heat transfer from a cylinder in an acoustic standing wave

8/20/2019 Heat transfer from a cylinder in an acoustic standing wave

1/8

Heat transfer from a cylinder in an acoustic standing wave

George Mozurkewich

PhysicsDepartment,Ford Motor CompanyResearchLaboratory,Mail Drop 3028, P.O. Box 2053,

Dearborn, Michigan 48121-2053

(Received 4 February1995;acceptedor publication 4 May 1995)

Heat transferwas measured rom wires of various diameters ocated at a velocity antinode n an

acoustic tandingwave.A transientmethodwasused, n which the rate of heat ransferwas deduced

from the rate of changeof temperature fter heatingwas turned off. For fixed wire diameterand

acoustic requency,he dimensionlesseat-transfer oefficient Nusseltnumber)Nu showsa

distinctive variation with acousticamplitude. At high amplitude, Nu follows the well-known,

steady-flow, orced-convection orrelation, ime averagedover an acousticcycle, while at low

amplitude,Nu has a constant alue determined y naturalconvection. he transitionbetween hese

regimes,which occurs ather abruptlywhen the streamingReynoldsnumber (basedon wire

diameterand acoustic elocityamplitude) quals88, is discussedn terms of a heat-transfer

bottleneck hat s openedby acoustic treaming. n empiricalcorrelation s presented. pplicability

to heat exchangersor thermoacoustic eat engines s considered. 1995 AcousticalSocietyof

America.

PACS numbers: 43.35.Ty, 43.35.Ud, 43.25.Nm

INTRODUCTION

Thermoacoustic ngines or cooling and for soundgen-

eration have recently become a subject of technological

interest.In somenstances,hepracticalityf thermoacous-

tic enginesdependson efficient operation.A major factor

contributing o overall efficiency s the effectiveness f heat

transferbetween the soundwave and the heat exchangers.

Optimal designwill dependon an understanding f the me-

chanicsof the heat-transfer rocess.

Thermoacoustic eat exchangers an be broadly classi-

fied into those involving internal flow in ducts or porous

geometries nd those nvolving external low aroundobjects

that are genericallycylinders.The first class s represented

by parallel-plateand honeycomb tructures hile the second

includes wire screens and reticulated foam. Work in this

laboratoryhas employedscreens nd foam and has assumed,

for designpurposes, hat the heat-transfer oefficientsmay

be obtained rom well-known, steady-flowcorrelations or

cylinders.The present nvestigation xplores he validity of

this assumptionor acoustic low aroundcylinders.

The influence of sound or vibration on heat transfer is a

problem that has generatedconsiderable xperimentaland

theoretical nterest.A valuable, early review was given by

Richardson,who, ettingside uoyancy,appedheprob-

lem in termsof two parameters, amely, he ratios of acous-

tic displacementor vibration)amplitude l and of viscous

diffusion ength r5 to cylinder diameterd. To avoid the se-

rious complicationof boundary-layerseparation,most theo-

retical treatments ave beenrestricted o the small-amplitude

limit, xi/d•l. Those investigationshave identified the

steadylow nown sacoustictreaming'4 s hemechanism

that cartes heat away from the vicinity of the cylinder, as

suggestedn early xperimentalork.-7Mostexperiments

with stationary ylinders n acoustic ields or with vibrating

cylinders n nominally stationary luids have been performed

in this imit. In the opposite,arge-amplitudeimit,heat

transferhas been ascribed o "convectionby time-average

steady-flowequivalent."The present nvestigationwill sug-

gest that this ascription s incomplete. Buoyancy effects,

whichgreatly omplicateheanalysis,anbe neglected'9

whenGr/Re is small nough,s n this nvestigation.Gr

and Re are the Grashofand Reynolds umbers; eebelow.)

