The theories of turbulence

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Title:
The theories of turbulence
Series Title:
NACA TM
Physical Description:
163 p. : ill. ; 28 cm.
Language:
English
Creator:
Agostini, L ( Léon )
Bass, J ( Jean )
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
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Subjects / Keywords:
Turbulence   ( lcsh )
Aerodynamics -- Research   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: The report includes a discussion of the kinematics of statistical mediums, particularly those which are isotropic. A mathematical study is made of the applications of Navier's equations to turbulent motion. Physical theories involving similarity are dealt with. Review is made of much of the work in turbulence. The theoretical discussions are illustrated by some correlation and spectrum curves based on measurements taken in the wind tunnel at the laboratory of the mechanics of the atmosphere at Marseille.
Bibliography:
Includes bibliographic references (p. 150-153).
Statement of Responsibility:
by L. Agostini and J. Bass.
General Note:
"Translation of "Les théories de la turbulence." From Publications Scientifiques et Techniques du Ministère de L'Air, No. 237, 1950."
General Note:
"Report date October 1955."

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University of Florida
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oclc - 127355482
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PREFACE


The theory of turbulence has made so much progress during these last
years that it is of interest to state exactly the obtained results, the
hypotheses on which these results are based, and the directions in which
new research is being conducted.

Messrs. Bass and Agostini have undertaken this work and have summar-
ized our actual knowledge of turbulence in a series of conferences which
took place at the Sorbonne, within the Paris Institute of Mechanics. The
reader will find in the following pages the text of these conferences,
perfected and revised by Mr. Bass.

In view of the magnitude of the subject and its simultaneously phy-
sical and theoretical aspects it had seemed advisable to entrust this
work to a team formed by a mathematician and a physicist. Mr. Bass had
taken the responsibility for the theoretical part, Mr. Agostini for the
physical part.

Initially, the report was intended to contain three theoretical and
two physical chapters followed by a chapter on the technique of the
measurements and on the appratus used: anemometers and statistical-
measurement apparatus.

The unexpected death of Mr. Agostini in August 1949 unfortunately
made modifications of the original project necessary. This premature
death deprived us of a highly valuable physicist, at the peak of intel-
lectual maturity, whose current work on these problems showed particu-
larly remarkable promise.

Mr. Agostini had only just begun drawing up the two last chapters;
Mr. Bass had to take up the editorial work and to complete it according
to Mr. Agostini's notes. The chapter on the technique of the measurements
has been omitted and will form the object of a later publication.

In order to make up for the gap in the experimental part, the text
was supplemented by some curves furnished by Mr. Favre which will allow
utilization for numerical calculations and will enable the reader to
judge the agreement between theory and tests.




A. Fortier,
Professor at the Sorbonne






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TABLE OF CONTENTS


INTRODUCTION . . . 1


CHAPTER I

GENERAL ASPECTS OF TURBULENCE. THE STATISTICAL METHOD

1. Definition of turbulence . . 6
2. Average values statistics . . 8
5. Random variables and the laws of probability . 10
4. The concept of random point velocity
field turbulent diffusion . . 15
5. Equations of development of the laws of probability 15
6. Random velocity. Hydrodynamic equations . 25
7. Systems of molecules . . ... 29


CHAPTER II

CORRELATIONS AND SPECTRAL FUNCTIONS

8. Introduction correlations in
space homogeneity, isotropy. . 33. 5
9. Properties of the functions f, g, a, b, c.
Incompressibility. . . .. 357
10. Spectral decomposition of the velocity . 45
11. Spectral tensor and correlation tensor . .... 45
12. Spectral tensor of isotropic, incompressible turbulence 50
15. Energy interpretation of the spectral function F(k) 55
14. Relations between spectral function F(k)
and correlation functions f(r) and g(r). . 58
15. Lateral and longitudinal spectrum . .... 62


CHAPTER III

DYNAMICS OF TURBULENCE

16. Introduction . . .. 64
17. Fundamental equation of turbulent dynamics . 65
18. Case of isotropic turbulence . .... 68
19. Local form of the fundamental equation . 75
20. Solution of the fundamental equation, when the
triple correlations are disregarded . 76
21. Solutions involving a similarity hypothesis . 80






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22. Transformation of the fundamental
equation in spectral terms . ... 86
23. First theory of Heisenberg . .... 89
24. First theory of Heisenberg (continued).
Space-time correlations . . 95
25. Second theory of Heisenberg . . 99


CHAPTER IV

THEORY OF LOCAL ISOTROPY AND STATISTICAL EQUILIBRIUM

26. Introduction . . ... 106
27. Definition of local homogeneity and local isotropy 108
28. Similarity hypotheses. Statistical equilibrium 111
29. Case of high Reynolds numbers . . 114
30. Validity of the similarity laws .. . 118
31. Interpretation of the laws of statistical
equilibrium in spectral terms Weizsacker's
and Heisenberg's theories . . 121


