Statistical Theory of Turbulence I.pdf

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Statistical Theory of TurbulenceAuthor(s): G. I. TaylorSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 151, No. 873 (Sep. 2, 1935), pp. 421-444Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/96557.

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421

Statistical Theory of TurbulenceBy G. I. TAYLOR, F.R.S.

(Received July 4, 1935)INTRODUCTIONAND SUMMARY OF PARTS I-IV

Sincethe time of OsborneReynolds t has beenknownthat turbulenceproducesvirtualmean stresseswhichare proportionalo the coefficientof correlationbetweenthe componentsof turbulentvelocityat a fixedpoint in two perpendicular irections. The significanceof correlationbetween hevelocityof a particleat onetimeand thatof the sameparticleat a later time, or between simultaneous elocitiesat two fixed pointswasdiscussedn 1921by the presentwriter n a theoryof " DiffusionbyContinuousMovements." The recent improvementsn the techniqueof measuringurbulence ave madeit possibleactually o measure omeof the quantities nvisagedn the theoryand thusto verifysome of therelationshipshen put forward.The theoryhas also been developed n severaldirectionswhichwerenot originally contemplated. The theory, as originallyput forward,provideda methodfor defining he scaleof turbulencewhen the motionis definedn theLagrangianmanner,andshowedhow this scale s relatedto diffusion. It is now shown that it can be applied either to theLagrangian r to the Eulerian onceptionsof fluidflow.Where urbulences producedn an air streamwith a definite calebymeans of a honeycombor regularscreen,eitherconceptioncan be usedto definea length which is relatedto certain measurablepropertiesofflowand is a definite ractionof the mesh-length,M, of the turbulence-producing creen.The Lagrangian onceptioneadsto a length11,whichis analogous othe "Mischungsweg" of Prandtl. Experimentson diffusionbehindscreens,PartIV, showthat 11= 0-1 M. The Eulerianconceptioneadsto a definite ength12whichmightbe regarded s the average ize of aneddy. Correlationmeasurementswith a hot wire, Part II, show that12 is about equal to 0-2 M.Thetheoryappliedn the Eulerianmanner o these correlationmeasure-ments also contains mplicitlya definitionof x, "the average ize of thesmallesteddies,"whichare responsibleor the dissipationof energybyviscosity.

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422 G. I. TaylorIt is proved hat

W-==5(u2/X2,),whereu2 s the mean squarevariation n one componentof velocityandW is the rateof dissipation f energy. Thisrelationships verified xperi-mentallyPartII).Therelationship etweenXandM is discussedandit is predicted hatturbulencen an air streammovingwithvelocityU willdie downso that

IVu2 Mprovidedhat thescaleof turbulences determinedy the mesh-lengthMwhereA is a universalconstantand B dependson the choice of theorigintakenfor x (the down-streamo-ordinate);u is a componentofturbulent elocity. This theoretical elationships comparedwith resultsof experimentsarriedout in windtunnels n Englandand in America.The theoryis appliedin Part III, to determine he distributionofdissipation crossthe sectionof a parallelwallchannel two-dimensionalpipe)and it is shownthatin the regionnearthewallsturbulent nergy sproducedmorerapidlythan it is dissipated. In the centralregion thereverse s the case.In PartIV the resultsof diffusion xperimentsmade n Americaand atthe NationalPhysicalLaboratoryare discussedand it is shown that acompleteset of such measurementsan give Vv2, 11,and a length X,whichmay be regarded s a measureof the "smallestsize of eddy" intheLagrangian ystem. ?nisconnected, hroughheLagrangianquationsof motion,with the average patialrate of change n pressure, amely

V j _ ) 2by theformula,

:\/E@i) A2 p-,;*.Finally t is shown thatthe theory eadsto the predictionhat Xa is aconstantmultipleof X. The only set of experimentswhich exists at

presentgives?, 2? approximately.All the above resultsaresubjectto the restriction hat the "ReynoldsNumberof Turbulence," amelyl V/u2/v, is greater han somenumberwhichmustbe determined y experiment.



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Statistical Theory of Turbulence 423

PART IAt an earlystagein the development f the theoryof turbulenceheideaarosethatturbulentmotionconsistsof eddiesof moreorlessdefiniterange of sizes. This conceptioncombinedwith the alreadyexistingideasof the KineticTheoryof Gases led Prandtland me independentlyto introduce he lengthI whichis oftencalled a " Mischungsweg and sanalogous o the " mean freepath" of the KineticTheory. The lengthI could only be defined n relation to the definitebut quite erroneousconception hat lumpsof air behavelike moleculesof a gas,preservingtheiridentitytill some definitepoint in theirpath, whenthey mix withtheirsurroundingsnd attainthe samevelocityand otherpropertiesasthe mean value of the corresponding roperty n the neighbourhood.Such a conceptionmust evidentlybe regardedas a very rough repre-sentationof thetrue state of affairs. If we considera numberof particlesor small volumesof fluidstarting romsome definite eveland carrying,say, heat in a direction ransverseo themean stream ines,theiraveragedistancefrom the level at which they startedwill go on increasing

