schwingungen zu versetzen, Annalen der Physik 107 (1859), 339.

[112] P. Riley, Pot boiler, Design Engineering (2007), 29–30,http://secure.theengineer.co.uk/Articles/303620/Pot +boiler.htm .

[113] H.S. Roh, W.P. Arnott, J.M. Sabatiera, and R. Raspet, Measurement and calculationof acoustic propagation constants in arrays of small air-filled rectangular tubes, Journalof Acoustical Society of America 89 (1991), 2617–2624.

http://www.thermoacousticscorp.com/pdf/10.pdf

http://www.win.tue.nl/~sjoerdr/papers/boek.pdf

http://secure.theengineer.co.uk/Articles/303620/Pot+boiler.htm

182 Bibliography

[114] H.S. Roh, R. Raspet, and H.E. Bass, Parallel capillary-tube-based extension of thermoa-coustic theory for random porous media, Journal of Acoustical Society of America 121(2007), 1413–1422.

[115] N. Rott, Damped and thermally driven acoustic oscillations in wide and narrow tubes,Journal of Applied Mathematics and Physics 20 (1969), 230–243.

[116] , Thermally driven acoustic oscillations. Part II: Stability limit for helium, Jour-nal of Applied Mathematics and Physics 24 (1973), 54–72.

[117] , The influence of heat conduction on acoustic streaming, Journal of AppliedMathematics and Physics 25 (1974), 417–421.

[118] , Thermally driven acoustic oscillations. Part III: Second-order heat flux, Journalof Applied Mathematics and Physics 26 (1975), 43–49.

[119] , Thermoacoustics, Advances in Applied Mechanics 20 (1980), 135–175.

[120] N. Rott and G. Zouzoulas, Thermally driven acoustic oscillations. Part IV: Tubes withvariable cross-section, Journal of Applied Mathematics and Physics 27 (1976), 197–224.

[121] , Thermally driven acoustic oscillations. Part V: Gas-liquid oscillations, Journalof Applied Mathematics and Physics 27 (1976), 325–334.

[122] O.V. Rudenko, C.M. Hedberg, and B.O. Enflo, Nonlinear standing waves in a layerexcited by the periodic motion of its boundary, Acoustical Physics 47 (2001), 525–533.

[123] W.V. Slaton, No moving parts electrical power generation for down-well environmentsor other remote locations, Technische Universiteit Eindhoven, Eindhoven, 2003.

[124] W.V. Slaton and J.C.H. Zeegers, Acoustic power measurements of a damped aeroacous-tically driven resonator, Journal of Acoustical Society of America 118 (2005), 83–91.

[125] , An aeroacoustically driven thermoacoustic heat pump, Journal of AcousticalSociety of America 117 (2005), 3628–3635.

[126] C. Sondhauss, Ueber die Schallschwingungen der Luft in erhitzten Glasrohren und ingedeckten Pfeifen von ungleicher Weite, Annalen der Physik 79 (1850), 1.

[127] S. Spoelstra and M.E.H. Tijani, Thermoacoustic heat pumps for energy savings,Proceedings of Boundary Crossing Acoustics (2005), 1–23,http://www.ecn.nl/docs/library/report/2005/rx05159. pdf .

[128] M.R. Stinson, The propagation of plane sound waves in narrow and wide circular tubes,and generalization to uniform tubes of arbitrary cross-sectional shape, Journal of Acous-tical Society of America 89 (1991), 550–558.

[129] M.R. Stinson and Y. Champoux, Assignment of shape factors for porous materials hav-ing simple pore geometries, Journal of Acoustical Society of America 88 (1990), S121.

[130] G.W. Swift, LANL Thermoacoustics Animations, Los Alamos National Laboratory,http://www.lanl.gov/thermoacoustics/movies.html .

http://www.ecn.nl/docs/library/report/2005/rx05159.pdf

http://www.lanl.gov/thermoacoustics/movies.html

Bibliography 183

[131] , Thermoacoustic engines, Journal of Acoustical Society of America 84 (1988),1146–1180.

