Revista Mexicana de f(sica (Suplemento) 32 No. 51 (1986) 53.548

ON THE HISTORY OF THESTATISTICAL THEORIES

OF TURBULENCE

G. Battimell i

Universita di Roma, Italia

RESU¡'¡E~

53

Presentamos en estas notas una discusión de la introducción deherramientas estadísticas en el estudio de la turbulencia. El punto fo-cal es el período entre las dos guerras mundiales, poniendo especial atención a las llamadas teorías semi-empíricas y en la teoría de G.I. Tayloren 1935. El contexto institucional, que da a estos desarrollos su signi-ficado histórico, se menciona brevemente. Las notas continúan con unaexcursión a los años que siguieron a la Segunda Guerra MUndial, cuandoapareció la teoría de Kolmogorov y también las formulaciones equivalentesde Heisenberg, von Weizsacker y Onsager. -

ABSTRACT

In these lectures the introduction of statistical tools in thestudy of turbulence will be discussed. The focus will be on the interwarperiod, with attention especially paid to the so-called semiempirica1theories deve10ped in the Twenties and to the main shift in techniquesand language occurring with the statistical theory of G.I. Taylor in 1935.The institutional context, which gives these developrnents their full his-torical meaning, will also be sketched. An excursion wi11 follow iotothe years irnmediately after the Second World War, when Kolmogorov's theo-ry, and its equivaleot formu1ations by Heisenberg, von Weizsacker aodOnsager appear 00 the scence.

54

l. FLUID MECHANIC5 ANO KINETIC THEORY: EARLY THEORIE5 OF TURBULENCE

1.1. A Turning Point: 'Statistical' VE 'Empírical' Theories of Turhulence

The four-part paper on "rhe Statistical Theory oí Turhulence" by Geof-frey TayIer, Yarrow Research Professor at Trinity College, Cambridge, pub-lished in 1935 in the Proceedings oí the Royal Society(l). 15 unanimouslyreferred to as the most important turnin~ point in the history oí the the-oretical investigations on turbulent flows. Modern writers just considerit as the starting point fer the introduction oí statistical thinking inthe study oí fluid dynamics. To his contemporaries, it marked a strikingdeparture fram the maín concepts aud ideas which dominated the field atthat time, aud it was regarded as the introduction of a totally. new way oflooking at the problem of turbulence.

Turbulent flow had been for about half a century the largest singlestumbling block to the proper understanding of the properties of the be-haviour of fluids for engineers and applied mathematicians. Despite thesizeable amount of work done on the subject, very little had been accom-plished, up to the mid-Thirties, on the way of understanding the fundamental physical processes underlying the "hopelessly complicated"(2) proper-=-ties of the behaviococof actual turbulent flows, and no satisfactory wayhad been found to deal with the random fluctuations which chardcterize aturbulent velocity field. In order to render the subject more amenable IDmathematical investigation, Taylor limited himself to a simplified versionof the problem, introducing the idea of the isotropic turbulence, i.e~ enein which "the average value of any function of the velocity components. d~fined in relation to a given set oi axes, is unaltered if the axes of reference are rotated in any manner,,(l) or, roughly speaking, one in which -the intensity components in all directions are equal.

The importance of Taylors's work lies not so much in his derivationof new results, as in the fact that, using this simplified model of turbulence, he clearly defined the statistical quantities significant to thedescription of the state of turbulence, i.e.,the correlation functions b~tween velocity components at differenc times or different points in space,and that, by relating his model to the state of isotropic turbulence actually produced in a wind tunnel, he opened the way for meaningful expe~

ss

tal investigations of che relevant statistical properties oi a turbulentflow. This started a new trend in theoretical and experimental researchas well, and froID that poine on ie was customary to refer to theories ofturbulence by drawing a distinction between 'empírical' or 'phenomenol-ogical' (Le. pre-1935) theories, and 'statitical' (Le. post-1935) theo-

ries.

Stated simply chis way, chis distinction can be misleading. The 80-

ealled 'semi-empírica!' theories of turbulence Di che mid-Twenties and earIy Thirties were often often referred to (both by their founders and byla ter authors) as being based on analogies between a turbulent flow andche kinetic theory of gases. This clearly sugges~ that sorne kind of sta-tistical ideas were employed in these'theories. too. It is interestingtodiscuss the nature of the statistical concepts which are present in theseearly theories of turbu1ence, and the extent of the mentioned analogy~ththe kinetic theory. Before doing that, it seems appropriate ta say some-thing about the scientific community in which these ideas were producedand circu1ated.

1.2. Aplied Mathematics and Mechanics: a New Sector oE the ScientificCornmunity

As a problem in classical mechanics, the study of the properties oEfluid motion had for a long time attracted the interest and the effortsafa wide range of scientists, which ene can variously label as classicalphysicists, mathematical physicists, applied mathematicians - and, in sanecases, theoretically-minded engineers. The coromon concern fer the mathe-matical investigation of problems in classical mechanics which are of fundamental importance in practical applications slowly comes to identify,toward the beginning of the Twentieth Century, a peculiar figure af scientist,which more and more diversifies himself both from the physicist - no lon-ger interested in any problem in classical mechanics where, in principIeat least,there i5 (supposedly) nothing new to discover- and from the puremathematician - interested in developing new fieIds of mathematieal inve~tigation, and shunning the mere application af old and consolidated meth-ads of analysis to problems in ltechnieal physics1

• The archetypal modelof this professianal figure is shaped. in the early years of the century.

S6

in GOttingen under the influence of Felix K1ein. Ludwig Prandtl. whose(3)fundamental paper on boundary layer had been almost total1y ignored at

the Third International Congress oí Mathematics in 1904. carne to embodythe particular blend oC pure and applied science fostered by Klein and associated with GOttingen. The rnodel spread through Germany, mostly throughthe stream oi Prandtl1s students which carne out oí Gottingen and, withmore or les s delay and the permanence oi nation~characteristics. a simi-lar trend was visible a11 ayer Eurape and -even if later 00- on the othersirle oE the Atlantic.

In the years following the end oí the First World War. there areclear signs oí a widespread feeling for the need oí a more marked charac-terization of the peculiar nature of this kind of scientific activity atthe institutional level, and the scientistis engaged in the fie1d striveto hui Id up a stronger identity for their profession. After the Naturforscherversammlung held in Nauheim in 1920, some of the leading exponentsof app1ied mechanics in Germany (Prandtl, von Kármán, von Mises, Trefftz)agreed that it would be desirable to have at the next meeting sorne ses-sions specifically devoted to prob1ems in applied mathematics and mechan-ics, as distinct from the sessions for mathematics and physics(4). In1921, edited by R. van Mises, professor of app1ied mathematics at the University of Ber1in, a new journal makes its appearance, the Zeitschriñ fürangewandte Mathematik und Mechanik (Z~), which imediately becomes themain reference point for scientists working in the field.

Not sharing the strong nationalistic views of the majority of hiscolleagues, made stronger by the situation of the defeated Germany in thepost-war years, T. von Kárman had more ambitious plans. K3rman, che mostbril1iant of Prandtl's students, at the time director of the AerodynamicsInstituce in Aachen, devoted his main efforts towards che aim of recon-structing the international scientific cooperation broken by the war. In1922, together with the distinguished mathematician T. Levi-Civita fromRome. Kármán organized a conference on Hydro-and Aerodynamics in Inns-bruck(S). There the prospect was discussed of holding at regular inter-vals international meetings of scientists working in the more generalfield of applied machematics and mechanics. Notwithstanding the difficu!ties and the resistances (sometimes che boycott: Kármán often complained

57

about the 'Katz-und Mausspiel' between German and French scientistis(6».the program succeeded. The first International Congresa of Applied Mec~ies (ICAM) was held in Delft in 1924, aud ao International Committee wasformed with the task oi planning the future meetings. These were held ~ularly every iour yeara untíl the breaking oi World War 11, in Zürich(1926). 5tockholm (1930), Cambridge, U.K. (1934) aud Cambridge, Mass.(1938). Ihis international body gave consistence to the emergence audconsolidation oi a new sector oi the scíentific comunity.

1.3. Aplied Mechanics aud the Problem oi Turbulence.

Bordering with the fields of mathematics. physics and engineering.but claiming and identity oi its own, this group of scientists dealt witha heavy body of problems in the mechanics of solids and fluids which re-quired for their solution a degree of mathematical investigation far be-yond the capabilities of the cut-and-try approach of the practical engi-neers. From the beginning. turbulence was one such problem, andils weightkept growing in the interwar periodo At each ¡CAH one of the general le£tures was devoted to a review of reeent advancements in the problem ofturbulence, and a steadily growing number of papers and cornmunieationswere discussed on the subject. The main contributions to ~serniempiricaltheories of turbulenee were presented, respectively by L. Prandtl and T.von Kármán at the International Congresses in 1926 and 1930.

At that time, the problem of turbulence was usually eonsidered asconsisting of two different parts. The first problem was 'the onset ofturbulence', i.e., how and under which conditions in a given boundary thetransition oecurs froro the regular, laminar flow to the irregular, chaotie pattern charaeteristie of turbulent behavior. The problem was at-taeked by well-established mathematieal methods, studying the stabilityof the flow superimposing to the laminar solution of the equations smalldisturbances and determining for whieh values of the parameters eharae-terizing the flow these small disturbances kept growing instead of dyingaway. Even if few definite results had been obtained, the line of inves-tigation to be followed seemed to be clearly drawn: more carefu! experim-ental va!ues to be ehecked against the results of heavy manipultation ofstandard tools in mathematical analysís(]).