indefinitely o that we can only considera " Mischungsweg in relationto somearbitraryime of flightduringwhichwe must consider hat theparticlespreserve heir individualpropertiesdistinctfromthose of theirsurroundings.Clearlythis is an arbitraryconception and if pursuedlogicallyprobably eads to a definitelywrongresult. The only way inwhicha smallvolumecanlose its heatis by conductivity o its surround-ings. A decreasen molecularconductivitywould therefore ead to anincreasing ime duringwhich the small volume would retain its heatdistinct from its surroundingsand consequentlya decrease in con-ductivitywouldnecessarilyead to an increase n the " Mischungsweg."In all theorieswhichmakeuse of I it is assumed hat 1dependsonly onthe dynamical onditionsof the fluid and is nearly ndependent f suchphysicalconstantsas thermalconductivity.In all applicationsof " Mischungsweg theoriesthe lengthI is con-sideredonly in relation to further,more or lessarbitrary, ssumptionsconcerninghe effectof turbulence n the mean motion or of the meanmotion on turbulence. It appearsas a fictitiouslength,the existenceof which s detectedonly by observationsof the distributionof meanvelocity,temperature,tc.The difficultyof defininga " Mischungsweg,"r scaleof turbulence,without recourseto some definitehypotheticalphysical processwhichbearsno relation to realitydoes not arise in such applications. The



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424 G. 1. Taylordifficulty,however, still exists and it led me, some yearsago, to introducethe idea* that the scale of turbulence and its statistical properties ingeneralcan be given an exact interpretationby consideringthe correlationbetween the velocities at various points of the field at one instant of timeor between the velocity of a particle at one instant of time and that ofthe same particle at some definite time, i, later. Some generalrelationsapplicable to either of these two aspects of the turbulent field were dis-cussed, and the application of the definitionsused in the second of themto diffusion in one dimension was worked out in detail. In this applica-tion of the theory the particles are conceived to move irregularlybutwith continuous velocity, v and v2is supposed to be independentof time.The diffusion of particles starting from a point (y- 0) is shown todepend on the correlation Rt between the velocity of a particle at anyinstant and that of the same particle after an interval of time i. Incontinuous turbulent movements Rt must be a function of i such thatRt =1 when i 0 and Rt -> 0 when i is large.If Y2 is the mean square of the distance through which the particleshave diffused in time t it was proved that

dtY2)= (1)If the time of diffusion is small so that Rt has not departedappreciablyfrom its initial value I 0, (1) becomes

(d(Y2) =v2so that

VY2 v't, (2)where v'=If the diffusion is taking place in a stream of air moving with velocityU and if the spreadis observedat a small distance x down-streamfromthe source t x/U so that

_ V (3)x UJ.If the irregularmotion is of such a characterthat it is possible to definea time T such that Re = 0 for all values of i greaterthan T, so that thereis no correlation between the velocities of a particle at the beginningand end of the time interval T, then

Yv - v2 Rt (4)A* 'Proc. Lond. Math. Soc.,' vol. 20, p. 196(1921).



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Statistical Theory of Turbulence 425Yv is thereforeonstant or allvaluesof t > T in spiteof thefact that thevalue of Y2?is continually increasing and v2 s constant.

Underthesecircumstancest is possible o definea length11, uchthat_ T d11Vv2= v2JR- = dt (5)It will be seenfrom(5) that the length11,definedas

11 - v2 Rf dt (6)bears the same relationship o diffusionby turbulentmotion that themean free path does to moleculardiffusion. In this sense it is verysimilarto the "Mischungsweg," , but with this importantdifferencethatthe questionof mixturedoes not arisein defining t.As is pointedout above,theorieswhichdependessentially n the ideaof mixtureby subdivisionand ultimatemoleculardiffusion ead to theexpectation hat the " Mischungsweg will dependverygreatlyon themoleculardiffusivepower of the fluid. In the theoryof diffusionbycontinuousmovements he length11bearsno relationto any processofmixture, ndeed it is equally valid if mixture never takes place. Theeffectof moleculardiffusionwouldbe to prevent hefluidfrombecomingever increasingly "spotty," i.e., it would tend to prevent a continualincreasen the deviationsof the measurable roperties f the fluidfronitheir meanvaluein the neighbourhood. Mixturehas no effectin thistheoryon the diffusivepowerof turbulentmotion.

CORRELATIONN THE TURBULENT FIELD WHEN DESCRIBED N THEEULERIAN MANNERIn a loose way it has beenthoughtthat the "Mischungsweg" engthI is relatedto, and evenmay be takenas a measureof the averagesizeof the largereddiesin turbulent low. It will be noticedthat in theoriginal" Mischungsweg theories,and also in the theoryof diffusionby continuousmovements, verythings definedn a Lagrangianmanner,i.e., by following he pathsof particles. When a fieldof eddying lowisconsidered s an entity n itself,apartfrom its effectas a diffusiveagent,it is moreusualto think n termsof the Eulerian onception f fluidflow,i.e., a field of stream ines conceived o exist in spaceat one instantoftime. Any ideas we may have about " the size of an eddy" are likelyto be formulatedn the Eulerian ystem. For this reasonit would not'VOL. CLI.-A. 2F