For more recentwork, see Davidsonø and Peterkaand

Richardson. M nother review that shouldbe of interest for

the case of internal-flow heat transfer has recently

appeared.2

Thermoacoustic nginesoperate on the borderlinesbe-

tween hese imits. Because he oscillating luid conveysheat

betweenheheatexchangernd hermoacoustictack,the

longitudinal dimensionof the heat-exchange tructuremust

be less than or roughly equal to the acousticdisplacement.

Hence, for external low aroundwire screens r through e-

ticulated oams,x •/d•> 1. Good thermoacoustic esignof the

stack equires ransverse imensions f the pores o be a few

times rSv. ssuming hat the heat-exchangetructure oesnot

have a grosslydifferentscale, rS,/d-1.

This investigationwas undertaken o characterizeheat

transfer rom acoustic low aroundcylinders n this theoreti-

cally awkward regime that is also not well exploredexperi-

mentally.The intent was to producea correlation hat could

be applied o the designof thermoacousticeat exchangers.

The correlation hat was obtained s not directly applicable o

thermoacoustic evices,however, because he geometry of

this experimentdiffers in two important respects rom the

physical situationrelevant to thermoacoustic ngines with

external-flowheat exchangers. irst, in the presentcase the

cylinder s isolated,while in engines he heat exchangers re

adjacent o a porousbody, the stack. Second,any practical

heat exchangermust consistof a plurality of cylinders hat

would interactwith eachother n a complexway. Hence this

investigation epresents nly a first step toward solving the

practicalproblem.

2209 J. Acoust.Soc. Am. 98 (4), October 1995 0001-4966/95/98(4)/2209/8/$6.00 ¸ 1995 AcousticalSocietyof America 2209

istribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 164.41.123.180 On: Wed, 27 May 2015 18:35:47


2/8

lb)

Moveablelunger

•ne

(c)'

Loudspeaker

FIG. 1. (a) Circuit or measuringesistancen presence f biascurrent. b)

Acousticapparatus, howing esonance ube with loudspeaker,moveable

plungerwith microphone,ndwire under est. c) Detailof wire mountedn

Teflon holder with voltageprobes.

I. EXPERIMENTAL METHOD

The experiment onsisted f placinga heatedcylindrical

wire at a velocity antinodeof a standingacousticwave and

measuring he rate of transferof heat from the wire to the

acoustic medium. This section describes the methods of mea-

suringheat transferand of applying he standingwave.

The cylindrical wire servedas heater and thermometer.

Jouleheatingwas obtainedby passing large current rom a

dc source hrough the wire. Temperaturechangeswere de-

duced from a four-probemeasurement f the wire's resis-

tanceusinga smallalternatingurrent 1 mA at 16 kHz) and

a lock-in amplifier Fig. l(a)]. Titaniumand rheniumwere

chosenas wire materialsbecause hey have relatively large

yet strongly temperature-dependentlectrical resistivities.

Diametersranged rom 0.125 to 2.0 mm.

The heat-transfer coefficient h was deduced from the

decay of the wire temperatureTw toward ambient T O after

turningoff the Jouleheatingwhile maintaining ixed acoustic

amplitude.Assuming hat the temperaturenside the wire is

radially uniform and that the ends are maintained at fixed

temperature,Tw is only a functionof the axial coordinate .

Let the wire have length 2L along z between its voltage

contacts,diameterd, perimeter I=vrd, and cross-sectional

areaA = vrd2/4, nd et thematerial ave hermal onductiv-

ity ks specificheat cs, densityPs, and thermaldiffusivity

xs=ks/psc Thedecay beys3

o2 w o( w to)

ksA 9z-hII(Tw-Tø)=PscsA9t ' (1)

The Appendixpresentshe solution.To leadingorder,which

is good enough for all practical purposes, he relation be-

tween h and time constant r is

(2)

The first term arises rom heat lost radially through he cy-

lindrical surface,and the second rom heat flowing parallel

to the cylindricalaxis. The second erm usuallyamounted o

a smallcorrection


3/8

being a velocity node and the driver roughly an antinode, he

tube then contained3/4 wavelengths,with a velocity antin-

ode at the wire location. The effect of the driver location's

not being a perfectantinode ouldbe compensated y replac-

ing the wire by a second1/2-in. microphoneand adjusting

frequencyand plungerposition o obtain a pressure ode at

the wire location.Higher frequencies ould be obtainednear

odd multiplesof 141 Hz, or, by moving he plunger,at vari-

ous other requencies. he highest requencyusedwas 2391

Hz.