[132] , Analysis and performance of a large thermoacoustic engine, Journal of Acous-tical Society of America 92 (1992), 1551–1563.

[133] , Thermoacoustic engines and refrigerators, Physics Today (1995), 22–28.

[134] , Thermoacoustic natural gas liquefier, Proceedings of the Department of En-ergy Natural Gas Conference (1997),http://www.netl.doe.gov/publications/proceedings/97 /97ng/ng97_pdf/

NG7-1.PDF .

[135] , A unifying perspective for some engines and refrigerators, Acoustical Societyof America, Melville, 2002.

[136] G.W. Swift and R.M. Keolian, Thermoacoustics in pin-array stacks, Journal of Acous-tical Society of America 94 (1993), 941–943.

[137] G.W. Swift and W.C Ward, Simple harmonic analysis of regenerators, Journal of Ther-mophysics and Heat Transfer 10 (1996), 652–662.

[138] G.W. Swift and J. Wollan, Thermoacoustics for liquefaction of natural gas, Los AlamosNational Laboratory (2002), 21–26,http://www.lanl.gov/thermoacoustics/Pubs/GasTIPS.pd f .

[139] K.W. Taconis, Vapor-liquid equilibrium of solutions of 3He in 4He, Physica 15 (1949),738.

[140] E. Tward, M. Petach, and S. Backhaus, Thermoacoustic space power converter, SpaceTechnology and Applications International Forum, AIP Conference Proceedings654 (2003), 656–661.

[141] Y. Ueda and C. Kato, Stability analysis of thermally induced spontaneous gas oscil-lations in straight and looped tubes, Journal of Acoustical Society of America 124(2008), 851–858.

[142] Y. Ueda, T. Kato, and C. Kato, Experimental evaluation of the acoustic properties ofstacked-screen regenerators, Journal of Acoustical Society of America 125 (2009),780–786.

[143] M. Van Dyke, Slow variations in continuum mechanics, Advances in Applied Me-chanics 25 (1987), 1–45.

[144] L. van Wijngaarden, Symposium on Applied Mathematics, dedicated to the late Prof.Dr. R. Ti, ch. Nonlinear acoustics, pp. 51–68, Delft University Press., Delft, 1978.

[145] G. Walker, Stirling engines, Clarendon, Oxford, 1960.

[146] W.C. Ward, J. Clark, and G.W. Swift, Design Environment for Low-amplitudeThermoacoustic Energy Conversion, Los Alamos National Laboratory (2008),http://www.lanl.gov/thermoacoustics/DeltaEC.html .

http://www.netl.doe.gov/publications/proceedings/97/97ng/ng97_pdf/

NG7-1.PDF

http://www.lanl.gov/thermoacoustics/Pubs/GasTIPS.pdf

http://www.lanl.gov/thermoacoustics/DeltaEC.html

184 Bibliography

[147] W.C. Ward and G.W. Swift, Design environment for low-amplitude thermoacoustic en-gines, Journal of Acoustical Society of America 95 (1994), 3671–3672.

[148] R. Waxler, Stationary velocity and pressure gradients in a thermoacoustic stack, Journalof Acoustical Society of America 109 (2001), 2739–2750.

[149] A.G. Webster, Acoustical impedance, and the theory of horns and of the phonograph,Proceedings of the National Academy of Academic Sciences 5 (1919), 275–282.

[150] J.C. Wheatley and A. Cox, Natural egines, Physics Today 38 (1985), 50–58.

[151] J.C. Wheatley, T. Hofler, G.W. Swift, and A. Migliori, An intrinsically irreversiblethermoacoustic heat engine, Journal of Acoustical Society of America 74 (1983), 153–170.

[152] , Understanding some simple phenomena in thermoacoustics with applications toacoustical heat engines, American Journal of Physics 53 (1985), 147.