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426 G. 1. Taylorbe possibleto connect directly he size of an eddy, even if it could beaccurately efined,with the value of I or of 11as definedby (6) in theLagrangiansystem. At the same time it seems to be a matter ofconsiderableheoretical nterestto investigate he statisticalpropertiesof a field of turbulent low whendescribedn the Eulerianmanner,witha view to defininga lengthwhich may representn some definiteway the"size of an eddy."The correlation heorydeveloped n my paper,"Diffusionby Con-tinuousMovements,"s equallyapplicablen this caseandmaybe usedto formulateanother definitionof the scale of turbulence. It is clearthat whateverwe maymeanby the diameter f an eddya highdegreeofcorrelationmustexist between he velocitiesat twopointswhichare closetogetherwhen comparedwith this diameter. On the otherhand, thecorrelation s likely to be small betweenthe velocity at two pointssituatedmany eddydiameters part. If, therefore,we imagine hat thecorrelationR. between he valuesof u at two pointsdistanty apartinthe directionof they co-ordinate as been determinedor variousvaluesof y we mayplot a curveof R, againsty, and this curvewill represent,fromthe statisticalpoint of view, the distribution f u alongthey axis.If R. fallsto zeroat, say,y - Y, thena length 2 can be defined uchthat

12 jR dy =0Ryddy. (7)o oThis length 12may be regardedas the analogue n the Euleriansystemof 11,whichis defined n the Lagrangian ystem. It may be taken as apossibledefinitionof the "average ize of the eddies."

EXPERIMENTALMETHODS FOR MEASURING 11 AND 12The compensated ot wireis capableof beingusedto measure everalof the quantitieswhicharenecessarilyonsideredn anystatisticalheoryof turbulence.(1) u2can be measuredby meansof a hot wireanemometer. If theamplifieddisturbancesrepassedthrougha wirethe heatproduced angive rise to a currentn a thermojunction,hich will cause a deflectionin a galvanometer roportionalo u2.(2) If two hot wires are set up at a distancey aparttransverse o astreamof air and the currentsproducedby variationsn U at the twopoints are sent throughthe two coils of an electricdynamometer,heresultingdeflectionwill be proportionalo u0U'where u0 andu, are the



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Statistical Theory of Turbulence -427velocities at the two points. In this way Rz,u0ujzu can be measured.By repeating these measurementsfor a number of different distances ofseparation y between the two hot wires, Rv can be determined for allvalues;of y and hence by integration12can be found. The (R., y) curvehas already been obtained in certain cases by Messrs. Simmons andSalter at the National Physical Laboratory by this method (see fig. 1of Part II).Another method is to arrangetwo equal hot wires on two arms of aWheatstone bridge thus measuring (Ul - uO)2. If H2 and u12 aremeasuredindependentlyat the two stations, uoul can be found from therelationship

uo2+ 17- (uO u)2- 2uOu. (8)Yet another method due to Prandtl* is to pass the currentsfrom thetwo hot wires through coils which cause deflections of a spot of lightin two directions at right angles to one another. If the two hot wiresare identical and so close together that the correlation is nearly 1 0, thespot of light moves over a very elongated elliptic area, the long axis ofwhich is at 450, to the deflections caused by either of the wires in the

absence of disturbancesfrom the other. By measuring the ratio of theprincipal axes of the elliptical blackened areas produced on a photo-graphic plate by the moving spot of light during a prolonged exposure,it is possible to calculate RV. This method is specially suitable formeasurements when the correlation is very high, i.e., 1 - RVis small.It is not so suitable for small correlations as the electric dynamometer,method. Correlation measurements made in this way are shown infig. 1 of Part III of this paper.(3) By introducingheat at a concentratedsource or a line source in anair stream and measuring the spreading of the heat to leeward of the-

source it should be possible to measure the quantity - Y which occursrt~~~~~~~din (1) and hence to find Rt dt for various values of t. If this reaches

-a constant value at some distance down-stream then 11can be found.This method was suggested in my paper on " Diffusion by ContinuousMovements." Up to the present, however, the theory has only beenapplied to cases like that Qf diffusion in the atmospheret where there isno a priori eason to suDDose hat any definite scale of turbulence can be

* Prandtl and Reichardt," Einfluss von Wiirmeschichtung uf die Eigenschafteneiner TurbulenterStromung." Deutsche Forschung,p. 110 (1934).+ gV,ttrn 'Prnir Dnir Znc: ' A unl 11 n 1zA (101IN

2 F 2



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428 G. I. Taylordefined. Indeed,Mr. 0. G. Suttonhas shown hat the bestrepresentationof diffusion n the air near the ground s obtainedby assumingRSoc -nso that Rt does not vanishhowever great i may be. In fact j'R4 dtoincreased ontinuouslywithincrease n t so that 11,definedas in equation(6),wouldhave no definitevalue.Theturbulencewhich occurs n windtunnels s producedor controlledby a honeycombwith cellsof a definite ize. In a windtunnel, herefore,there s an a priorireasonwhy the turbulencemightbe expectedo be ofsome definitescale. In fact, it might be expectedthat both 11and 12would be some definite ractionof the mesh of the cells. Under thesecircumstanceshe diffusion quations 1) and (6) reduce o

dty2 = llv* (9)Thisexpressionsvalidwhen hedistance of the pointsatwhichmeasure-ments of Y2 are madefrom the point or line source of diffusion s sogreatthat Rt 0 where i x/U and U is the meanspeedof the airstream.