The velocity amplitudeat the wire, u , was calculated

from the pressure mplitudeat the plunger,P l, according o

u =Pl/pgC, wherepg andc arethe density ndspeed f

sound in the gas. At 141 Hz, distortion of the sinusoidal

wave form at the plungerwas insignificantup to the largest

amplitudeattempted,correspondingo a 22-m/s peak. Dis-

tortionwas greaterat higher requencies. or example,many

experiments were performed at 840 Hz where second-

harmonicdistortionas large as p2/pl=O.15 was observed.

However, the distance between the wire location and the

plungerwas always an even numberof quarterwavelengths

for even harmonics.Therefore u2-0 at the wire location,

leavingonly higherharmonics hat were correspondingly e-

pressed n magnitude.

II. RESULTS

Resultsrepresentedn dimensionlessorm. 3Let the

gashave hermal onductivityg, density g,heatcapacity

Cp, and hermal iffusivityKg=kg/pgCp. eat transfers

characterized y the Nusselt number

Nu=hd/kg (3)

Velocity s characterized y a ReynoldsnumberRe , based

on acoustic elocity amplitudeand wire diameter:

Rel=Uld/V, (4)

where v is the kinematicviscosityof the gas.For the purpose

of describingnatural convection, he temperature ifference

is characterized y the product of the Grashof and Prandtl

numbers,

Gr=d3/• g(Tw- To)/V , (5)

Pr= v/Kg. (6)

Here,/• is the gas's hermal expansivity, nd g is the accel-

erationdue to gravity. The productGrPr is called he Ray-

leigh number.)For forcedconvectionn a steady low, Re

and Pr suffice to determineNu. In an oscillating low, one

more parameter must be introduced o account or the fre-

quency of flow eversals.DefiningheStokes epth, r

viscous diffusion length during an acoustic period, by

8•,=v/rrf)1/2Swift'sotationl),his dditionalarameters

taken to be the ratio 6v/d.

The experimentalmethodwas testedby measuring atu-

ral convection i.e., without sound) rom a 0.25-mm Re wire

overa rangeof temperatureifference w- To. ResultsFig.

2) areconsistentithpublishedecommendedalues.7 he

figure also shows hat heat transfer n the presence f intense

sound s independent f temperature ifference.

10

,1 i i i i i i i i

0.001 0.01 0.1 1

GrPr

FIG. 2. Nu vs GrPr for 0.25-mm rheniumwire. Open circles:natural con-

vectionmeasurementsno soundwave); he ine s a smooth urve hrough

recommendedalues seeRef. 17). Filled circles:measurementsith arge

soundwave; line is a guide for the eye.

The characteristic variation of heat transfer with acoustic

amplitude xhibits hree egimes Fig. 3). For smallampli-

tude, heat transfer s independent f amplitudeand equal to

the free-convection alue.For large amplitude,Nu is propor-

tional o Re•/2, eminiscentf thecorrelationor laminar

steadylow. 8The ransitionetweenheseimiting ehav-

iors is fairly abrupt n that, over a substantialangeof Re1

Nu is smaller han would be expected rom the square-root

variation.This behavior n the transition egion suggestshe

existenceof a bottleneck n the pathway for heat removal.

With increasing requency, he transitionregime moves to-

ward argerRe1 (Fig. 4).

The square-rootbehavior at large Re 1 has been ex-

lO

o vertical •J•

,_, 1/2 -•

............•e ..../

..-

ß

ß

..

,..