CHAPTER V

DECAY OF THE TURBULENCE BEHIND A GRID

52. History '. . . .. 124
35. Initial and final phase of turbulence . 126
54. Concepts regarding the structure of the
final phase of turbulence . . 152
35. The concept of "dynamic statistical equilibrium" 155
56. Synthesis of the results relating to the
structure of the spectrum of turbulence . 141


INDEX OF PRINCIPAL NOTATIONS, FUNDAMENTAL FORMULAS,
AND DIMENSIONAL EQUATIONS . . 145

APPENDIX . . ... . 147
Some experimental results . . 147

REFERENCES. . .. . 150







































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TECHNICAL rMEM1ORA1IDUM 1377



THE THEORIES OF TURBULENCE*

By L. Agostini and J. Bass


INTRODUCTION


The theory of turbulence reached its full growth at the end of the
19th century as a result of the work by Boussinesq (1877) and Reynolds
(1893). It then underwent a long period of stagnation which ended under
the impulse given to it by the development of wind tunnels caused by the
needs of aviation. Numerous researchers, mathematicians, aerodynamicists,
and meteorologists attempted to put Reynolds' elementary statistical
theory in a more precise form, to define the fundamental quantities, to
set up the equations which connect them, and to explain the peculiarities
of turbulent flows. This second period of the science of turbulence ended
before the war and had its apotheosis at the 1938 Congress of Applied
Mechanics.

During the war, some isolated scientists von Weizsscker and
Heisenberg in Germany, Kolmogoroff in Russia, Onsager in the U.S.A. -
started a program of research which forms the third period. By a system
of assumptions which make it possible to approach the structure of tur-
bulence in well-defined limiting conditions quantitatively, they obtained
a certain number of laws on the correlations and the spectrum. These
results, once they became known, caused a spate of new researches, the
most outstanding of which are those by the team Batchelor-Townsend at
Cambridge.

The analysis of these works became the subject of a series of lec-
tures at the Sorbonne in February-March 1949, which subsequently were
edited and completed. The mathematical theory of turbulence had already
been published in 1946 (ref. 3) but practically ignored all publications
later than 1940. Since the late reports have improved the mathematical
language of turbulence, it was deemed advisable to start with a detailed
account of the mathematical methods applicable to turbulence, inspired
at first by the work of the French school, above all for the basic prin-
ciples, then the work of foreigners, above all for the theory of the
spectrum.

"Les Theories de la Turbulence." Publications Scientifiques et
Techniques du Ministere de L'Air, No. 237, 1950.





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The first chapter deals with the precise and elementary definition
of turbulence (sections 1 and 2) and describes the tools of mathematical
statistics on which the ultimate developments are based. Starting from
paragraph 5, chapter I, the reader should be familiar with the defini-
tions of the calculus of probabilities, the theorem of total probabili-
ties, and the theorem of compound probabilities. This chapter is entirely
theoretical, and its aim is to review the methods suggested by the theory
of random functions and which seem likely to be applied to turbulence.
Only the use of Navier's equations has, so far, produced positive results,
and chapters III, IV, and V are largely devoted to it. However, it should
be pointed out that there are theories less familiar to hydrodynamicists
which have been proved in other branches of physics (kinetic theory
of gases, quantum mechanics). The purely random method is described in
paragraph 5, and its adaptation to molecular systems (according to Born
and Green) in paragraph 7. The statistical method has the advantage of
furnishing a remarkable demonstration of the general equations of hydro-
dynamics (paragraph 6) and of providing an exact classification of the
statistical parameters of turbulence (paragraph 4), which is interesting
to keep in mind when studying the foreign reports, too exclusively devoted
to spatial correlations. In any case, the reader who wants to read
chapters IV and V can pass up most of chapter I, except perhaps para-
graphs 1 and 2, without major trouble.

Chapter II deals with the kinematics of statistical mediums and,
particularly, isotropic mediums. It seemed practical to include at the
same time the velocity correlations, the theory of which has been given
in almost final form by KArmAn, at the end of the second period, and of
the spectrum, the theory of which, due to Taylor's initiative, has only
been achieved very recently. Only paragraphs 10 and 11 refer to statis-
tical functions, and their detailed knowledge is not indispensable for
reading the rest of the chapter. The results and the formulas of
chapter IT are constantly applied in the subsequent chapters, but it is
not necessary to know the proofs which are, in most cases, a simple matter
of calculation. The most important of these formulas are, moreover, com-
piled in a special section following chapter V.