APPLICATIONOF DIFFUSION EQUATIONWHEN TURBULENCEISDECAYING

In the air streambehinda grid or honeycomb he turbulences notconstant. It decreasesas the distance down-streamncreases. Thepreceding heorycannotthen be'applied withoutfurther nvestigation.If vl is considered s a functionof t the diffusion quations

Ct ~~~~dt21 2 vt t_-di (10)for Y vt-t di and t(Y) is the rate of increasein Y2 at time t afterthe beginningof the diffusion roma concentratedource.If tR,_eis the coefficientof correlationbetween he velocityat timet andthatat time t-i, (10) maybe written

i_-y2=t0 vt- t(tRt-t)dis 11dt ~ ~~~~ 2where v'%,'t are written for V/v2t,A/vWhenthe averageconditionof the turbulentmotionis constantwithrespect o time tRt-t is the sameas tRt+ or Rt and is a functionof i



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Statistical Theoryof Turbulence 429only, so that (11) is identical with (1). When v' is not constant, it is notpossible to proceed beyond (11), but the existing experimentalevidenceseems to show that turbulent diffusion is proportional to the speed, sothat if matter from a concentratedsource is diffused over an areadown-streamfrom the source, anincrease in the speed of the whole system (i.e.,proportional increases in turbulent and mean speed) leaves the dis-tributionof matterin spaceunchanged (though the absolute concentrationis reduced). The condition that this may be so is that tR,t is a functionof n only where d- = v'd~ (v'/U) dx (12)and x Ut is the distance down-stream from the source.The equation which represents the lateral spread of matter or heatfrom a concentrated source is therefore

1U dd V2) HR,d- (13)where nz fV' dx, (14)

and R, is the correlation between the velocities of a particle at times t,and t2 when B = v'dt. If R. falls to zero at a finite value of -, say

=, and remains zero for all greater values of R, XRd-ms finite.0If 11be writtenfor R,dn then (11) becomeso 1U d -1521Vx (Y2) 11. (15)This is the same expression as that found for turbulencewhich is notdecaying.*It is worth noticing that (13) may be expressed n the form

i d y2) - inzed-m. (16)w~(Y2)= x ~P,dWhen x is small so that R= 1 over the range from 0 to x, (16)-

becomes - - (17)The integralof (17) is

y2= 2 or Vy2 (18)* See equations (1) and (6).



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430 G. 1. TaylorWhenthe turbulences constantn xv'/U so that(18) reduces o thepreviousexpression 3) for the spreadof matternear a concentrated

source. If theturbulences not constantand f Y2and v'/U aremeasuredU d-at a number of values of x, then both N and i v'dx (Y2) can be found.Thus R,,dNcan be plottedagainstN and R),can be foundgraphicallyfromthisexperimentalurve.

MICRO-TURBULENCEND DISSIPATION F ENERGYBesides the motions which are chieflyresponsible or the diffusivepowerof turbulencehewholefieldmaybe in a stateof micro-turbulence,i.e., theremay exist very small-scaleeddieswhich,though they play averysmallpart n diffusion, etmaybe theprincipal gents n thedissipa-tion of energy. Theymay also be the principal ausesof the effects ofturbulenceon the boundary ayer in wind tunnel work becausetheabsolutemagnitude f the spacerates of change n pressuremaydependon them.

DISSIPATION F ENERGYThe rate of dissipation f energy n a fluidat anyinstantdependsonlyon the viscosity, t, and on the instantaneousdistribution f velocity.If, therefore, he representationf the essentialstatisticalproperties fthevelocity ieldcanbe expressed y theR, curveandsimilarcorrelationcurves t mustbe possible o deduce romthem the rateof dissipation fenergy. This would in generalinvolvea complicatednalysis,but theproblemcan be muchsimplifiedf the fieldof turbulentlow is assumedto be isotropic.

ISOTROPICURBULENCEIn isotropic urbulenceheaverage alueof anyfunctionof the velocitycomponents,definedn relation o a givenset of axes,is unalteredf theaxes of referenceare rotatedin any manner. That thereis a strongtendency o isotropy n turbulentmotionhas long beenknown. It has

beenshownby FageandTownend,*for instance,hat the average aluesof the threecomponentsof velocityin the centralregionof a pipe ofsquaresectionare nearlyequalto one another. In the atmospherehesamephenomenon as beenobserved; hough,as mightbe expected,he* Townend, ' Proc. Roy. Soc.,' A, vol. 145 (1934) (see fig. 15, p. 203).



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Statistical Theoryof Turbulence 431vertical components are smaller near the ground than the horizontalones, this inequality decreases with height above the ground.*

The assumption of isotropy immediately introduces many simpli-fications both into the statistical representation of turbulence and intothe expression for the mean rate of dissipation of energy.The general expressionfor the rate of dissipation is(au\2 av7,\2 aw 27av a9U,2W {2ax 2ax 2\aa 2+ TX ayi2

+ y + az az axw (19)Making the assumption that the turbulence is statistically isotropic, therelations a) 2 at 2 aw 2and (auj2 au2 a2 / 2 = 'aw2 aw2 (20)and

avau awav auawaxay ay az az axare immediatelyobtained so thatxxfa 2 aua) av aup-6(ax) + ay axay (21)Equation (21) contains three types of term. It will now be shown thatthese are all related to one another so that if the value of one is knownthe other two are known.That relationships can be found between the mean values of squaresand products of au/ax, au/ay, av/ax, .., etc., is obvious. The simplestrelationship is obtained as follows. The condition of continuity isau+av+ aawso that

U)+ (-v) + ( -Wp - 2 (ax ay + avaw+ (22a\ ay, z a ay aZ azTx)The conditions of statistical isotropy therefore lead to the relationship

(aU 2 2 au av (23)* Taylor, 'Q. J. R. Met. Soc.,' vol. 53, p. 210 (1927).