, •. ...... i ........ i , , , , •1

1 o 1 oo 1 ooo

Re

1

FIG. 3. Nusseltnumberversus coustic eynolds umberat fixed requency

(142 Hz) for 3/4-mm Ti wire. Resultsare shown or horizontaland vertical

sound ropagation.ymbols: ata.Solid ines:correspondingits o Eq. (8).

Dotted ine: Eq. (7).

2211 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 George Mozurkewich'Heat transfer n an acousticstandingwave 2211



4/8

lO

, i i i

014,5z

• 842

, i i , i , , i

10 100 1000

Re

FIG. 4. Nu vs Rel at three requenciesor 3/4-mm Ti wire. Lines are fits to

Eq. (8). With increasingrequency,he transitionrom small o largeNu

moves oward larger Rel.

plainedas "convectionby time-average teady-flow quiva-

lent," as demonstrated y an early experimentusinga heated

cylindern water.9•ukauskas8recommendsu=0.51

Prø'37 e '5 or a cylindern steady rossflow,or 40


5/8

TABLE I. Experimentalesults. irst threecolumns escribe xperimental onditions. ext threecolumnsist

parametersbtainedrom its o Eq. (8). Lastcolumn ivesexpectedalues f Nuna(seeRef. 17)'

Wire f (Hz) •,,/d K Re•it Nuna Nh'thy

' •*nat

1/8-mm Re 145 1.440 0.280 8.45 0.46 - 0.48

841 0.599 0.255 19.0 0.42

1/4-mm Re 145 0.722 0.368 25.6 0.65 0.60

840 0.299 0.271 29.6 0.60

3/4-mm Ti 143 0.238 0.327 44.2 0.90 0.95

278 0.171 0.353 60.2 0.88

422 0.138 0.319 69.7 0.91

842 0.095 0.378 141.0 0.91

2391 0.057 0.270 186.0 0.98

2-mm Ti 144 0.090 0.425 214.0 1.16 1.50

840 0.038 0.425 517.0 1.11

One experimentwas performedwith a mixture of 20

mol % argon in helium at approximately1 bar, using the

0.75-mm Ti wire at 230 Hz. Nu vs Re• showed the same

qualitativeeaturesndwas it to the same orm,yielding

Nunat=0.41,=0.27,andRe•it:34.7.

In order o illustrate he qualityof the fit, all data (ex-

cluding he He-Ar mixture)are plottedon a singlecorrela-

tion curve n Fig. 6. These data cover a broad range of pa-

rameters, including 1


6/8

fore the parcelreturns o the vicinity of the wire. Yet a fourth

possiblemechanisms vortex shedding, s the vorticescarry

energy. Whatever mechanism s effective, the important

point s that the rate limiting step s the transferof heat from

thewire nto hemoving as,which ccountsor heRe /2

dependence.Thereafter, sufficient time elapses for the

"other" mechanism o remove hat heat from the gas before

the gas returns o the wire's vicinity. If its effectiveness e-

pends n Re morestronglyhanRe /2, then he other

mechanismwould becomerate limiting below some ower

value of Re1.

A clue to the identity of the mechanism hat impedes

forced-convectioneat transferat smallerRe1 couldbe pro-

vided y heobservednverseroportionalityetweene•it

and 6v/d. It is important o realize that the set of dimension-

lessnumbers hat was chosen or this analysis orms a com-

pleteset within he assumptionf incompressiblelow near

the wire): any otherdimensionlessumber hatcouldpossi-

bly be physically elevant o the presentproblem Strouhal

number, roudenumber, r whatever) an be formedby an

appropriate roductof membersof the chosenset. By sub-

stituting q. (9) into Eq. (8), one inds hat he parametriza-

tion becomesparticularly simple in terms of the product

Re1 6v/d, a combination hich suggestscoustic treaming,

as will now be shown.