58clear view oí turhulence by satisfactory mathematical penetration •...orin the 501utioo will come only by breaking the boundaries of classicalmechanics and turning to statistical considerations. Fol1owing the im-posing and truly amazing results which have been accomplished by physicalstatistics in the last years, ene will probably take the second peint ofview, which, if it proved to be true, would be of inestimable, fundamen-tal meaning fer the general interpretatían oí mechanics."

lt could be pointed out that here van Mises í8 both suggesting atechnical step to be taken aud expressing the marked sympathies towardindeterminism aud rejection of classical, causal mechanics which he clearIy showed in other occasions just at the same period, as P. Forman indi-cates in his study of Weimar

h.. (10)german p YS1C1stS • Less

culture and the repudiation of causality bysensitive than the physicistsl community to

philosophical considerations, and essentially untouched by the ongoingconceptual revolution of quantum mechanics, the applied mathematicianstook sorne time to accept and develop von Mises's technical suggestion.The first paper to attract wide recognition and to inspire a line of re-search, in which statistical concepts taken from the kinetic theory ofgases were introduced, was L. Prandtl's report at the rürich ICAM in1926, "Uber die ausgebildete Turbulenz,,(ll). Preceded by a shorter com-munil¡ation in the ZAMM in 1925 (12), it claimed to be "still far from con-clusive", but to "concern, rather, the first steps in a new path which 1hope will be followed by many others".

1.4. Fluid Mechanics and Kinetic Theory: L.Prandtl

Right at the beginning of his paper, Prandtl c1early defines thelimits of the task he sets to himself:

The researches on the prob1em of turbulence which have beencarried 00 in ~ottingen for about five years have unfortu-nate1y 1eft the hope of thorough understandiog of turbu1entflow ver y sma1l. The photographic and kinetographic pic-tures have shown us only how hopelessly complicated thisflow is •••.Such pictures have hitherto been utilized on1y for statistical information on the magnitude of the ve10city variations.Otherwise, we have not been ab1e to learn much from them.That which 1 would call the "great prob1em of developed tuE.

59

bulence". involving a thorough understanding and a quantitativecalculatían of the processes through which nev phenomena are co,!!.stantly being developed and a1so involving the determinatían ofthat mixing force which i5 produced in each individual instaneeby the cantest of new development and damping, i8 not likelytherefore to be soon solved.It i5 always possible, however. if we forega a deeper underst~íog of the phenomena of turbulence. to £0110w theoretically va~iOU5 phenomena in a logical way controlled by experiments, esp~cially regarding the mean motion in a given turbulent flow. ~the determinatían of the mean velocity, as a funetion of theplace. i5 a technically very important task. The first step inthis direction may be so characterized that the apparent visco~ity forces, produced by the mixing, will be represented in sucha form that they can be introdueed into the hydrodynami~ ~tffe~ential equations for the mean motion of turhulent flows 1 .

The apparent viscosity forces here mentioned by Prandtl are thosearising in a turbulent flow as a con sequen ce of the fluctuating, irregu-lar velocities. Let us consider, as Prandtl does, a two-dimensional flow,such as the flow between two parallel pIates; one is interested in de termining the distribution of the mean velocity in the x direction as a function of the distance y from the wall. The Navier-Stokes equation for thex-component of the velocity u is

dU+UdU+VdUat ax ay _ 1. .£E.+ ~ ra2u + a2u]p ax o 2 ,2

lOX oy

where v i5 the y-component of the velocity, P the pressure, p the massdensity and ~ the kinematic viscosity coefficient.

This equation may be easily sol ved (assuming steady conditions) inthe case of laminar flow where V is zero and no fluctuations around themean values are present; the result is a parabolic profile for the ve-locity. In the case of turhulent flow, following Reynolds, one canwrite u • U + u, V '" v where U is the mean value oí the x-component oíthe velocity and u and v the fluctuating parts oí the x and y componentsof the velocity respectively. Taking into account the continuity equa-tion and time-averaging according to the rules

u • U u '" v '" O

510

[ ] [ ] [ ] r ]

where the bars ayer the quantities denote the time-averages, ane obtains.fram the aboye equation

au + U auat ax"2 aupu - \J dX 1 apar r

~ au -ay puv

The fluctuating velocity components do nat disappear fram the~quation after the averaging process due to the presence of a non linearterm, aud they appear as addítional 'turbulent viscous forces1

, in theforro oi the factor puv showing the existence of a nonzero correlationbetween the fluctuating velocity components.

The existence of tu~bulent shearing stresses, much larger than themolecular viscous force \J(dU/ay), had long been known fram empírical evidence. Fol1owing the French engineer J. Boussinesq(14) they were oftenrepresented in the forro eedU/ay) where E, known variously in the liter-ature as I Austauschgrosse' or 'eddy viscosity', is a quantity dependenton the particular turbulent flow and varying from point to paint. Prandtlsproblem was to find a way to relate the turbulent stress puv to the meanvelocity in order to allow the closure of the hydrodynamic equations andthe determination of the velocity profile for a turbulent flow.

To do so, he pictured the motian of the turbulent flow as a mixingprocess of colliding fluid particles, introducing an analogy betweentransport phenomena in kinetic theary and turbulent exchange:

At this point it is essential to introduce the characteristiclength for the turbulent condition, which here plays a similarrole to that of the mean free path in the kinetic gas theory.lt can be considered as the diameter of the simultaneouslymoved fluid masses, but also as the distance traversed by sucha fluid mass before it loses its individuality by mixing withneighbouring masses .•. Following the second definition, weviII designate this length as the 'mixing path' (or 'mixinglength') and represent it by l. If we assume that a movingfluid mass, situated in a current with velocity decrease cros~wise to the current, possesses a velocity equal to the mean

511

velocity at the. point ..•••here it originated. and that it 1Sshifted by the mixing length 1 transversely to the current.its velocity wil1 then differ from the mean velocity at thenew point, and this velocity difference, in its first ap-proximation, equals l(aU/ay) •..•.•hen the mean direction of£10\01 relative to the x-axis 15 chosen. The mean oscillationu can therefore be pur proportional to l(aU/ay). The trans-verse motion v can be considered as being produced in themanner that two fluid masses with different u. which findthemselves in one another's presence. either come tORetheror move farther aparto The velocities v thus produced cantherefore be ~t proportional to u. The apparent shearin~stress (t = puv) therefore becomes

when the proportionality factors of u and v and also the cor-relation factor, which would come into the formation of the(15)mean product, are collected into the still unknown length 1

A comparision with the Boussinesq formula 1 e £(autay) shows thatone can express the 'eddy viscosity' £ in the form

plv.

In the kinetic theory, molecular viscosity is given by the product PAc,where A is the mean free path and c is the average velocity of the heatmotion; the mixing length and fluctuating velocity thus play an anal-ogous role in turbulent flow theory as the mean free path and the heatmotion do in the kinetic theory of gases.

Prandtl was well aware that his formula, though qualitatively satisfactory (it gives resistances proportional to the square of the velocity.as it was empirically known to be the case for turbulent friction), suf-fered from serious drawbacks. The whole line of reasoning. and the veryidea of a 'mixing length'. rest on a questionable picture of the rnixingprocess in turbulent flow; a 'fluid particle' is a rather ill-definedconcepto Moreover. very little can be said from the theoretical pointof view about the mixing length 1 (one can just say that it must go tozero as Qne approaches the limiting walls. where transversal motion is

su

prevented by the boundary conditions). It has to be determíned fer everyparticular f10w configuratían froID empírical data. The statístical pro£erties oí the turbulent velocity field, which are embodied inta the míx-ing length, still elude theoretical investigatían. And, since 1 15 atheoretically unknown quantity, ane 15 still not able to salve the hydr~dynamic equation and derive the velocity profile. Prandtl's propasal,as he himself admitted, was just a first suggestion. The next step,leading to ",hat i5 still regarded as the roain result oí the semiempirical theories, was taken by van Karman in 1930(16). -

1.5. Van Karman's Sirnilarity Hypothesis

Van Karman's aim was "to make the laws Di turbulent flow in channelsmore accessible to calculation with a minimum of arbitrary assumptions".A first assumption consists in neglecting the viscous terro in the hydr~namic equation; this is certainly a wrong approximation in the immediateproximity of the walls, where viscous force domina tes over the turbulentstress, but experimental evidence suggests it to be a reasonable assump-tion a11 over the remaining section oí the channel. A second assumption,oí essentially statistical character, is made about the properties ofthe fluctuating velocities. It is assumed that "the state of flow inthe neighbourhood of two points corresponding to difíerent va1ues oí ydifíer írom each other only by the scale oí velocities and length", or,in other words, one assumes lIsimilarity of the state of fluctuation independent oí the position oí the considered point in respect to the meanmotionlT•

The average velocity around a point at ypowers oí y as

o can be developed in

u2

U'y + U" L +o o 2

where U and U" are the first and seeond partia1 derivatives of U witho orespect to y evaluated at y = O. Accordingly, the stream funetion~ = ~(x,y) defined by the relations

ao/u =-ay

can be expressed as

v e

513

o/(x,y) = t U~ / + i U~i + 000 + 1jJ(x,y) (1)

where W(x,y) i5 the stream function of the fluctuating moticn. If nowene sets

x = U; y 1jJ(x,y) (2 )

the similarity condition requires that only 1 and A be dependent on thePoint. Le. on U' u" etc. while the function f(f,;.n) be independent.o' o •The Navier-Stokes equation written by means of the stream function. thevorticity equation, i5

Neglecting the viscous terro and keeping only the first terms in expres-sion (1) for~. we may write

[, ~lU Y + ,o ay.£M' _ ~ U"dX dX O

Usiog the relations given by (2) this last express ion may be written as

Ulll1o o

This last equation i5 independent of A, 1, U'. U", as requested byo othe similarity hypothesis, if the following relations hold true:

U'l ex ~o 1

514

or

u'1 ex ....£.