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432 G. 1. Tayloror au av

axa= (24)In otherwordsthere s a definitecorrelation oefficient etweenau/axand av/ay equalto 2.MEAN VALUEOF GENERALQUADRATIC UNCTION OF au/aX, aViaX,

aU/ay, ..., ETC.Consider he most generalpossibleexpressionor the meanvalue ofany quadraticunctionof the nine quantities

au av aw au av aw au av awTX TX FXSayyS -y az zI TzXIn generalthere are 36 possiblecombinations f 9 thingstaken2 ata time. Thus the most generalquadratic xpression ontains45 terms,namely he9 squaresof thequantitiesoncerned nd the36combinationsof 2.When the motion is statisticallystotropicthe 45 termsfall into 10groups,eachof whichcontains 3 or 6 meanswhich are equal to oneanother;for example,one groupcontaining3 equal termsconsistsof

(aU)2\2 )2 and aWAnothercontaining6 equaltermsconsistsof

auav auaw avau avaw awau awavaxa: axayI ayaz' ayTax ay azaThe 10 possibleindependentmeanvalueswill be denoted by a,, a2,. a1o accordingto the scheme laid out in Table I where the top row ofthe tablegivesthe typetermand all othertermsof the sametypecan beobtainedby permutingymmetricallyhe elementsof the typeterm.The symbolwhichrepresentshe meanvalue of any term of a type isgivenin the second row and the numberof independenterms n eachgroup s given n thelastrow.In termsof thesesymbols 21)becomes

W/ - 6a1+ 6a3+ 6a8. (25)



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Statistical Theoryof Turbulence 433

Z-

.

(~~~~~~1C~ ~ '

I( 1 o E

I-Ga 1'


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434 G. 1. TaylorI now propose to prove that the 10 values, a,, a2, ... al0, are inter-connected, so that if the value of any one of them, which is not zero, is,

known all the rest are known. For this purpose it is necessary to prove9 linear relationships. One such relationship has already been proved(see equation (23)). Expressed in the symbols of Table I (23) may be,writtena, - - 2a6. (26),

Further relationships may be obtained as follows. Take any one ofthe 45 possible terms in the most general quadratic expression involvingthe 9 partialdifferentialsof (25). Transformu, v, w, x,y, z, by rotation ofthe axes to u' v', w', x', y', z'. The transformed expressionwill still be,quadratic but will contain terms of other types than the original one.When the mean values of the terms in the transformed expression areconsidered it is a necessary consequence of the definition of isotropy thatthe value of each is equal to that of the type term in the group in whichthey are classed. A simple transformation is obtained by rotating theaxes through 450 about the axis of z so that

IV/x' x + y a2 = u + uV/2y'=-x+y v/2v U+v (27>

z = z w wHenceAu 1 '@uu Iau' u v' _

aX, au, l axu'av t+ a at'x-V XU aVX.,Z'\ a aU V aU' rtax -j - - -~X - --aX 2 au' aX' ayt~ ey av aut aXt au, a`j\ax a/ x' ay y y T/ x' _ -Fy7 -

aw I awf A u 1 ( aw wl aw

az T-2; +z + az(28)Ajw w'Take, orexp au By qa

Take orexamle ;83a) l. By squarin hetransforedexpressio



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Statistical Theoryof Turbulence 435for au taking the mean value and substituting he symbol for thecorrespondingypetern fromTableI it willbe foundthatau,)2aut2avi2ui avia x)/= al = 4; tAx'}+ ( aX a+ ax' I

= (al+a3-a5-a +a,+a8-a2-a5). (29)Similarly(7;)2= a,. =1 (aja+ a5 +2, + a6+ a8+ a2+ a5), (30)(y) = a3 = (al + a3 -a5 + a2-a6-a8+a2-a5), (31)(aV)= a3-i (a + a3+ a5-a2-a6-a.a8-a2+ a5). (32)

Fromtheseequationst willbe foundthata2 -a5 O (33)and a, -a - a, Ia C(

No further elations an be derivedby transforminghe typetermscorre-spondingwith a2,a5, a6,or a8. Proceedingo terms nvolvingw or zau=-a10 + u V -(a3- a3) _0, (35)auav 1ay -a9-= (a2-a9+ a4-a7 -ia7-a4+ a9-a2)- 0, (36)au av 12x8z=a7= 2,(a- -4 7 ,+7,-a4- a9+a43* (37)hence since

a2 a9 =0 a7 (2--1)+a4=O (38)a =a4= Ha2- aID;-a4- a - a7 + a4- ag + a2),

and hence a4 (2-1) + a7-0 (39)combining 38)with(39) a4= a7= O. (40)Summingup the resultsso far obtained6 of the 10 independentypes



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436 G. I. Taylorof mean are zero, namely, a2, a4, a5,a7, a9, al0 and there are two inde-pendent relationships between the remaining 4 means, namely,

a, =-2a6 and a-a3- a6-a8 ?O.Of these the first depends on incompressibility and isotropy. Thesecond depends only on.isotropy.One furtherrelationship can be obtained by volume integrationof thegeneral dissipation expression (19). This integration is well known*:it issrr r: au\ za2 aw\ law avJJJ\Yax ya: 2 ax 4- 2v) + 2z + T + azy