[153] J.C. Wheatley, G.W. Swift, and A. Migliori, The natural heat engine, Los AlamosScience (1986),http://www.fas.org/sgp/othergov/doe/lanl/pubs/00326 875.pdf .

[154] T. Yazaki, T. Biwa, and A. Tominaga, A pistonless stirling cooler, Applied PhysicsLetters 80 (2002), 157–159.

[155] T. Yazaki, A. Iwata, T. Maekawa, and A. Tominaga, Traveling wave thermoacousticengine in a looped tube, Physical Review Letters 81 (1998), 3128–3131.

[156] T. Yazaki, A. Tominaga, and Y. Narahara, Stability limit for thermally driven acousticoscillations, Cryogenics 19 (1979), 393–396.

[157] A. Zaoui, Homogenization techniques for composite media, ch. Approximate statisticalmodelling and applications, Berlin, 1987.

[158] L. Zoontjens, C.Q. Howard, A.C. Zander, and B.S. Cazzolato, Numerical comparisonof thermoacoustic couples with modified stack plate edges, International Journal of Heatand Mass Transfer 51 (2008), 4829–4840.

http://www.fas.org/sgp/othergov/doe/lanl/pubs/00326875.pdf

Index

Aacoustics . . . . . . . . . . . . . . . . . . . . . . . . 1, 39, 74adiabatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6airconditioning . . . . . . . . . . . . . . . . . . . . . . . . 12approximation

boundary-layer . . . . . . . . . . . . . . . . . . . 96short-stack . . . . . . . . . . . . . . . . 54, 96, 161wide-channel . . . . . . . . . . . . . . . . . . . . . 56

asymptotics . . . . . . . . . . . . . . . . . . . 12, 34, 161attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

BBernoulli

effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112equation . . . . . . . . . . . . . . . . . . . . . . . . . 136

bucket-brigade effect . . . . . . . . . . . . . . . . . . 25Buckingham π theorem . . . . . . . . . . . . . . . 31

Ccoefficient of performance . . . . 19, 101, 103

Carnot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19relative . . . . . . . . . . . . . . . . . . . . . . .19, 103

composite expansion . . . . . . . . . . . . . . . . . 144conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 28

energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28momentum . . . . . . . . . . . . . . . . . . . . . . . 28

convective derivative . . . . . . . . . . . . . . . . . . 28critical temperature gradient . . . . . . . . . 103cross-sections

cylindrical . . . . . . . . . . . . . . . . . . . . .14, 73parallel-plate . . . . . . . . . . . . . . 14, 37, 96pin-array . . . . . . . . . . . . . . . . . . . . . . 14, 73rectangular . . . . . . . . . . . . . . . . . . . . 14, 73triangular . . . . . . . . . . . . . . . . . . . . . 14, 73wired-mesh . . . . . . . . . . . . . . . . . . . . . . .73

cryocooling . . . . . . . . . . . . . . . . . . . . . . . . . 4, 11cycle

Brayton . . . . . . . . . . . . . . . . . . . . . . . . 7, 20

Stirling . . . . . . . . . . . . . . . . . . . . . . . . . 7, 20thermodynamic . . . . . . . . . . . . . . . . 7, 20

DDarcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 121DeltaE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16DeltaEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16devices

heat-driven . . . . . . . . . . . . . . . . . . . . . . . . 8sound-driven . . . . . . . . . . . . . . . . . . . . . . 8

dimensional analysis . . . . . . . . . . . . . . . . . . 12dimensionless parameters . . . . . . 13, 15, 31dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

thermal-relaxation . . . . . . . . . . . 68, 100viscous . . . . . . . . . . . . . . . . . . . . . . .68, 100

down-well power generation . . . . . . . . . . 10

Eefficiency . . . . . . . . . . . . . . . . . . 1, 20, 119, 125

Carnot . . . . . . . . . . . . . . . . . . . . . . . 20, 119relative . . . . . . . . . . . . . . . . . . . . . . .20, 119

entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Ffood refrigerators . . . . . . . . . . . . . . . . . . . . . 12Fourier transform

continuous . . . . . . . . . . . . . . . . . . . . . . 140discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Fourier’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Ggas liquefaction . . . . . . . . . . . . . . . . . . . . . . . 11Green’s function . . . . . . . . . . . . . . . . . . 73, 169