U"o

U,312A O U'a--U,,2 O

O

The shearing stress 15 thus given by

T

The terro in parenthesís in the far right hand sirle oí this last expres-sion 15 naw a constant characteristic oí the particular turbulent flow.Prandtl's result comes up again

T a p¡2 U,2o

which comes as no surprise, since, as von Kármán remarks

Praudtl set the campanent oi fluctuatían v propartiona! to thelength 1 aud the absolute value laU/ayl oí the velocity gradient,from which we obtain T = pI2( aU¡3y)2 in agreement with (Kármán'sresult). Considered from the standpoint oí the fluctuatían theo-ry, this assumption signifies that the correlation between u andv can be taken as independent Di the positíon. Evidently this assumption corresponds to OUT assumption of similarity and, there--fore, it is not to be wondered that we confirm this expression.

Our similarity consideration, however, gives essentiallyrnorethan this in that we obtain an equation determining the mixinglength, which to the first approximation appears as the ratio ofthe two first differential quotients of the mean velocity. Itremains to be seen what the consequence(of this relation wi11 bein respect to the velocity distribution 17).

In the case of flow in a channel of width 2h, if To represents theskin friction at the surface, the value of T at a distance y froro themiddle of the channel is given by

T To...:Lh

515By putting together this express ion with those previously obtained fer 1aud -r ene has

T :1o h

or

u"U2 •

,th 1k--- 0-

(To! p IY

where k i5 a constant. This expression may be integrated «au/ay) goingto infioity at y = h. i.e., at the wall)

u. 12k

1,th- ¡y

Integrating again, ene obtains

for the velocity as a funetían oí the distance from the wall (given by

h-y) o

The quantity k stil! remains in the formula as ao unknown constantto be experimentally determined. Nonetheless. ene now had a theoreticallysatísfactory result to substitute fer the various power laws which hadbeen empirically suggested to account for existing experimental data.

In the final summary oí his papero von KÁrmán pointed the limits oíhis results, showing at the same time bis personal satisfaction:

Although the determination of the characteristic length offluctuation 1 with the help oi the similarity assumption hasjustified itself to sorne extent. the assumption should bebetter founded. The prooi of the existence of a solution ofthe flow equations limited to a relatively narrow width offluctuation. This calculatíon should also give the value of

516

the constant k. It 15 a1so questionable, whether or not theassumption i$ applicable to other more complicated cases. 1wish to emphasize that this i6 not at al] certain ..•.

It 15 possible, however, that through the introductionof the equations of flow. the way to a rational theory ofturbulence will be opened, leading to a rational treatmentof many problems associated with turbulent flow ..• (lB)

Karmán was clearly proud to have accomplished more thao his oIdteacher Prandtl had been able to do, by rederiving his results in a moresatisfactory way, without using dubiaus hypotheses on the mixing process,aud pushing the aualysís to the final goal of obtaining a theoretical expression for the velocity profile. Surprisingly, he maintained that histheory was basically a refined version of a kinetic theory of turbulence.Thus "Kinetische Theorie der Turbulenz.1 was the original title he hadenvisioned for his paper, and only in the final version did he change itto "Mechanische Ahnlichkeit und Turbulenz': T'¡ritin~at the end of 1929 toJ.M.Burgers, the leading Dutch fluid dynamicist in Delft, he said: "1have tried something like a kinetic theory of vortexes, and 1 use theword 'kinetic' as opposed to 'statistic' just as the kinetic theory ofgases following Maxwell is opposed to the statistical viewpoint followingGibbs and the modern statistics(19)1I. Burgers, a former student of PaulEhrenfest, was at the time attempting to introduce the ideas of statistical ensembles in the study of a collection of vortexes(20) audKármán was stressing the difference between Burgers' approach and his own.As fer as the kinetic theory is concerned, however, it seems fa ir to saythat the actual relationship between the kinetic theoryPrandtl's and Kármáfi's 'kinetic' theories of turbulence

of gases andis nothing more

than a very crude analogy. These theories, anyway, opened the road to acertain degree of theoretical insight into the study of turbulent shearstress and resistance, which was their ultimate goal; and much work wasdone in the following years trying to adjust them to the most various flowconfigurations and to obtain experimental data allowing to determine thenumerical constants which sti11 appeared as unknown quantities in the eq~tions.

5171.6. Atmospheric Turbulence, Diffusion aud Random Walks, i.e., G.I.

Taylor.

In 1932 G.I.Taylor presented his version of a 'mixing length' the-ory, which differed from Prandtl's in the fact that in the mixing processTaylor considered vorticity instead of momentum as the conserved quan-tity(20). Taylor had become interested in the problem oi resistance audfluid friction through his acquaintance with the Cambridge aerodynamicistB.M.Jones aud the experimental work which was conducted at the AeronauticULaboratory at Cambridge aud at the National Physical Laboratory in Ted-dington. With the emergence oE Prandtl's aud Kármán's theories he was ledto Qevelop a theory oE his awn. taking over again ao idea which he hadintroduced many years earlier. when his interest in turbulent phenomenahad first been aroused in a completely different contexto Taylor's preo~cupation with turbulent flow dates in fact from the period immediatelypreceding the first World War. A.Schuster,professor ofphysics atManchester,had given money to Cambridge University in order to establish a tempararyreadership in dynamical meteorology; Taylor was appointed in 1911 andstarted doing research 00 turbulent traosfer processes in the lower atmo-sphere. Fo110wing the tragedy of the Titanic in 1912. the British Government sent Royal Majesty's ship 'Scotía' for a scieotific expeditían tolook for iceberg s in the Northern At1aotic and report about their position.Tay10r was on the Seo tia as the meteor010gist. and spent six months a1on~the Banks of Newfoundland £lying instrument-carrying kites from the deckto co11ect data about vertical temperature and wind ve10city distribution.In the resulting paper, publíshed in 1915(22), and devoted to theturbu1enttransfer of heat and momentum in the atmosphere. he had introdueed thesame idea of a mixing 1ength which appeared ten years later in Prandtl'spapers. As it dealt with atmospheric phenomena, Taylor's work was notpaid mueh attention by the applied mathematieians busy with their problemsof friction in channels and pipes. Taylor, meanwhile, was stillattraetedby atmospheric problems, and turned his attention to the properties oí di!fusion oí partieles in air produced by f1uctuating turbulent veloeities.Stimulated by his knowledge of K.Pearson's work in statisties, he envisOOnedturbulent diffusion as a kind of random process, referring it to statisti-

518

cal problems widely studied by Pearson, such as the drunkard's walk orthe random migratían oí insects. The result was a very elegant paper onIIDiffusion by continuous movements". published in 1921(23).

Befare discussing the continuous version oí the problem oí randommigratían in ane dimension, Taylor proceeds to a 'slight extension' oíthe discontinuous case. Imagine a paint moving along a straight linewith uniform velocity v which, aíter a time interval T, either reversesits motían or continues moving forward. and let the process be repeatedn times. Thea time nt=T

mean squareX2 wil1 ben'

value oí the distance traveled by the point after

where the bracket indicates that the mean value i5 taken. Henee

with

d vT.

It there is no correlation between any of theand

x vlTT,n

,xs' [x ,x]

r so for all ., s

a familiar result showing that the standarodeviation is in this case pro-portional to the square root of the total time. Taylor now proceeds toconsider the case in which there is correlation between successive dispuc~menta:

Actually in a turbulent fluid or in any continuous motion thereis necessarily a correlation between the movement in any oneshort interval oi time and the next. This correlation will evidently increase as the interval oi time diminishes. tillo when-the time is short compared with the time during which a finitechange in velocity takes place, the coefiicient oi correlationtends to the limiting value unitY(24).

519

Let us assume that xr and xr+1 are correlated by a correlation coef-ficient e, so that the correlation between xr and xr+s i8 given by thecoefficient eS Then ane has

X2 __nd2 2 [ 2 (2) 3 n]+ 2d nc+(n-l)e + n- e + .•. +cn

Summing the series, and putting TI = TIT d v l, ane gets

2 { [~~~] 2 n 2 }x2 TT - 2e (l-e )Tv 2n (l-e)

If we now let T become indefinitely small, e will tend to its limitingvalue oE l. Calling A the 1imít to which the quantity l/el-e) tends ast + O. e + 1, ane sees that X2 has the limiting value

n

For sma!! values of T this reduces to xn = vT. which i5 what ane wouldexpect since the correlation coefficient between successive displacementshas nat appreciably departed from the value oE l. When T i5 large, on thecontrary. ane has Xn = v 12AT. The memory oE the correlation has be enlost.

Taylor now goes on the investigate the statistical properties of thecurve of any function varying irregularly with time, say p(t). character-ized by the constancy of the standard deviation lfP1T over successive pe-riods of time. Obviously the value of [p2] does not identify the curve;one has to specify the values of [ (dp/dt)2], [(d2p/dt2)2], etc. as well.Taylor shows that the successive derivatives of the curve p(t) are corre-lated between each other in the following way

[~Jp 2ndt

o

520

The correlations to which p aud its differential coefficientsmust be subject in arder that their standarddeviations may beconstant, have now been established. We can, therefore. nowuse these standard deviations to define sorne statístical pro~erties oi the curve.

In analysing aoy actual curve, it may be very difficultand tedious to obtain these standard deviations. There is,however. another method oi defining the statistical propertiesoi the curve which i5 equivalent to that given aboye, butwhich i5 likely to be much more manageable in practice. Thismethod will now be considered(24).

This other method consists in considering the correlation coefficienthetween values oi the curve at two different times. This 1S defined by

which reduces, s1nce the standard deviation of p 1S constant, to

R~

Now expand Pt+~ in powers oí ~

? ¿Pt+~ P + ~.'!I'. + ~- + ...t dt 2T dt2

Hence,

[P t Pt+~] [ p2] + <~J .c [p BJ + ...+ 2~

and

R~ 1 -(2 [[*-l} ~4 [(~JJ + ...