Acoustictreamingnducedy thewire Ms a likely

candidate or the bottleneckmechanism.n the perturbative

limit, '22he haracteristictreamingelocitys nthevicin-

ityof hewire sus- u•/ood,orwhich streamingeynolds

number can be defined: •

Res=

l= 2

:• d ' (10)

to 1.4d. Buoyancy does not seem to be involved because

Re•it swellcorrelatedo 6•/d withoutllusiono empera-

ture difference. his might be regarded s a weak argument

because he temperature ifferencewas not varied widely

and because the orientation of the tube showed some effect.

Nevertheless, ven if buoyancyplays somesubsidiary ole,

thepredominantependencef Re•iton •/d demandsn-

other interpretation.Thermal diffusion does not seem to be

involvedbecause, y the following argument, his also m-

plies x 1-d for the critical case.Let L be the lengthof the

wire,Tg the emperaturef a typicalgasparcel, ndb the

thickness f the thermal ayer adjacent o the wire. In order

of magnitude,7 he ateat which e//t s removedrom he

wire s kgLd(Tw-Tg)/b,and he nteractionime s d/u1.

Thus the total amount of heat abstracted is

kgLd[Tw- Tg)/b d/u1- Similarly,he otalheat ransferred

from the parcel to its cooler surroundings is

kgLXl[(Tg-To)/b]Xl/U . The latter expressionssumes

that the distance ver which he heatmustdiffuse s compa-

rable to b, which follows from conservation f mass.By

equatinghese xpressionsndsolvingorTg, one indshat

the transitionromcoolgas o hot gas,Tg=(Tw- T0)/2,

occurswhen x 1-- d.

Results or the helium-argon mixture (Pr=0.40) are

more or lessconsistent ith those or air (Pr=0.71). Exten-

sive experimentswith air alone are, of course,unable o dis-

cern nydependencef K, Re•it,orNunaonPr; hechosen

parameterization,q. (8), assumes forced-convectione-

pendenceroportionalo Prø'37. hereashemixture'salue

K=0.27 falls just one standard eviation rom the average

for air (Table ), thatassumptioneemso be valid.As for the

transition,he mixture'sxperimentalalue,Re•rit=34.7,

falls a little more hanone standard eviationbelow he pre-

dictedvalueof 39.2_+3.9 ased n Eq. (9). However,onecanCombining he last form with Eq. (9), one finds that the

transitions characterizedyRes=88.his impleesult generaten nfiniteet fexpressionshat re quivalentn

air to Eq. (9), suchas

suggestshat streamingplays a key role in breaking the

bottleneck.

Acoustic streaming s a physicallyreasonablemecha-

nism becausehe consequentteady low is capableof car-

rying heat to greatdistancesrom the wire. Yet the physical

significancef Res=88 (or of theequivalentormx1=6.6

is notclear.t maybeworth otinghatBertelsen3observed

an nstabilityearRes= 00of thestreaminget24 manating

from a vibrating ylinder. he fact thatRes hasbeendefined

usinga perturbativeorm for the streaming elocity aisesa

concern. The ratio of streaming o acousticvelocity is

us/u 1--Res/Re1,hich quals 8/Re•itat the ransitional

value of Re1. Thus n theseexperimentshe calculated alue

of Us/U1 at the transition angedup to -10, where he per-

turbativereatmentails?Furthermore,t seemsmplausible

on physicalgroundshat us could substantiallyxceedu

Nevertheless, the data indicate that the behavior is controlled

by the numerical alueof Res as defined n Eq. (10).

To strengthenhe argument hat streamings important,

an attemptwill be madehere o systematicallyliminateal-

ternativemechanisms. ortexshedding oesnot seem o be

involvedbecause oundary-layereparation ecomes con-

cernwhen •-d, 26 n conflict ith heempiricalesulthat

the ransition ccurs hen 1--8• for 8v angingr6m0.04d

=(15.8-+1.6)

(11)

eachof whicha,rnountso a differentassumptionbout hePr

dependencef Re•it.Themixture'sxperimentalaluealls

nearly two standarddeviationsabove the predictedvalue

29.3-+2.9 basedon Eq. (11). Therefore,of these wo forms

the one basedon 6•/d seems o be preferred,although he

presentdata permit some ntermediatePr dependence,ike

Re•ritøc[(/• v)1/2/d]-1.