+ U+ aW2+ a'V 2 d ydauaW\2 ?( + ) dx dydz= + 2 )dxdydz- an q2)dS + 2 u v w dS, (41)

wherei=(wlay) -- (av/lz), etc.

and the integrals are taken over the cloud surface S and through itsvolume. If the closed surface is large compared with the scale of theturbulence the surface integrals are small compared with the volumeintegrals which may therefore be neglected. Taking the mean value ofall the quantitiesin (41) and expressingthe resultfor isotropic turbulencein terms of the symbols of Table I, (41) becomes

W/ , = 6a, + + 6a+ 6a8-6a3-6a8. (42)Hence a, + 2a8 0, (43)solving (26), (34), and (43) it will be seen that

al = 1a3 -2a6=--2a8. (44)Three obvious corollaries to this result may be noticed:(1) The correlation coefficientbetween Au/ax and av/ay is - 2(2) The correlation coefficient between au/ayand avlax is-4.(3) When the mean value of any one of the four possible types ofquadraticterms which are not zero is known all the rest are known, sothat the mean value of any quadratic unction of the spacerates of change

* See, for instance,the chapteron viscosityin Lamb's" Hydrodynamics."



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Statistical Theory of Turbulence 437of velocity is also known. In particularthe dissipationmay be expressedin terms of The correctexpressionisW/u - 6a1 + 6a3 + 6a8 - 3a3 + 6a3 -1 5a3 = 75 4

STATISTICALEPRESENTATIONF MICROTURBULENCEThe value of () is clearly related to the way in which the value of

R, falls off from its initial value I 0 as y increases from zero. I haveproved,* in fact, that l 2 (au L 1Y4 2U2 .\2 (46)The curvature of the Rv curve at y 0 is therefore a measure of (auso that

2u Lt 2R (47)-y ?/-*O \y2(7

The significance of the expression (47) can best be appreciated bydefining a length X such that1-2 Lt (1 (48)

2 iS then a measure of the radius of curvature of the R, curve at y - 0.If the curve is drawn on such a scale that its height is H (correspondingwith R -1 at y 0) the radius of curvatureat y 0 is ?X2/2H.Another interpretationof X may be found by describing the parabolawhich touches the Rv curve at the origin. This parabola will cut the

axis R, = 0 at the point y = X. Xmay roughly be regarded as a measureof the diameters of the smallest eddies which are responsible for thedissipation of energy.

CONNECTIONETWEEN ISSIPATION F ENERGYAND CORRELATIONFUNCTIONR.,

Combining (45) with (47) and (48), the dissipation is related to thecorrelation function R, by the equationW =15 Lt l ,Rv (49)y->

* ' Proc. Lond. Math. Soc.,' vol. 20, p. 205, equation (14), (1921).



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438 G. 1. Tayloror W-15pu /X2. (50)

Since u2 and R. can be measured directly by means of the hot wiretechnique referred to earlier, the relationship (49) can be verified ifW can be measured by other means. The way in which this can bedone and the comparison between this statistical theory and the resultsof observationwill be discussedlater. In the meantimeit may be noticedthat if the Reynolds's stressesin geometrically similar fields of flow areproportionalto u2 or u'2,W s proportionalto u'3, so that Xis proportionalto (u')-, and since ? is proportional to the curvatureof the RVcurve aty 0 we are led to the prediction that the curvatureof the R, curve atits summit,y - 0, will be proportional to 1 u'. In the limit for veryhighvalues of u' the RI,curve may be expected to have a pointed top.

SUGGESTIONFOR EXPERIMENTALTEST IN WIND TUNNEL OFPREDICTEDCORRELATIONRELATIONS

It has been shown how measurements of correlation between thereadings of two hot wires at points close together in a transversesectionof a pipeor wind tunnelcangivethe valueof (u . If similarmeasure-ments could be made in a line parallel to the main stream, values of(>)2 could be obtained in the same way. Equation (44) shows that

(u 2 (au)2

and referringto equation (47) which is equally true when x is substitutedfor y, it will be seen that for the correlationto fall a given amount fromits coincidencevalue 1 .0 the separationof the two hot wires must be 2times as great when one lies up- or down-streamfrom the other as it iswhen they lie across the stream.This is a definite new theoretical prediction which could be tested.If difficulty s found in working with one hot wire down-streamfrom theother, measurements might be made with the two wires mounted at afixed distance r apart on a rotating holder, and the variation in thecorrelationR as the holder is rotated might be found.The correlationbetween the values of u observedat two points situatedat a shortdistance, r, aparttina line makingan angle 0 to the wind directionis*r2X,u 2(1Sv~~~~~~ 2U1 - - ) (51)

* Compare equation (47).