2 ~ [ p2] 4~ [p2]

After these preliminaries one can discuss the problem of diffusion

521

by continuous mevements:

The theorems which have just been proved vil! now be used tofiod out what are the essential properties of the motion oia turbulent fluid which makes it capable of diffusing throughthe fluid properties such as temperature. smoke content,colouring matter or other properties which adhere to eachparticle oi the fluid duriog its motion.

Consider a condition in which the turbulence in a fluidi5 uniformly distributed so that the average conditions oievery pointin the fluid are the same. Let u be the velocityparallel to a fixed direction, which ve viII cal1 the axis oix. oi the particle on which our attention i5 fixed. lt viIInow be shown that the statistical properties which were de-fined aboye (now in relation to u instead of p) are suffi-cient to determine the law of diffusion. Le .•the law whichgoverns the average distribution of particles in~¡ially con-centrated at one point. at any subsequent time(2 .

2Suppose that one knows u and R~ (this is now the correlation coeffi-cient between the value of u for a particle at any time and the value ofu for the same particle a time la ter ~). Now

r [ut uf;l d~o

and since [u21 does not vary with t and R~ is an even function of ~.

2 ft[u I R~ deo

One a1so has

fr ut u¡} d~ = [utI: u~ d~J= [ utxlo

where X is the distance traversed by a par tic le in time T. Hence

and

2 ft[u] R~ d~o

I ] 1 d [X2]utX = "2dt

522

Thus the problem oí diffusion, in a simplified type of turbulent motian,i5 reduced to che consideratían oí a single quantity, the correlationcoefficient.

If £0110w5 immediately that, when the timefiOt differ appreciablyoi che last express ion

or

from unity. the2equals T /2. so

integralthat

T i5 so small that R~ does

on the right hand sirle

On che other bando it seems reasonable to as sume that after a certaintime interval TI the correlation coefficient ",.il1fa11 to zero. In chiscase

lim J.t R~ d~t~ o

will be finite and equal to l, sayo Then,for 1>11

and

so that the standard deviation i5 again proportional to the square roct oíthe time. As Taylor remarks. "this, therefore, la a property which acontinuous eddying motian may be expected to have which is exact1y ana10-gous to the properties oi discontinuous random migration in one dimension'~

523

1.7 A turning peint fer a turned problem

It i5 evident that all the roots Di Taylorfs approach to the statis-tical theory oí turbulence oi 1935 are ta be found in the paper we havediscussed. The 'uniformly distributed' turbulence oi 1921 leads directlyta the more rigorous definitían of isotropic turbulence. and the ideathat the fundamental characteristics oí turbulence are expressed by thecorrelation functions i5 clearly laid out in this work. Que might wellwonder why fourteen years were necessary for these ideas to be takenagain aud force their way through a wider audience. As far as Taylorhimself i5 concerned, ao explanatían could be found in his personal styIeoí work aud habit oí research. He defined himself a classical physicist,and often stated that he was willing to pursue atine of theoretical inves-tigation only if at the same time the chance was given oi checking itagainst the results of simple experiments he could perform by himself.Due to the rather abstract character of his study of 1921, he explicitlyhanded over the problem to the mathematicians, publishing the paper inthe Proceedings of the London Mathematical Society. He took the matterover again in 1935 when, with the deve10pment in the ear1y Thirites of.the technique of the hotwire anemometer, it was possib1e to mea sur e withsufficient precision the fundamental quantities he was interested in:correlations.

Why Taylor's paper on diffusion failed to have any effect upon thecommunity of applied mathematicians working on the prob1em oi turbulenceis not difficult to see if one compares it with the typical products oi

Prandt1 and van Kármán. lt is quite clear that Taylor's prob1em is actually a completely different one. His 'simplified and uniform1 turbulencehas very little to do with the inhomogeneous turbulent flow in a channelwhich is the central problem far the leading hydrodynamicists of the day.Only much later, in 1935, will these ideas spread and produce a dramaticchange in the approach to the problem of turbulence. This will in turnbe possible beca use the problem itself will have by that time modified itsnature. The discussion of this shift i5 the subject of the next chapter.

S24

2. STATISTICAL THEORIES OF TURBULENCE; G. TAYLOR'S TURNING POINT IN 1935

The year 1935 is usually considered as a crucial moment in the development oi the theories of turbulence, due to the publicatían, in theProceedings Di the Royal Society, of a four-part paper by G. l. Tay10r,oi Trioity College, Cambridge, entitled "Statistical Theory of Turbu-lence" (27) . This paper had a great impact on the conununity of applied mat..!!.

ematicians working in the field oi fluid dynamics and the scientists io-volved with experimental research on the properties oi turbulent flows.It started a new trend Di copious production of theoretical and experi-mental results as well. From that moment on, it was customary to referto theories of turbulence by drawing a clear distinction between 'empiri-cal' or 'phenomeno1ogica1'.

In 1935 Tay10r approached the prob1em in much the same way as he haddone in 1921. At that time, his work had passed almost unnoticed. Four-teen years later the same ideas aroused widespread interest. We wi11 discuas the reasons for such a sharp change of attitudes towards Tay1or'stheoretica1 proposa1.

Tay10r begins by pointing at the conceptual difficulties encounteredin the various 'mixing 1ength' theories existing at that time. lt isconvenient to quote directly from the intraduction of his paper:

At an early stage in the develapment of the theary of turbulencethe idea arase that turbulent motion consists of eddies of moreor less definite range of sizes. This eonception combined withthe already existing ideas of the Kinetie Theary of Gases ledPrandtl and me independently to introduce the length 1 wieh isoften ca11ed a "Mischungsweg" and is analogous to the "mean freepath" of the Kinetic Theory. The length 1 could on1y be definedin re1ation to the definite but quite erroneous conception thatlumps of air behave like mo1ecules of a gas, preserving theiridentity till sorne definite point in their path. when they mixwith their surroundings and attain the same velocity and otherproperties as the mean va1ue of the corresponding property in theneighbourhood. Such a conceptíon must evidently be regarded asa very rough representatían of the true state of affairs. If weconsider a number of particles of small volumes of fluid startingfrom sorne definite level and carrying. sayo heat in a directiontransverse to the mean stream lines. their average distance fromthe 1eve1 at which they started will go on increasingindefinitelyso tbat we can only consider a "Mischungsweg" in relation to &Xnearbitrary time of f1ight duriog which we must consider that thepartieles preserve their individual propertíes distinct from those

525

Di their surroundings. Clearly this i5 an arbitrary conceptionand if pursued logically probably leads to a definitely wrongresult ...The difficulty ••. led me, sorneyears ago, to introduce the ideathat the scale Di turbulence and its statistical properties ingeneral can be given au exact interpretatían by considering thecorrelation between the velocities at various points "Di the fieldat ane instant of time or between the velocity Di a partiele atene instant of time and that Di the same partiele at sorne defi-oite time, S. later(28).

The work Taylor is referriQ.g to is, clearly, his "Diffusion by Cootínuous

Movements" Di 1921. There he had shown that in a field Di "uniformlydistributed" turbulence two quantities were sufficient to characterizethe diffusive properties of the turbulence, namely its intensity ~u21,defined as the mean square root of the velocity fluctuations (supposedconstant throughout space), and the correlation coefficient between val-ues of the turbulent velocity of the same fluid particle at two differentinstants of time.

After surnmarizing the main results of the 1921 paper (see chapter Ü.Taylor introduced length 11 defined by

11 • ~ JTR~ d~o

which could be interpreted as giving a measure of the scale of turbulence,similar to the mixing length of the older theories. with the importantdifference that the new length was now tied to a fundamental statisticalfeature of the turbulent field, bearing no relation to ambiguously def~mixing processes. It could be defined even if there was no mixture atall.

Shifting from a Lagrangian to an Eulerian viewpoint, another typicallength could be introduced, considering now the correlation coefficientbetween velocities at the same time at two different points in the fluid

526

A and B spaced a distance y apart R • defined byy

RY

where uAand uB denote the velocities at paints A and B respectively. Itis to be expected that this correlation coefficient, •.••.hich is equal tounity foc y = O, will fall to zera as y increases. If Y is a distancesuch that R

y= O for aay y > y a length 12 may be defined by

RYdY

analogous in the Eulerian system to the length 11 in the Lagrangian sys-temo Taylor suggests this length as au "average size Di the eddies".

Taylor then looked into the general problem oi utilizing the funda-ffi;ntal statistical properties oi turbulence just defined in arder tocharacterize the significant physical features Di the system. In a tur-buleat £low, due to the presence oi viscosity, the relevant quantity isthe rate of dissipation of energy:

The rate of dissipation of energy in a fluid at any instantdepends only on the viscosity. ~. and on the instantaneousdistributian of velocity. If, therefare. the representationof the essential statistical properties of the velocity fieldcan be expressed by the ~ curve and similar correlationcurves it must be possible to deduce from them ttE rate of dissipatian of energy. This would in general involve a complicatedanalysis, but the prablem can be much simplified if the fieldof turbulent flaw is assumed to be isotropic.In isotropic turbulence the average value of any function ofthe velocity components. defined in relation to a given set ofaxes, is unaltered if the axes of reference are rotated inany manner(29).

The general expression for the mean rate oE dissipation of energy [W]. obtained from the equations oi motion. is given by

527

[wJ = V {2 [[~~ n + 2 [~; r} 2 [[~; n+ [[~: + ~~] 2J + [[~~ + ~~n+ [[~~ + ~:] 2J}

In the case of isotropic turbulence Taylar investigated the mean valuesof aoy quadratic function oí the nine quantities (dU lax.), Which aontainsa 1

forty five terms. With ingeniaus calculations he showed that when theturbulence i~isotropic all these terms are simply related to each other,so that it 15 sufficient to know just Qne oí them, say [ (du;ay) 2]. Hence.

the express ion for the mean rate oi dissipation reduces to

The quantity [(du/ay)2] is related to the way in which the value oi RY

falls off from its initial value of 1 as y increases from zero. As wesaw befare. Taylar had proved in 1921 that the correlatían coefficientcould be expressed as a power series

RY

Therefore

lim)'+0

l-R-----.:.L

2Y

and ane may introduce a new length A defíned by

1~=

l-R1im ---:i....)'+0 2

Y

528

so that

Iwl 15~

Since the length A 15 related to the correlation length R I the mean rateyof dissipation of energy is expressed as a function Di the intensity audseale Di turbulence.