The dependencef heat ransfer n the magnitude f Res

that s found n theseexperiments eems o be unanticipated.

Richardson'salculations,applicable hen (5•/d0.3, wheremost

of thepresentmeasurementsall. The space,of l/d vs 8v/d

is shown n Fig. 7 along with the region occupiedby the

presentexperiments nd variousborderlines nticipated y

Richardson.Thesolidineshows es=88.t is apparent

that the experimental ehaviorchangesqualitativelywhere

no changewas expected,and that no qualitativechange s

observedwherechangeswere expected.

2214 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995.

George Mozurkewich: eat transfer n an acoustic tandingwave 2214



7/8

103

2

1

10 0

-1

-2

10 3 ........ i o .I .,I

10-3 10-2 10-• 10 0 10 •

FIG. 7. Coordinate pace s dividedup into various egions dashedines)

by RichardsonseeRef. 2): convection y inneracoustic treamingA), by

outeracoustic treamingB), by time-average teady-flow quivalent C),

and rivial effects D). Circles:ocations f datapoints.Solid ine: Res=88.

IV. CONCLUSION

Heat transfer rom a heatedwire at a velocity antinode

in an acoustic standingwave was found to fall into three

regimes. For small acousticvelocity, free convectiondomi-

nates. For large acousticvelocity, the heat transfer s deter-

mined by its time-average teady-flow quivalent. n this re-

gime, the rate determiningprocess s the abstraction f heat

from the wire into adjacentparcelsof flowing gas; whatever

the process hat eventually carries the heat to infinity, it is

fast enough that it does not limit the overall heat-transfer

rate. The transitionbetween he two limiting regimesoccurs

for Res=88. In the vicinity of this Res that other process,

which appears o be acousticstreaming, s the rate limiting

step.

In this experiment, the wire interacted hermally only

with gas in an open tube, while in thermoacoustic ngines,

the heat exchangers re located mmediately adjacent o the

stack. From the standpointof thermoacoustic pplications,

the most importantpracticalquestion nvolves the effect of

the proximity of the stack. Provided that the gas parcels

move far enough o interactwith the stack, he stack s pre-

sumablycapableof transferring eat to or from the parcelsas

necessary.f so, the presenceof the stack would open the

bottleneck n the intermediate egime, and the heat-transfer

coefficientwould increase o the value expected rom time-

average steady-flow equivalent. Therefore, provided that

x• >>d, the relevant heat-transfer ate could presumablybe

estimatedrom Eq. (7).

Several directions or further investigationmay be sug-

gested. All these measurementswere performed with the

wire located at a velocity antinode. By locating the wire

elsewhere,one could explore the heat-transfer ffect of the

streamingpattern induced by the resonant ube's walls, as

well as any possible nfluenceof a superimposed scillatory

pressure. n this work only one gas mixture was used.More

extensive measurements with mixtures are needed to sub-

stantiatehedependencef •' crit

•e• on Pr. While theseexperi-

ments were motivated by a practical problem in thermoa-

coustics, the results are not directly applicable there.

Measurementsare needed with more realistic geometries,

suchas wire screens, nd n the proximity of someheat sink,

preferably a thermoacoustic tack.

ACKNOWLEDGMENTS

The author is grateful to L. C. Davis, D. F. Gaitan, A.

Gopinath,S. R. Murrell, W. L. Nyborg, V. W. Sparrow,and P.

J. Westervelt or helpful discussions nd communications.

APPENDIX

This section resentshe solution o Eq. (1), which de-

scribes he axial temperature istribution n a cylinder sub-

ject to convectiveheat transport hrough ts cylindrical sur-

face. Based on the argument in the text, the radial

temperaturevariation within the cylinder is assumednegli-

gible; the only relevantcoordinates z, which lies along the

cylinder axis. The wire initially attainsa steady emperature

distributionby being subjected o steadyJoule heating, and

the transientbehavior hat ensuesafter turning off the Joule

heating s to be determined.The wire's ends at z =- L will

be assumedo be maintained t temperature •,.