Hharmonics

first . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35higher . . . . . . . . . . . . . . . .15, 95, 133, 161second . . . . . . . . . . . . . . . . . 14, 35, 61, 86

186 Index

Iindustrial waste heat . . . . . . . . . . . . . . . . . . 11inner solution . . . . . . . . . . . . . . . . . . . 144, 148integral equations . . . . . . . . . . . . . . . . 73, 170integrating factor . . . . . . . . . . . . . . . . . . . . .149irrotational flow . . . . . . . . . . . . . . . . . . . . . . 134

KKuznetsov’s equation . . . . . . . . . . . . . . . . 134

Lleast-squares solution . . . . . . . . . . . . . . . . 122Los Alamos National Laboratory . . . 11, 16

MMathieu functions . . . . . . . . . . . . . . . . . . . . 143method

homogenization . . . . . . . . . . . . . . . . . . 14matched asymptotic expansions . 144slow variation . . . . . . . . . . . . . . . . . . . . 37

Nnondimensionalization . . . . . . . . . . . . . . . . 13nonlinearities . . . . . . . . . . . . . . . . 13, 100, 134

Ooscillations

heat-driven . . . . . . . . . . . . . . . . . . . . . . . . 8sound-driven . . . . . . . . . . . . . . . . . . . . . . 8

outer solution . . . . . . . . . . . . . . . . . . . 144, 148

Ppulse-tube refrigerator . . . . . . . . . . . . . . . . . .4

Rreflection condition . . . . . . . . . . . . . . . . . . 134regenerator . . . . . . . . . . . . . . . . . . . . . . . 5, 9, 24resonance . . . . . . . . . . . . . . . . . . . . . . . 139, 141

frequency . . . . . . . . . . . . . . . . . . . . . . . 134Rijke tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

SSCORE stove . . . . . . . . . . . . . . . . . . . . . . . . . . 12self-consistent models . . . . . . . . . . . . . . . . . 14self-oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 15

above . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15below . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15through . . . . . . . . . . . . . . . . . . . . . . . . . . .15

shock waves . . . . . . . . . 15, 95, 133, 155, 158

side branch . . . . . . . . . . . . . . . . . . . . . . . . . . . .11singing flame . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Sondhauss tube . . . . . . . . . . . . . . . . . . . . . . . . 3speed of sound . . . . . . . . . . . . . . . . . . . . 2, 165stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4, 9, 22standing wave . . . . . . . . . . . . . . . 6, 9, 20, 162

nonlinear . . . . . . . . . . . . . . . . . . . . . . . . 133standing-wave . . . . . . . . . . . . . . . . . . . . . . . . 93statistical modeling . . . . . . . . . . . . . . . . . . . .14streaming . 14, 15, 35, 59, 85, 112, 123, 127,

161Gedeon . . . . . . . . . . . . . . . . . . . . . . . . . . . 59inner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Rayleigh . . . . . . . . . . . . . . . . . . . . . . 29, 59