Taylor then turned to the problem of the decay of turhulence. Thekinetic energy per unit volume oí a turbulent field 15 given by

122 2"2p[u +v +w I

which reduces to

3 2"2 p[u ]

fer isotropic turbulence. The rate Di decay of the kinetic energy perunit volume is therefore given by

3 d 2--p-Iu]2 dt

Consider now a field of turbulence produced in a wind-tunnel downstreamof a square-mesh grid in a flow with uniform translational speed u. Therate of decay in tbe regian downstream froID the grid. wbere it may beshown that the turbulence 15 actually isotropic. can be written as

and equated with the mean rate of dissipation [W]. That is.

3 d 2--pu-Iu]2 dx

529

Furthermore Taylor suggested that in the field oí turbulence produced by

the grid, the acale oí turbulence would be dictated by the length oí thesquares oí the square-mesh grid M, aud that a relatían should thereforeexist between Aaud M. He found that

AM

with u' = ~u~ aud A a universal constant fer a11 grids of a definitetype. Usiog this relatían ane finds the following simple law fer thedecay of turbulence

U 5xul '" A2M + constant.

where x denotes the distance froro the grid.This 15 essentially the content oí Part 1 of Tay1erls papero Parts

11, 111 aud IV contain no new ideas, but are devoted to comparison oítheoretical predictions with experimental wind-tunnel data, discussionoí measurements aud suggestions for further experimental work. We willcome back 1ater to this aspect of Tay1or's papero and turn now to ashort discussion of the roots of his ideas on isotropic turbu1ence andhis main source of inspiration.

We know that what 1ed Tay10r to his paper on diffusion in 1921 washis interest on heat diffusion prob1ems in atmospheric turbu1ence andhis acquaintance with K. Pearson's work on statistics, especia11y onrandom walks. From 1921 to 1935 Tay10r did not pub1ish aoything aloogthe same lines, but there is clear evideoce that this prob1em was continu3us1y present in his miod. In the iotroduction to the paper 00 dif-fusion he said:

In the course of the work 00 discussion of the convergency ofthe series used is attempted. The work must therefore be re-garded as incomplete. The author feels that such questionsmight be examined with advantage by apure mathematician, andit is in the hope of interesting one of them that he w;shesto offer this paper to the London Mathematica1 Society{JO).

530

The pure mathematician which became interested to the problem ""asNorhert Wiener. 1 .••.•as first attracted to investigatet<liener's role in

the story by a remark which can be found in his autobiography; speakingoí his early work, when he occupied himself ""ith Lebesgue integratíanaod measure theory, Wiener ""rote:

1 ""as an avid reader of the journals, aod in particular oi theProceedings oi the Lendan Mathematical Society. There 1 S8"" apaper by G.I.Taylar .. concerning the theory oi turbulence. !bisi5 a field oí essential importance for aerodynamics aod aviation.aod Sir Geoffreyhas for many years been a mainstay oi Britishwork in these subjects. The paper ""as allied to my own inter-ests, inasmuch as the paths oi air particles in turbulence arecurves and the physical results of Taylor's papers involveaveraging or integration over families of such curves.With Tay10rls paper behind me, I carne to think more and moreof the physical possibilities of a theory for average s overcurves. The problem of turbulence was too complicated for im-mediate attack, but there was a re1ated problem which I foundto be just right for the theoretical considerations of thefield which 1 had chosen for myse1f. This was the problern oíthe Brownian motion. and it was to provide the subject oí myfirst major mathematical work(31).

From what 1ittle i5 left of the correspondence between the two men inWiener's papers, it seerns that things went actually the other way round:it was Taylor who sawWWner's paper on Brownian motion (puhlished in theProceedings of the London Mathematical Society in March of 1924(32~, andwrote him sending a reprint of his paper in April 1924:

I take the 1iherty oí sending you the enclosed paper on thechance that it might be possible to extend your methods givenin a recent paper 11The average va1ue oí a functiona111• Proc.Lond. Math. Soco Vol. 22 p. 454. to the problem which I thereconsidered. My paper was written entirely from a physicalpoint of view and at the time 1 wrote it 1 was not aware oíEinstein's work. It appears to be about a similar proh1ern toyours and Einstein's but the results seero to be oí quite adifferent type. The difference perhaps lies in the condition3 p. 455, namely that the probabílíty that the particle con-sidered wanders a given distance in a given time is iodepend-eot oí the "directioo io which it wanders" (33).

Wiener,a5 Einstein, was considering the special case in which there i5 nocorrelation between 5uccessive displacements. '1Hener's answer i5 not pre-

531

served in his correspondence, but from a second letter from Taylor datedJune 1924 we can ínfer that he showed interest in his work, and senthim more reprints of his papers 00 irregular motion.

Taylor kept £0110.•••.iog Wiener's more abstraet work; in a 1etter datoo

January 193~he writes to Wiener that he i5 studying his paper on ActaMathematica (this i5 ene Di Wiener's fundamental papers, the one on "Gen-

eralized Harmonic Analysis"), and inquires about the possibilities oEusing r,.]iener's methad in "solving the problem Di twc-dimensional turbu-

lence .•••.hen attacked in a similar manner to that used for ene-dimensionalturbulence in my paper 'Diffusion by continuous movementsl 11(34) . 1932

i5 the year in which Taylor published his vorticity transfer theory,whieh is of mixing length type; it is clear that he was also thinking atthe same time of extending his old method using eorrelations. The influenee of Wieners work on harmonie analysis was even more apparent in asubsequent development of Taylor's theory. of whieh we wil1 speak later.

There is. therefore. no quantum jump from 1921 to 1935, but rathera slow maturation of ideas and refinement of tools from Taylor's partoThere is. on the contrary. a quite marked quantum jump fram the earlysemiempirieal theories oi turbulenee and Taylor's statistical approachin 1935. Ii one loaks at the main eharacteristies of Prandtl's and vanKármán's theories in the late Twenties. ando most important, at the kindoi problem they were attempting to solve, it is clear that with Taylor'stheory we enter a different worldj when one refers to the problem of tur-bulenee, one is speaking, in both cases, of two very different things.uThe great problem of developed turbulence", as Prandtl define:! it, was inthe Twenties the problem of deriving sorne theoretical express ion for thevalues of the tangential stresses (the Reynolds stresses) responsiblefor the transport phenomena in a fluid flowing along a given boundary.and for the resistanee experieneed by a body immersed in the fluid. Theterms "problem of developed turbulence" and "problem of resistanee" wereoften utilized as synonimous. One attempted to find a way of introducinginto the mean flow the effect~ of the tangential stressps due to the pre~ene e of eorrelations between different eomponents of the turbulent veloeity at the same point in the fluid, and to understand how this affected

532

the shape of the velocity profile in particular flow configurations withmarked characteristics of inhomogeneity and anisotropy. No wonder thatTaylor' s paper on diffusion failed to attract much attention, referringas ir was to the "simplified, uniformly distributed turbulence". Thefirst reference to Taylor's work on diffusion appears in von Karman'sgeneral lecture on turbu1ence at the Fourth International Congress ofApplied ~echanics in 1934(35), mentioned as being "a radica11y new ap-proacH'that accounts for the slowness of its acceptance. The shift fromthe problem of developed turbulence as it is conceived in the late Twen-ties and early Thirties, to the problem of the statistical properties ofturbulence as emphasized in the late Thirties implies not only a chan¡;tein the technical instrumeots invo1ved but a drastic shift in the oatureof the problem itse1f. Dne does not 100k any more to the characteristicsof the mean flow but focuses on the randoro patteros oí the fluctuations.rhe asymmetric, inhomogeneous state of the f10w a10ng a fixed boundaryis replaced, as the object of the analysis, by the isotropic state ofturbu1ent flow where no boundaries are presento The key words change.From 'shear stresses', 'wa11 distance', 'law of resistance', 'turbulentmixing'. one goes to 'corre1ation functions'. 'random walks'. 'isotropicturbu1ence', 'Fourier-component analysis'. It is a shift in technicallanguage which is highly revealing of the changing character of the prob-1em under investigation as much as of the kind of matheroatics and ideasinvolved. This can be immediately rea1ized by pointing at the mere factthat in isotropic turbulence there are no tangential stresses. rhe"great problem of turbulence". as defined by Prandtl, simply disappears.

From this point oí view. the question one is 1ed to ask is not somuch "why did ir take so long for such ideas to be widely accepted anddeveloped", but rather "why. then, were they accepted and deve10ped inthe mid-Thirties. if they were so radically new?".

Let me first substantiate the statement that Taylor's ideas wereaccepted and deve10ped. by giving a short account of the content oí twoimportant papers published in 1938, one by T. van Kárman and L. Howarth,the other by Taylor himself.

Karman was great1y impressed by Taylor's work and soon saw the pos-sibility for improvement. In the academic year 1936-37 he worked close1y

533

with the young British mathematician Leslie Howarth, a student oi S.Goldstein in Cambridge. which carne to Calteeh as a King'q College Research

Fellow (according to W. Sears,h ,,)(36)teach me tensor t eory • .