Effectsof Jouleheatingq* per unit volumemay be in-

cluded y adding * to the eft-hand ideof Eq. (1). Defining

O(z,t) rw(z,t)-r 0 andX-(ksA/hlI)u2, t becomes

092 1 1 090 q*

•z2 h20- . (A1)

s 8t ks

Upon omitting the first term on the right-hand side and de-

finingU=q*X2/ks, hesteady-stateolutions easilyound

to be

cosh(z/X)

O(z,O)--(U+ o-rb) osh(L/X) (A2)

The subsequent ransient solution was found by Laplace

transformation. The result is

cosh(z/X)

O(z,t)=(rbTo) osh(L/k)

n=O ½/'(/•-21-)q-½/.2(/•-21_)2(•k2/L)

Xexp Ks + L2 t . (A3)

The reader may confirm that it satisfiesEq. (A1) (with

q*=0), the boundary onditions, nd the initial condition.

[For the initial condition, onsiderhe Fourier-seriesepre-

sentation f Eq. (A2) on an intervalof length4L.] The ex-

periment measuresnot the temperaturedistributionbut the

resistance. aking the resistanceo be linear in T, its variable

part R(t) is proportionalo the spatial ntegralof Eq. (A3)

from -L to L. Thus

2215 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 George Mozurkewich: Heat transfer in an acoustic standing wave 2215



8/8

n=0 yr2(n «)2 1+ yr2(n «)2(X2/L2)

Xexp gs '•+ L2 t . (A4)

The amplitudesof the successiveerms fall off initially

like (n+ 1/2)-2;comparedo n=0, n= 1 is alreadyeduced

bya factor f 9. Once2(n + 1/2)2>L/X, hey alloffeven

fasten (In the experiments, X.Thereforeany temporalvariationof T• appears nly as

a correction to the correction.

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1180 (1988).

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Am. 32, 337-338 (1960).

8j. Kestin, rans. SMEJ. HeatTransfer3, 143-146 1961).

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streaming," . Fluid Mech. 30, 337-355 (1967).

1øB. . Davidson,Heat ransferroma vibratingircularylinder,"nt. J.

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13 . J. Chapman,undamentalsf HeatTransferMacmillan,ewYork,

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14Ref. 3,p. 119.

15V.S.Arpaci, onductioneatTransferAddison-Wesley,eading, A,

1966),p. 53.

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19F.K. Deaver,W. R. Penney,ndT. B. Jefferson,Heat ransferroman

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21Ref. , Fig.5.

22H.Schlichting,oundary-LayerheoryMcGraw-Hill,ewYork,1979),

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23 . F. Bertelsen,AnexperimentalnvestigationfhighReynoldsumber

steadystreaminggeneratedby oscillatingcylinders," J. Fluid Mech. 64,

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24B. .DavidsonndN. Riley, Jetsnducedyoscillatingotion,".Fluid

Mech. 53, 287-303 (1972).

25W.P.Raney, . C. Corelli, ndP.J. Westervelt,.Acoust. oc.Am.26,

1006-1014 (1954).

26. r. Stuart,Double oundaryayersn oscillatoryiscouslow," .Fluid

Mech. 24, 673-687 (1966).

27 his rgumentollowsn spirithatof C. P.Leeand . G.Wang, Acous-

tic radiation orce on a heatedsphere ncludingeffectsof heat ransferand

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28Ref. 5,p. 301;Ref.13,p. 140.

2216 J. Acoust.Soc. Am., Vol. 98, No. 4, October1995 George Mozurkewich: eat transfer n an acoustic tandingwave 2216

heat transfer from a cylinder in an acoustic standing wave

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