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Statistical Theoryof Turbulence 439wherex' - x cos 0 + y sin 0. Since

aU= cos0 au+ sin 0LUax' Ax ay(51) becomes, n the notationof TableI,

r2R=- 1-2 2 (acos20 + a3sin20 + 2a2cOS O in 0). (52)When the turbulences isotropic hisis

(2au,1 -R= = -2 (C S2 0 -V+sin2 0).2u2 aYi

hencefromthe definitiontof X.. ~~~~~~~~~21- R= (COS2 0 + sin2 0). (53)It appears, herefore,hat 1 - R shouldvary in the ratio 2: 1 as theholder is rotatedfor the maximum to the position to maximumto.minaimumorrelation.DIMENSIONALELATIONSHIPETWEEN AND, SCALEOF TURBULENCEIt has beenshownby v. Karman hat if the surfacestress n a pipe isexpressed n the form T pVx2 then

U0 U f(a) (54)vx ~awhereU0 is themaximum elocity n the middleof thepipeandU is thevelocityat radiusr. Thisrelationships associatedwith the conceptionthat theReynolds'stressesareproportionalo thesquares f theturbulent-componentsf velocity. It seemsthat the rateof dissipationof energyin sucha systemmustbe proportional,o far as changesn lineardimen-sions, velocity, and density.are concerned, to pu'3/l, where 1is some linear4imension definingthe scale of the system. For turbulenceproducedby geometricallyimilarboundariesherefore

W - constant 15.For suchsystems herefore/2 = IV (55)

t See equation(48).



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440 G. 1. Taylorwhere C depends on the position relative to the solid boundaries of thepointat which observationsaremade and on the elementused for defining1.

APPLICATIONTO AIR STREAMBEHIND REGULAR GRIDS ORHONEYCOMBS

Formula (55) is specially well adapted for discussing the decay ofturbulence n an air stream behind a grid or honeycomb, because it hasbeen found that at a certain distance down-stream the streambecomesstatistically uniform, i.e., the "wind shadow " of the grid disappearsand the mean velocity becomes uniform. Under these circumstancesitseems that the C of formula (55) must be a constant for any definite.form of grid. The researchesof Schlichting*have shown that at a shortdistance behind a cylindricalobstacle the wake assumes a definite form.The width of the wake and the velocity of the air in the middle of thewake depend on the drag coefficient of the obstacle so that obstacles,of very varied cross-sectionsproduce identical wakes provided their dragcoefficients are identical. For this reason it may be expected that if aregular grid or honeycombis constructedthe scale of the turbulentmotionproduced by it at any distance down-streambeyond the point where the"'wind-shadow" has disappearedwill depend only on the form and meshsize of the grid, and not on the cross-section of the bars or sheets fromwhich it is constructed. On the other hand, the velocities of the turbulentcomponents will certainly depend on the drag coefficient of the bars,themselvesas well as on the distancedown-streamfrom the grid at whichmeasurementsare made.These considerations lead to the prediction that if only one-form ofmesh is considered,say a square mesh, and if the length I in (55) is takenas M, the mesh length, i.e., the side of each squareof the mesh, then theconstant C in (55) will be an absolute constant independent of the formof the bars of the grid. We are thus led to a definite expression forx/M namely,

-=A \ (56)

where A is an absolute constant for all grids of a definite type, e.g., forall s-iiiare-mesFh grids or honevcomhs.* IIngen. Arch.,' vol. 1, p. 533 (1930).



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Statistical Theory of Turbulence 441PREDICTION OF LAW OF DECAY OF TURBULENCEBEmND GRIDS

AND HONEYCOMBSWe are now in a positionto predict he way in which turbulencemaybe expected o decay when a definite cale hasbeen given to it as the airstreampasses througha regulargrid or honeycomb.Therate of loss of kineticenergyof the turbulence er unit volume sipU d_ I2 +V2+W2),

which n an isotropic ield of turbulences- pU (u2)

Thismustbe equalto the rateof dissipationW, so that3 U d (u - 15.S22 (57)

This equation is capable of experimentalverifications ndependentlyof therelationship56)betweenXand M because,as hasbeenshown,XisconnectedwithR. through 48)and R, can be measured nstrumentally.Ontheotherhand, f therelationship 56)betweenXand M is assumedto hold it is possibleto calculate he law of decayof turbulence. Sub-stituting or Xfrom (56), (57) becomesU d(u 2) 10 (58)u'3dx MA ' 58

and integrating 58) the followingvery simple aw of decayis predicted,u' =A2M -+ constant. (59)

Thisexpression houldbe applicableo all cases wherethe turbulenceis of a definite cale. Thelinear aw of increasen U/u' shouldthereforeapplyto all wind tunnelswherethe scale of turbulences controlled y ahoneycombor grid,and the valueof the constantA determined xperi-mentally,using (59),should be universalor all squaregrids. Thus, theturbulencebehinda square-sectionhoneycomb with long cells shouldobey the samelaw of decayas thatproducedby a square-mesh rid offlatslatsor a square-mesh ridof roundbars,andthe valuesof A shouldbe identicaln all thesecases.VOL. CLI.-A. 2 G



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442 G. I. TaylorFor other types of grid or honeycomb, e.g., with hexagonal or tri-angular cells or a grid of parallelslats or plates, the constantA determined

experimentally by applying" 59) to observed values of u' at differentdistances down the air stream might be expected to assume other values.EXPECTED LIMITATIONSTO PREDICTED LINEAR LAW OF DECAY OF