Sutherland’s formula . . . . . . . . . . . . . . . . . . 29

TTaconic oscillations . . . . . . . . . . . . . . . . . . . . . 3thermoacoustic

couple . . . . . . . . . . . . . . . . . . . . . . . 96, 157devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1heat pump . . . . . . . . . . . . . . . . . . . 1, 7, 19prime mover . . . . . . . . . 1, 7, 8, 108, 133refrigerator . . . . . . . 1, 7, 8, 19, 100, 157

thermoacoustics . . . . . . . . . . . . . . . . . . . . . . . . 1spacecraft . . . . . . . . . . . . . . . . . . . . . . . . .12weakly nonlinear . . . . . . . . . . . . . . . . . 12

thermodynamics . . . . . . . . . . . . . . . . . .1, 6, 17laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

transition effects . . . . . . . . . . . . . . . . . . . . . . .15traveling wave . . . . . . . . . . . . . . . 6, 9, 22, 162turbulence . . . . . . . . . . . . . . . . . . . . . . . 2, 15, 95

Vvolume-averaging techniques . . . . . . . . . 14vortex shedding . . . . . . . . . . . . . . . . . . . . . . . . 2

Summary

This thesis addresses the mathematical aspects of thermoacoustics, a subfield withinphysical acoustics that comprises all effects in which heat conduction and entropy vari-ations of the gaseous medium play a role. We focus specifically on the theoretical basisof two kinds of devices: the thermoacoustic prime mover, that uses heat to producesound, and the thermoacoustic heat pump or refrigerator, that use sound to produceheating or cooling.

Two kinds of geometry are considered. The first one is the so-called standing-wavegeometry that consists of a closed straight tube (the resonator) with a porous medium(the stack) placed inside. The second one is the so-called traveling-wave geometry thatconsists of a resonator attached to a looped tube with a porous medium (regenerator)placed inside. The stack and the regenerator differ in the sense that the regenerator usesthinner pores to ensure perfect thermal contact. The stack or regenerator can in principlehave any arbitrary shape, but are modeled as a collecting of long narrow arbitrarilyshaped pores. If the purpose of the device is to generate cooling or heating, then usuallya speaker is attached to the regenerator to generate the necessary sound.

By means of a systematic approach based on small-parameter asymptotics and di-mensional analysis, we have derived a general theory for the thermal and acoustic be-havior in a pore. First a linear theory is derived, predicting the thermoacoustic behaviorbetween two closely placed parallel plates. Then the theory is extended by consider-ing arbitrarily shaped pores with the only restriction that the pore cross-sections varyslowly in longitudinal direction. Finally, the theory is completed by the inclusion ofnonlinear second-order effects such as streaming, higher harmonics, and shock-waves.It is shown how the presence of any of these nonlinear phenomena (negatively) affectsthe performance of the device.

The final step in the analysis is the linking of the sound field in the stack or regenera-tor to that of the main tube. For the standing-wave device this is rather straightforward,but for the traveling-wave device all sorts of complications arise due to the complicatedgeometry. A numerical optimization routine has been developed that chooses the rightgeometry to ensure that all variables match continuously across every interface and theright flow behavior is attained at the position of the regenerator. Doing so, we can pre-dict the flow behavior throughout the device and validate it against experimental data.The numerical routine can be a valuable aid in the design of traveling-wave devices;by variation of the relevant problem parameters one can look for the optimal traveling-wave geometry in terms of power output or efficiency.

188 Summary

Samenvatting

Dit proefschrift is gewijd aan de wiskundige aspecten van thermoakoestiek, het vakge-bied dat bestaat uit alle akoestische verschijnselen waarin de warmtegeleiding en en-tropie variaties van het gas een rol spelen. In het bijzonder richten we ons op de the-oretische fundamenten van twee soorten thermoakoestische apparaten: de motor, diewarmte omzet in geluid, en de warmtepomp of koelkast, die geluid gebruiken om ver-warming of koeling te genereren.