Kárma.n taId him "1 .•••i11 get Leslie ta

In their joiot paper(37). they developeda general theory of isotropic turbulence, introducing the carrelatíantensor, wich describes the carrelatían coefficient between two arbitraryvelocity components at twa arbitrary points, and showing ha.•••in isotrupkturbulence it i5 completely determined by a single sea lar funetion. Theyrederived Tay10r's results in a more elegant .•••ay. and then turoed to thediscussion of the dynamics of isotropic turbulence. obtaining an equationfor the time development of the eorrelation funetion whieh eontains, dueto the non-linearity of the equations, non-vanishing terms deseribingtriple eorrelations. Using an approaeh similar to the one Kármán haddeveloped in his 1930 theory, that is, introdueing a simi1arity hypothe-sis, they suceeded in redueing the number of unknowns and deriving asatisfaetory differential equation.

Taylor's eontribution(38) introdueed the idea of the speetrum ofturbu1enee; he resolved the fluetuating ve10eity variation in a turbu-1ent fie1d at a fixed point ioto its harmonie eomponents, introdueing aspeetra1 funetion whieh gives the eontribution to the ve10eity fie1d humthe continuous range of frequeneies. He then showed how the speetrumcurve is nothing el se than the Fourier transform of the eorre1ation curve.From this moment the use of the speetra1 representation of a turbu1entve10city fie1d beeame eommon1y used. Once again, behind Tay10rls workon spectral analysis we find the influeoee of Wfuner; Taylor used (andmade explicit mention of the source) sorne theorems whieh had been provedby Wiener in his book on "The Fourier Integral" published in 1933 as adireet filiation of the work on generalized harmonic analysis.

These theoretical investigations found immediate ground for applie~tion in the wide range of new problems posed by wind-tunnel testing dur-ing the Thirties. With the growing importance, after World War 1, oiwind tunnel research as a tool for improving aircraft design, sorne puz-zling situations began to emerge from the huge quantity of empiric mate-rial. It was sooo found that experiments performed in geometrically ide~tical arrangements gave widely differeot results in different wind tun-

534

neIs. In 1923 the National Physical Laboratory began the circulatían oi

two airship models fer comparative tests in a large number oí the windtunnela oi the world. The results in the U.S. wind tunnels showed variations oí SO percent from a mean value. It was realized that the reasonfor these discrepancies was to be found in the degree oi turhulence pre-sent in the maín flow, which affected in a sensible way the measurements.It was found, for example, that the intesity oi turhulence in the maínflow was responsible fer the premature transitíon oí the boundary layeralong a wing section from laminar to turbulent; as a result the drag coefficient ter a given profile could vary by a factor oí tWQ.

Que had therefore interest in having wind tunnels with as low a turbulence factor as possible, in order to better approximate the conditi~in free air, and in finding a rational way of classifying wind tunnelsaccording to their turbulence factor, to make possible comparisons be-tween results obtained in different tunnels. A first step, suggested byPrandtl, was to classify wind tunnels in arder of magnitude of turbulencein terms of the curves of drag coefficient of a sphere. Qne defined asthe 'turbulence factor' for a given wind tunnel the ratio between theReynolds number at which the drag coefficient of a sphere assumed a givenvalue in free air, and the measured Reynolds number at which it assumedthe same value in the wind tunnel stream (due to the premature transitionto turbulent flow in the boundary layer, this happens at a Reynolds num-ber lower than the one calculated for free air). More refined measure-ments showed that things were more complexo the diameter of the sphereacted as an isolated, independent parameter. Qne had to take into acoountnot only the intensity, but also the scale of the turbulence in the flow.

The apparatus which allowed to obtain reliable direct measurementsoi the intensity of turbulence, the hot-wire anemometer, was developedand greatly improved in the early Thirties by the ~ottingen group underPrandtl. at the National Physical Laboratory in Great Britain, and in theU.S. at the National Bureaus of Standards by H. Dryden and his coworkers.Working in cooperation and with the financial assistance oi the NationalAdvisory Comittee for Aeronautics, this group improved in the early Thir-ties the hot-wire anemometer to the point that they were able to followfluctuations in the velocity to the upper frequency of about 5000 cycles

535

per secando Qne of the maio results oi the measurements performed withthis instrument in arder to study the characteristics oi wind-tunnelhntulence was to show that the turbulence sorne distance downstream froro

a regular array of grids ~ highly isotropic. The mean value of thefluctuatían oi the velocity component was independent oi the particulardirection studied.

Just at this moment, aud when it had become clear that the seale oi

the turbulence played a role as important as its intensity, Taylor's th~ory oi isotropic turbulence appeared. It carne at the right moment toperfectly fit a practical need. Correlations, which are the basie ingr~dients of Taylor's definitían oi seale, can easily be measured by ade-quately coupling two hot~wi.reanemometers. Taylor' s theoretical framer,,¡orkmade it possible to compare results, design new experiments, and makesense of existing data. Parts 11, 111 and IV of Taylor's paper give justthe detailed prooi that the numerical results obtained in wind tunne1s inthe previous years at Gottingen, Caltech, at the NPL, and by the Drydengroup, can a11 be explained by the theory exposed in Part l. In particu-lar, the law oi decay of turbulence behind grids was confirmed in a spec-tacular way. The NBS-NACA group immediately performed a series. of ac-curate measurements uti1izing Tay10r's theory as a guide(39). In Dryden's~.,ords

these measurements give information about the decay of turbu-1ence and change in sca1e, which has an interest from the pointof view of the theory of turbu1ence and a practica1 be~ring onthe design of wind tunnels to secure low turbu1ence(40).

It is significant that the main advertising campaign for what wassometimes called 'Taylor-Kármán's statistical theory' was conducted nei-ther by Tay10r, which was quite a shy person, nor by Kármán, which cer-tain1y was not a shy person, but by Hugh Dryden, in a long series oi tecbnica1 papers, general lectures and review artic1es(41).

We have commented on a wide range of factors which come together inexplaining the development and the reception of the new statistical theo-ry: the pure mathematician developing the appropriate too1s to deal withthe intricacies of chaotic motion; someone ab1e to translate these ideas

S36

into a simpler mathematical language, applying them to a specific prob-lem aud pointing to its practical value; a scientific community withtheoretically minded people receptive to new ideas aud willing to pur~new lines oE research; a group of experimentalists ready to appreciatetheoretical contributions which can shed light on their specific prob-lems; a growing concern with the specific problem. which is in turodictated by wider external factors; the existence of experimental facil-ities aud the development oi an appropriate technology; an institutionalsetting allowing these factors to merge aud synthetize. Al1 these thingsgrow aud slowly converge in the early Thirties. Taylor's theory oE 1935is Taylor's work; it marks a turning point in the history of the theoriesof turbulence.

3. KOLMOGOROV. HEISENBERG. VON .EIZSACKER. ONSAGER: A CASE OF SIMUL-TANEOUS DISCOVERY

3.1 Introduction

In the early Forties, the Russian mathematician A.N.KolmogoInv. theGerman physicists W. Heisenberg and C. ~on Weizsacker. and the Swedishchemical physicist L. Onsager proposed independently of each other, whatis basically the same theoretical model to deal with the problem of dev-eloped turbulence. By means of different formulations and techniquesthey were led to the same fundamental results. These contributions are

altogether referred to, in the literature that followed. as "Kolmogorov'stheory". It is worth to discuss them and bring forth their coromon chara~teristics, since they represent an important turning point in th~ complexitinerary of the theoretical researches on turbulence.

3.2 Kolmogorov

First to be published are three short papers by Kolmogorov in the_ (42)Proceedings of the Soviet Academy of SC1ences ,dealing with the sta-

tistics of small-scale structures in isotropie and homogeneous turbul~e.The opening sentence of the first paper gives a clear idea oi the kiod ofapproaeh: "lo coosideriog the turbulence it is natural to assume the com-

537

ponents of the velocity ... as random variables in the sense of the the-ory oi probabilities". lt i5 just the case that such ao approach couldactually look "natural" only to a probabilist like Kolmogorov. The aharpjudgment expressed by von Kármán in 1937 towards the attempts to useprobability in attacking the problem of turbulence (" ••.Gebelein attemp£ed to salve the problem by looking at ii from the high tower oi the gen-eral theory of probability; 1 believe that this tower i5 too high toallow ene to see the simple facts,,(43» gives enough indicatíon as tohow "natural" such a language appeared to the applied mathematiciandealing with the subject.

Apart froID the formalized language, Kolmogorov's work exhibits arather simple strueture that can be easily summarized. The basie physi-cal meehanism of the problem are ineorporated as three hypotheses:

a) turbulenee is homogeneous and isotropie at sroall seale;b) at small seale the statistieal properties depend nnly on the

kineroatie viseosity V and the dissipated energy E; ande) in the inertial range ("small seales, but not too sroall") the

statistieal properties depend only on the dissipated energy.

FroID these three hypotheses it is possible to obtain the relevantstatistieal quantities by the simple use oE dimensional analysis. FrOIDhypothesis b) one has that the only meaningful length at small sea le n;known as Kolmogorov's length, is given by

This length represents the wavelength at whieh viseous effects becomedominant. Also, the only meaningful veloeity at sroall scale v, known asKOlmogorov's velocity is given by

538

and represents the typical velocity associated \lith vartexes oí scalen.

For r«L, where L i5 the length defining large scales, from hypot~esis b) and dimensional analysis,

2<CuCo) - u(r» >

where B i5 a universal funetían. For che inertial range. n «r« L,

che funetían B can be obtained with the help oí hypothesis e). One findsthat

2 2/3<CuCo) - u(r» > a (Er)

Using the same type oí arguments, ane finds that the energy spectrum E(k)i5 given by

fer l/t« k« lIno It i5 interesting to remark that this last result,known in the literature as the "5/3 1aw" or "Kolmogorov's la",,",does notappear explicitly in these papera, though it £0110\15 as a rather immediate consequence.