TURBULENCEThis is a very comprehensive prediction, but it is subject to certainlimitations. In the first place it cannot be expectedto apply when Mu'/vis small, for equations of the type (56) are not true when Mu'/v is small.In fact if (56) were supposed to hold when Mu'/v is small X would begreater than M, a condition which is clearly impossible at any rate nearthe grid.A second restriction is that the formula cannot be expected to applyin the region immediately behind the grid where the mean velocity isvariable, i.e., where the " shadow " of the grid is still distinct. It isfound experimentally that when the diameter of the bars of the gridis small compared with M the shadow may extend to as much as 20 M

or 30 M behind the grid, but when the bars are as broad as W M theshadow disappearsa few mesh lengths down-streamfrom the grid.A third limitation may be expected to operate when the turbulence isnot entirely due to the grid through which the stream passes. If, forinstance, a very turbulent stream passes through a grid consisting of thin-wiresarranged in a large-scalemesh the scale of the turbulence in thestream might hardly be affected by its passage through the grid.SUMMARYOF RESULTS AND THEORETICALPREDICTIONS

(1) When the turbulence of a definite scale is produced or controlledin a stream of air by a honeycomb or grid of regularly spaced bars thescale of turbulence can be investigated in two ways. If the Lagrangianconception of fluid motion is adopted the scale of turbulence can bedefinedin referenceto the correlationR, betweenthe velocity of a particleand that of the same particle at time . later. This conception is suitedfor `discussing experiments on diffusion of heat from a concentratedsource.(2) If the diffusive spread of heat or matter from a line source ismeasured near the source it is proportional to the distance from thesource and measures the transverse component of turbulent velocityindependently of the scale.



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Statistical Theory of Turbulence 443(3) If the diffusivespread s measuredat a numberof positions extendingfar down-streamfrom the source a length 11analogous to the mean free

path in kinetic theory of gases can be determined. It is anticipatedthatthis will be some definite fraction of the mesh.size M of the honeycombor grid.(4) Measurements of correlation between.. imultaneous values of thevelocity at points distributed along a line, can determine a length 12which measuresthe scale of turbulence rom the standpointof the Eulerianrepresentation of fields of flow.Both these lengths may be expected to be some definite fraction of themesh length M, at any rate when the turbulenceis not very small.(5) A third length X can. be defined in relation to the dissipation ofw -energy by the equation - 15 U This length may be taken to repre-sent roughly the diameters of the smallest eddies into which the eddiesdefined by the scales 11or 12will break up.(6) If the rate of dissipation is proportionalto the cube of the velocity,as it is where the Reynolds's stresses are proportional to the squares ofthe turbulent components of velocity, X is proportional to V/uIn turbulence due to a square mesh honeycomb of mesh length M,--= A /\/M I where A is a constant. This formula is inapplicablewhen Mu'/v is small.(7) Using this value for Xit is shown that the law of decay of turbulenceis such that U/u' increases linearly with x in accordance with equation(59).

(8) X is also directly connected with the correlation between simul-taneous measurements of velocity at fixed points separated by a smalldistance. This correlation can be measured by suitable apparatus sothat the theory can be verified experimentally.(9) In isotropic turbulence the mean value of any quadraticexpressionof the space rates of change in the velocity is known when the meanvalue of any one of the terms in it which is not zero is known. Thisleads to the prediction,which might be verifiedexperimentally,that if thecorrelation between a component of velocity at a fixed point 0 and thatat a neighbouring variable point P is measured, the surfaces of equalcorrelation are prolate spheroids with P as centre, the long axis is V2times the equatorial axis and is directed in the direction in which the;velocity component is measured. This statementis identical, n substance,though not in form with that given on p. 439. 2 G 2



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444 &GI. TaylorCONCLUDING REMARKS

Of these results and predictions (1), (2) and (3) are substantially denticalwith the conclusions put forward in 1921 in my paper, " Diffusion byContinuous Movements," where the suggestion that the diffusive powerof turbulence should be used for the:purpose of measuringthe scale ofturbulence and the turbulent components was first made. Recentlyexperimentsof this nature have been made by C. B. Schubauer* and (2)has been verified,as will be shownin Part IV of the present paper. Mr.Schubauer, however, worked quite independently of my previous workand indeed gives an empirical explanation of his experimental results.Conclusions (4) to (9) are, I believe, new. It will be shown inPart II that all these results, except (9), have now been verified experi-mentally and shown to be true. Experimentalwork is now in hand totest the truth of (9).

Statistical Theory of Turbulence-IIBy G. I. TAYLOR, F.R.S.(ReceivedJuly 4, 1935)

MEASUREMENTSOF CORRELATION N THE EULERIAN REPRESENTATIONOFTURBULENTFLOW

The methods described n Part I havebeenusedbyMr. L. F. G. Simmons,of the National PhysicalLaboratory, o find experimentally he correlationbetween the turbulent components of velocity uo and u, at two pointsdistant y apart in a direction transverse to the stream. The measure-ments were made at mean speed U 25 feet per second in a wind tunnelbehind a honeycomb with 0 *9-inchsquaremesh. The results are shownin fig. 1 where the ordinates are Ru Uo and the abscissae are theU2correspondingvalues of y. It will be seen that the RVcurveis apparentlyrounded at the top and that RVfalls to 0 08 at y = 0 38 inches. Nomeasurementswere made beyond this point, but extrapolation seems toshow that R =0 when y is about 0 5 inches, i.e., when y is slightlyar.2ntEr th2nS.n

* ' Rep. Nat. Adv. Ctee. Aero., Wash.,' No. 524 (1935).

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