We onderscheiden staande-golf en lopende-golf systemen. De staande-golf systemenmaken maken gebruik van een gesloten rechte buis (de resonator) waarin een poreusmedium, de stack, geplaatst wordt om warmte om te zetten in geluid of geluid in warmte.De stack is opgebouwd uit nauwe kanaaltjes met willekeurige doorsneden. Als het doelvan het apparaat koelen of verwarmen is, dan is het gebruikelijk om de buis met eenluidspreker te verbinden om het noodzakelijke geluid te genereren. In de lopende-golfsystemen wordt de resonator gecombineerd met een lusvormige geometrie en wordtde stack vervangen door een regenerator. De lusvormige geometrie is nodig om hetgeluidsveld in de regenerator het lopende golf karakter te geven. De regenerator is ver-gelijkbaar met de stack, maar heeft veel nauwere kanaaltjes om perfect warmte-contactmet het gas te garanderen.

Gebruik makend van een systematische aanpak, gebaseerd op dimensie analyse enkleine-parameter asymptotiek, hebben we getracht een algemene thermoakoestischetheorie af te leiden. We beginnen met een lineaire theorie, die het thermoakoestischegedrag voorspelt tussen twee parallelle platen. Vervolgens wordt de theorie uitge-breid door willekeurige drie-dimensionale stack kanalen te beschouwen met de enigebeperking dat de dwarsdoorsneden langzaam veranderen in axiale richting. Uitein-delijk wordt de theorie voltooid door het toevoegen van kwadratische niet-lineaire ef-fecten zoals hogere harmonischen, schok golven, of stromingseffecten. Bovendien latenwe zien hoe de aanwezigheid van deze niet-lineaire effecten de prestaties van de ther-moakoestische apparaten (negatief) beınvloedt.

De laatste stap in de analyse is de koppeling tussen het geluidsveld in de stack ofregenerator en het geluidsveld van de hoofdbuis. Dit is relatief eenvoudig voor destaande-golf apparaten, maar de lopende-golf apparaten vereisen een speciale aanpakvanwege hun complexe geometrie. We hebben een numerieke optimalisatie routine ont-wikkeld die voor elke sectie de juiste afmetingen uitrekent zodat alle variabelen continuaansluiten over elke interface en het gewenste geluidsveld bereikt wordt bij de regene-rator. Op deze manier kan men het hele apparaat doorrekenen en valideren met metexperimentele data. De numerieke routine kan een waardevolle houvast zijn voor hetontwerpen van praktische lopende-golf apparaten.

190 Samenvatting

Curriculum Vitae

Peter in ’t panhuis was born in Roermond, The Netherlands, on July 10th 1981. After fin-ishing his pre-university education in Sittard at the Serviam College (now Trevianum)in 1999, he started his studies in Technical Mathematics at the Eindhoven University ofTechnology that same year. During his studies he did an internship at the Marcus Wal-lenberg Laboratory in Stockholm, Sweden, entitled “Calculations of the acoustic endcorrection of a semi-infinite circular pipe issuing a subsonic cold or hot het with co-flow”. In April 2005 he obtained his master’s degree in Industrial and Applied Math-ematics after writing a master’s thesis entitled “Li-ion battery modelling” under thesupervision of prof.dr. Hans van Duijn and dr. Evgeniy Verbitskiy.

From May 2005 till June 2009 he has been working as a PhD student at the Eind-hoven University of Technology within the Applied Analysis group, which is part of theDepartment of Mathematics and Computer Science, and the Low Temperature group,which is part of the Department of Applied Physics. This project, entitled “High-amplitudeoscillatory gas flow in interaction with solid boundaries”, was sponsored by the Technl-ogy Foundation (STW) and was performed under the supervision of dr. Sjoerd Rienstra,prof.dr. Han Slot, prof.dr. Jaap Molenaar, and dr. Jos Zeegers.

In addition to writing this thesis, he has taught classes to students of various facultiesat the Eindhoven University of Technology. Moreover, he has participated in four studygroups “Mathematics with Industry” at universities in the Netherlands (Eindhoven,Utrecht) and Denmark (Odense, Lyngby).

Documents

Mathematical aspects of thermoacoustics - Pure · Mathematical Aspects of Thermoacoustics PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,