The very way in which Kolmogorov's work i5 structured i8 typicallymathematico-deduetíve: one starts wíth a few hypotheses and proeeeds tothe formal deduetíon of eonelusíons (the ealeulatíon of eorrelatíon funetiona). lt is remarkable how the justífieation of the hypotheses byphenomenologieal eonsiderations i5 eonfined to a footnote, showing acompletely different attitude from that of an engineer. 00 the otherhand. it ia easy to realize that, besides the language of probabilitytheory. Kolmogorov do~s not rely 00 particularly original pheoomenol~alideas, nor does he introduce new techniques (the whole deductioo of hisresults is based esseotíally 00 the use of dimensional analysis, dailybread for eogineers and applied mathematiciaos). Take. for ínstance. the

S 39

description Di the energetic balance in turbulence:

From the energetical point of view it i5 natural to imaginethe process of turbulent mixing in the following way: thepulsations Di the first arder absorb the energy oC the mo-tion aud pass it ayer successively to pulsations Di higherorders. The energy of the finest pulsations i5 dispersedin the energy Di heat due to viscosity.

This description i8 essentially identical to the ane suggested abouttwenty years earlier by the British meteorologist L.F.Richardson in a50rt Di poem:

Big whorls have little whorls, that feed on their velocity,aud little whorls have lesser whorls. aud so on to viscosity(in the molecular sense)(44).

3.3 Heisenberg and van Weizsacker

In the last months of 1945, while being held as "war prisoners"together with other German scientists near Cambridge, Great Britain,Heisenberg and von Weizsacker developed in close collaboration a theoryof small-scale turbulence, However, their results were published in twosuccessive papers only in 1948(45), even though Kolmogorov's, unknown tothem in 1945, had been spread in Western scientific circ les thanks mainlyto the work of G.K. Batchelor(46). Although different. the two theorieslead to the same physical results. The papers of Heisenberg and vonWeizsacker present the same theory in different styles: Heisenberg'streatment is more analytical and formal, while von Weizsacker's approachis more phenomenological introducing explicit connections with astrophy~ical problems.

Let us take a quick look at their theory. The first equation ofReynold's hierarchy in k-space for homogeneous and isotropic turbulence is

where T(k) is a term coming from the nonlioear part of the Navier-5tokesequation. This term contains contributions due to the triple correlationfunctions of the velocity field harmonics. The problem is how to write

540

T(k) ís terms oE the energy spectrum so as to 'clase' the equatíon; aproblem analogous to the aue faced by Prandtl in 1925 in his attempt tofind ao expression for the Reynolds stresses tied to the propertíes oE

the mean motían aud solved by the introduction oE the "mixing length".In analogy with Prandtl's procedure, Heisenberg and von Weizsacker pra-pased that

r T(k' )dk'o

k-2V

t(k)Lk' 2E(k' )dk'

where vt(k) is a sort oE "turbulent viscosity". The analogy with Prandtlsuggests that vT(k) must have contributions from wave numbers larger thank. From this and dimensional analysis, it follows that

f[ E(k') ]1/3e ----- dk'k,3

k

Qne obtains, combining these results,

Jk r Jro[ , ]1/2 -JJk-dt

o E(k')dk'~£. 2L

V+C k E~~3) dk' o k,2E(k')dk'

for k »l/L. The solution of the equation in the inertial range is

E(k) : k-5/3

It may be noted that Heisenberg and von Weizsacker's theory containno relevant technical novelties: the basic ingredients are the idea ofturbulent viscosity, mutuated from Prandtl, and, once again. dimensionalanalysis.

3.4 Onsager

. (47)Onsager's contribution appears in a manuscript sent to C. L~n •working at the time at Caltech's Guggenheim Aeronautical Laboratory as

541

van Karman's assistant. dated june 1945, aud in a lecture given thatsame year, au abstraet oi which is published in the Physical Review(48).A later paper in 1949(49) is less interesting because in the meantirnehe had become acquainted with the work oi Kolmogorov. Heisenberg aud vonWeizsacker, aud his presentatían shows signs oE the influence of theirtheories. However. the 1945 abstraet in the Physical Review containsenough to understand the kind oE propasal, aud the deep link withKOlmogorov's ideas:

... it has not been pointed out up to now that the subdivisionoi energy must be a stepwise process •..For such a cascarle mechanism that part oi the energy densityassociated with large wave numbers should depend only on thetotal rate of dissipation E. Dimensional considerations thenrequire that the energy for a component of wave nu¡ber k beequal, apart from a universal factor, to E2/3 k-ll 3.

A rather amusing incident gives a fair idea of the non usual character of Onsager's contribution for the fluid dynamics experts of thattime. In addition of the manuscript sent to Lin, Onsager also sent asort of surnrnaryof his ideas to von Karman, at that time in Paris as scientific advisor for the U. S. Air Force. von Kármán co~mented to Lin:"1 received a letter and a kind of manuscript from a certain Mr. LarsOnsager. 1 find his letter somewhat "screwy" so 1 would be glad to haveyour opinion whether the paper is worth while reading. Perhaps you cownindicate to me in a few lines what the idea is, if any"(50) .

3.5 A New Look on Turbulence

This incident, by itself, only shows that at the time von Kármán wasbusy with completely different matters, and that he therefore payed noattention to Onsage~s manuscript; however, the very fact that the latterwas unknown to him gives an idea of the distance separating their respec-tive scientific environments. Moreover, it seems reasonable to take vonKármán's reaction to Onsager's proposal as typical of the kind Di atti-tude adopted by scientists engaged at that moment in research on turbu-lence toward these new theories.

542

What canoot but disconcert the applied mathematicians of the timeare nat so much the maín techniques aud ideas used in these works (as wesaid, they are basically ingredients, aud combinations of them, familiarlong since to experts in the field), as the way to put the problem andthe languages it i5 dealt with, typical of different scientific tradi-tions. It i5 true that the concept of isotropic turbulence, the use ofcorrelation functions to define the statistical properties. aud theFourier transferm for the spectral representatían. had be en introdu~ by

G. Taylor aud quickly accepted by researchers in the fieId. However,Taylor had taken care to show the immediate relevance of his approach onapplicative and technological grounds. On the contrary? a common characteristic of the theories we are discussing is that they were presentedwithout any trace of applicative interests; they deal with the problembecause oi its intrinsic interest oi a fundamental character. It maybe defined as a typical approach of theoretical physics.

This "simultaneous discovery" is of interest because it comes tol~ght when, on the scence of turbulent research, new disciplinary sectDrsappear. It coincides with the moment when sorne sectors of 'pure' re-search show a return of interest towards a line of investigation that hadbeen abandoned for a long time, to be destined to the attention of dif-ferent professional competences.

3.6 Different Backgrounds of the Same Theory

The itineraries, through which our four scientists reached the for-mulations of their theories, are not only independent, but also very diiferent from each other, as different as their respective scientific andinstitutional backgrounds.

Kolmogorov comes froro the great tradition oi Russian matheroatics;the presentation oi his papers clear1y shows his origin as a leader inprobability theory (his text where the foundations of the axiomatic ap-proach are laid down goes back to 1933(51». To gain a better understan-ding of his contribution to our smry, a detailed investigation oi thecharacterístícs acquired by the re1atíon between pure and applíed mathe-matics in the Soviet Union during the period between the two World Wars

would be highly needed.Heinsenberg's contribution reveals the specific

acquíred working under Somrnerfeld's direction in his

S 43

competence he had. (52)doctoral thes~s •

that dealt with problems oí stability aud transition to turbulence influid mechanics. VanWeizsacker, a "second generatían" theoretical ¡ily~

icist after the quantum revolution was led to the study oí turbulence by

the oature oí the astrophysical problem he was dealíog with in the ear-Iy Forties, namely the elaboratían oí a theory oí the origin oí the solarsystem based on the hypothesis oí a condensatían oí planets starting froma nebula(53). The description oi the evolution oí 5uch sn object i5 aproblem of the dynamics of a viscous fluid at high Reynolds numbers. Theresults obtained through his collaboration with Heisenberg led vonWeizsacker to modify his original model (54).

Finally. Onsager was a chemical physicist. His interest in thetheory of turbulence came from non-equilibrium thermodynamics and nonlinear phenomena.

544

REFERENCE5

l. Taylor, G.l., Statistical Theories oí Turbulence, Proceedings of theRoyal Society oí London, A, 151, 421-478, (1935).

2. Prandt!, L., Über die ausgebildete Turbulenz, Proceedings of theSecond International Congress of Applied Mechanics, Zurich 1926,pp. 62-75.

3. Prandtl. L., Uber Flüsigkeitsbewegung bei sehr kleiner Reibung,Verhandl. der 111 Intern. l~th.-Kongr .• Heidelberg, 1904. pp. 484-491

4. Letter froID Praudtl to von Kármán, June 15, 1921, in von Karmán'spapers, Box 23, falder 41 (Microform edition, deposited at the Natíonal Air and Space Museum, Smithsonian Institution, Washington, D.C.,oí the original in possession oí the Archives oí the California 10-stitute oí Technology. Pasadena. California). Hereafter referred toas VKP.

5. The proceedings of the conference have been published: T. von Kármán.T. Levi-Civita, eds .• Vortrage auf der Gebiete der Hydro- und aero-dynamik. Berlin 1924.

6. On the subject of the international scientific relationships in theperiod following Wor1d War 1 see B. Schroeder-Gudehus, Les scientif-iques et la paix - La comrnunaute' scientifique internationa1e au coursdes annees 20. Montreal 1978.

7. Two c1assic papers on the state of this subject in the ear1y Twentiesare: F. Noether, Das Turbu1enzproblem. Zeitschrift für angewandteMathematik und Mechanik. l (1921), pp. 125-138.W. Heisenberg. Uber Stabi1itat und Turbu1enz von Flüssigkeitstromen.Anna1en der Physik. 2i (1924), pp. 577-627.

8. O. Reynolds, On the Dynamica1 Theory of Incompressible Viscous F1uidsand the Determination of the Criterion. Philosophica1 Transactions oi

the Royal Society of London, A. 186 (1895). pp. 123-164.9. R. von Mises. Uber die Aufgaben und Zie1e der angewandten Hathematik.

Zeitschrift für angewandte Mathematik und Mechanik. l (1921). p. l.10. P. Forman. Weimar Culture, Causality and Quantum Theory. 1918-1927:

Adaptation by German Physicists and Mathematicians to a Rosti1e lntellectual Environment, Historical Studies in the Physica1 Sciences, 3(1971), pp. 1-115.

545

11. L. Prandtl, Uber die ausgebildete Turbulenz, Proceedings oi theSecond International Congress oi Applied Mechanics. Zürich 1926, pp.62-75.

12. L. Prandtl, Bericht tiber Untersuchungen zur ausgebildeten Turbulenz.Zeitschrift für angewandte Mathematik und Mechanik, S (1925), pp.136-139.

13. L. Praudtl, Uber die ausgebildete Turbulenz, cit.; quot. irom theEnglish translation by n.M.Miner, Turbulent Flow, published asTechnical Memorandum No. 435 oi the National Advisory Cornmittee forAeronautics (NACA). Oct. 1927.

14. J. Boussinesq, Essai sur la Théorie des eaux courantes, Mém. Sav. Etr.Acad. Se., París, t. XXIII (1877), pp. 1-680.

15. L. Praudtl. Uber die ausgebildete Turbulenz. cit.; quot. pp. 3-50fNACA translation. 1 apported some correction to error s in the originaltrans1ation.

16.- T. van Kármán, Mechanische Ahnlichkeit und Turbulenz, Nachrichten vander Gesellschaft der Wissenschaften zu C~.ttingen, Math.-Phys. Klasse(1930), pp. 58-76; Hechanische Ahnlichkeit und Turbulenz, Praceedingsof the Third Internacional Congress of Applied Mechanics, Stockho1m1930, pp. 85-93. An eng1ish translation of the first paper by J. Varuer,with the title Hechanical Similitude and Turbulence, was published asNACA Technica1 Memorandum No. 611, March 1931. A more accurate trans-latioo (by von Kármán himse1f ?) is found in VKP, Box 119, fo1der 20.Quotations are taken from this traoslation.

17. T. von Kárman, Mechanische Ahnlichkeit und Turbu1enz, cit.; quot.from VKP translation, p. 9.

18. ibid., pp. 19-20.

19. T. von K8rmán to J.M.Burgers, Dec. 12, 1929, VKP, Box 4, fo1der 22.20. Burgers' resu1ts were published in a series of papers in the Procee-

dings of the Section of Sciences of the Amsterdam K. Akademie vanWetenschappen, 1i (1929), pp. 414-425: 643-657; 818-833: 36 (1933),pp. 276-284; 390-399; 487-495.

21. G.I.Tay1or, The transport of vorticity and heat through fluids in tu~bulent motion, Proceedings of the Royal Society of London, A, 135,(1932), pp. 685-7.

546

22. G.I.Taylor, Eddy motian in the atmosphere, Philosophical Transactionsof the Royal Society of London, A, ~ (1915). pp. 1-26.

23. G.I.Taylor. Diffusion by Continuous Movements, Proceedings of theLondon Mathematical Society. ser.2, lQ (1921). pp. 196-212. Reprintedin S.K. Friedlander. L. Topper, eds., Turbulence: Classic Papers onStatistical Theory. New York 1961, pp. 1-16. Following referencesare made to this reprint.

24. G.I.Taylor, Diffusion by Continuous Movements, p. 4.25. ibid., p. 9.

26. ibid., p. 1lo

27. G.I.Taylor. Statistical Theory of Turbulence, Proceedings of the RoymSociety of London, A, 151 (1935), pp. 421-478. Reprinted in S.K.Friedlander. L. Topper. eds., Turbulence: Classic Papers on Statistka1Theory, New York 1961, pp. 18-75. Fo11owing references are made tothis reprint.

28. G.I.Tay1or, Statistica1 Theory of Turbu1ence, pp. 20-21.29. ibid., p. 27.

30. G.I. Taylor, Diffusion by Continuous Movements, p. 3.31. N. Wiener, I am a Mathematician, Cambridge, ~~ss. 1956, pp. 36-37 of

the 1973 paperback edition.32. N. Wiener, The Average Va1ue of a Functional, Proceedings of the London

Mathematica1 Society, ser. 2, 22 (1924), pp. 454- 467.33. Tay10r to Wiener, Apri1 8, 1924, N. Wiener papers, MIT Archives, Box 1,

fo1der 25.34. Tay10r to Wiener, January 5. 1932, N. Wiener papers, MIT Archives,

Box 1, fo1der 35.35. T. von Kárrnán, Sorne Aspects of the Turbu1ence Prob1ern, Proceedings of

the Fourth International Congress of Applied Mechanics, Cambridge 1934,pp. 54-9lo

36. Quoted in W.R.Sears. M.R.Sears. The Kármán Years at Caltech, AnnualReview of Fluid Mechanics, 11 (1979), pp. 1-10.

37. T. van Kármán. L. Howarth, On the Statistical Theory of Isotropic tur-bulence. Proceedings oí the Royal Society of London, A, 164 (1938).pp. 192-215.

S47

38. G.I.Taylor, The Spectrum oí Turbulence, Proceedings oi the RoyalSociety oi Londan, A, 164 (1938), pp. 476-490.

39. Probably the mast relevant publicatían originated by these researcheswas the joint paper by H.L.Dryden, G.B.Schubauer, W.C.Mock, Jr., U.K.Skramstad, Measurements oi Intensity and Seale oi Wind Tunnel Turbu-lence and Their Relation to the Crítical Reynolds Number oí Spheres,NACA Technical Repart n. 581, (1937).

40. H.L.Dryden, Turbulence and the Boundary Layer, Journal oi the Aero-nautical Sciences, ~ (1939), pp. 85-100, quot. p. 90.

41. Among other, more technical, papers by H.L. Dryden in the years im-mediately following 1935 are:The Theory oí Isotropic Turbulence, Journal of the Aeronautical Sci-ences, ~ (1937), pp. 273-280.Recent Developments of the Theory of Turbulence, Journal of AppliedMechanics, ~ (1937), pp. 105-108.Turhulence Investigations at the National Bureau of Standards, Proc~

edings of the Fifth International Congress of Applied Mechanics.Cambridge, Mass. (1938), pp. 362- 368.Turbulence and Diffusion, Industrial and Engineering Chemistry, 31(1939) pp. 416-425.

42. A.N.Kolmogorov, The Local Structure of Turhulence in IncompressibleViscous Fluids for Very Large Reynolds Numbers, Compt. Rend.Acad.Sc.URSS, 30, (1941), 301-305.A.N.Kolmogorov, On Degeneration of Isotropic Turhulence in an Incom-pressible Viscous Liquid, Compt. Rend. Acad. Sc. URSS, 31 (1941),538-540.A.N.Kolmogorov, Dissipation of Energy in Locally Isotropic Turbulence,Compt. Rend. Acad. Se. URSS, li, (1941), 16-18.

43. T. von Karmán, Turhulence, Aeronautical Reprints, -ª2., (1937). The RoyalAeronautical Society, London; Hans Gebelein, who was acquainted withKolmogorov's work on probability theory had attempted a probabilisticapproach. See H. Gebelein, Turbulenz; physikalische Statistik undHydrodynamik, Berlin, 1935.

44. The "poem" appears in L.F.Richardson, Weather Prediction by NumericalProcess. Cambridge. 1922, p.66.

548

45. C.F. von Weizsacker, Das Spektrum der Turbulenz bei grossen ReynoldschenZahlen, Zeit, f. Phys., ~, (1948), 614-627.

',.J. Heisenberg, Zur statistischen Theorie der Turbulenz. Zeit. f. Physik,

124, (1948), 628-657.

46. G.K.Batchelor, Double Velocity Carrelation Funetían in Turbulent ~otion,Nature, 158, (1946), 883-884.

Comrnunication to the Turbulence Symposium. VI International Congress ofApplied Mechanics, París, September 22-29, 1946.C.K. Batchelor, Kolmogoroff's Theory of Locally Isotropic Turbulence,Proc. CamboPhil. Soc., ~, (1947), 533-559.

47. L. Onsager to C.Lin, june 1945. T. von Karmán Papers, National Aír andSpace Museum, Smithsonian Institution, Washington, D.C. (Micro ficheccpy of the original deposited at the Archives, California Institute ofTechnology. Pasadena). Box lB, fo1der 22.

48. L. Onsager, The Distribution of Energy in Turbulenee, Phys. Rev., ~,(1945), 286. Abstraet of communication presented at the Meeting of theMetropolitan Section of the American Physical Society, Columbia Univer-sity. New York, Novemeber 9-10, 1945.

49. L. Onsager. Statistical Hydrodynamics, Suppl. Vol. VI. Ser. IV, NuovoCimento. (1949), 279-287.

50. T. von Karman to C. Lin, August 23. 1945. von Karman's Papers, quo~ Box18, folder 22.

51. A.N.Kolmogorov. Grundbegriffe der Wehrscheinlieh Keitsrechnung. Berlin,1933.

52. W.Heisenberg, Uber Stabilitat und Turbulenz van Flüssigkeitsromen, Am.J. Phys., l..'!-, (1924), 577-627.

53. C.F. van Weizsacker, Uber die Entehnung des Planetensystem. Zeit. f.Astroph., ~, (1944), 319-355.

54. C.F. von Wcizsacker. The Evolution oí Galaxies and Stars. Astroph.Journ., 114, (1951), 165-186.