Chapter III is a mathematical study of the application of Navier's
equations to turbulent motion. Paragraphs 17, 18, 19, and 22 are funda-
mental. Their main purpose is to recall KArman's results of 1958 with
some improvements and some supplements of more recent date. The para-
graphs 20 and 21 review a certain number of physically reasonable solu-
tions of the Karman-Howarth fundamental equation. Some of these assume
particular importance in chapter V but, first, it seems advisable to give
an impartial view of the whole and to proceed progressively into the
domain of the concrete. The paragraphs 25 and 24 deal with an equation
by Heisenberg which involves time correlations and from which probably
not all possible results have been extracted. It is not indispensable
to have knowledge of this in order to continue. Paragraph 25 contains a







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mathematical account of Heisenberg's numerical theory of the spectrum,
which is taken up again in chapter V from a more physical point of view
and whose examination, contrary to paragraphs 25 and 24, proved useful
before attacking chapter V.

Chapters IV and V deal with new physical theories involving simi-
larity hypotheses and producing numerical laws. Chapter IV reviews the
works of Kolmogoroff and Weizsicker, chapter V those of Heisenberg,
Batchelor, and Townsend on the decay of turbulence created by grids.

Finally, in an appendix, the theoretical discussions of chapters III,
IV, and V are illustrated by some correlation curves and spectrum curves
measured directly in the wind tunnel by A. Favre, in the laboratory of
the mechanics of the atmosphere at Marseille, or derived from experi-
mental curves by elementary transformations.

An exhaustive study of modern theories of turbulence calls for some
knowledge of the calculation of tensors, probabilities, and statistical
analysis, besides the classical conceptions of differential and integral
calculus.

As regards the tensors, knowledge of the definitions and fundamental
operations with rectangular cartesian coordinates is sufficient. There
are a number of articles on this subject, but they generally lean toward
the tensor analysis with curvilinear coordinates for which there is no
need. (It should be noted that the tensor analysis plays, in contrast,
an important part in the theory of the boundary layer around an airfoil.)
Incidentally, there is available a little book recently published, by
Lichnerowicz, entitled: Elements of tensor calculus (collection
Armand Colin). On mathematical statistics, the book by Darmois, pub-
lished by Doin (1928), can be consulted. For the elementary theory of
random functions, consult the first part of Bass' report (ref. 3) and
the appendix to d'Angot's "complements of mathematics" (editions of the
Revue d'Optique), edited by Blanc-Lapierre. More detailed information
on statistical functions can be found in Levy's book: "Stochastic
Processes and Brownian Motion" (1948).

The present report deals only with general theories which are valid
whatever the physical or geometric causes of turbulence may be, as is
shown in chapter IV. Only in chapter V the assumptions are limited and
the study involves a problem of decay of turbulence that is compatible
with the turbulence in wind tunnels. These theories are probably appli-
cable to mediums of extremely diverse scales, from the microturbulence
to the terrestrial atmosphere (Dedebant and Wehrle), to stellar atmos-
pheres and to interstellar matter (Weizsacker). However, among the
hypotheses there is always that of the incompressibility, which precludes
the application to sonic or supersonic flows. This is not an indispensi-
ble hypothesis, but it simplifies the calculations considerably, and the






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consequences are easy to check. So the incompressible turbulent motions
are practically the only ones studied up to now.

Among the other hypotheses worthy of discussion, those referring
to processes of energy transport (paragraph 24 and chapter V) assume that
all turbulent energy comes from the motion of the whole, dissociated and
broken up by the obstacles of periodic structure. But nature furnishes
examples of different turbulences such as the so-called thermal turbu-
lence, for example, where the source of energy is not the fan of a wind
tunnel but the solar radiation suitably transformed into kinetic energy.
Kolmogoroff's theory of local isotropy applies probably to thermal turbu-
lence, but the forms of energy transport in the spectrum must be differ-
ent from those encountered in wind tunnels. The same applies to the
astronomic turbulent mediums alluded to previously. So the foregoing
remarks limit the scope of the recent theories, in a certain measure.

Omitted entirely was the problem of turbulent boundary layer which,
experimentally, depends on the same technique, but has been approached by
different mathematical methods. Furthermore, it is a complicated problem
where the turbulence is neither homogeneous nor isotropic.

In what measure are the results, suitably demarcated, defined? The
statistical character of the turbulent velocity seems a clear and well-
established notion and consequently the mechanics of turbulence will be
a statistical mechanics. As far as the kinematics (chapter II) are con-
cerned, we are therefore on solid ground. The dynamics of turbulence
(chapter III) itself is likewise well established, by means of the
hypothesis of the validity of the Navier equations. This hypothesis,
generally adopted because it is convenient and, one might say, necessary,
has however at times raised considerable doubts. However, while experi-
mental verifications do not contradict it, there is yet no occasion to
reject it.

But, what should be expected from experimental verifications? First
of all, it is found that the theories are still rather imperfect. In
fact, the theories are usually limited, acceptable in the limiting con-
ditions which are difficult to attain actually (very high Reynolds num-
bers, for example). No rigorous verification should be expected since
the true conditions are too far removed from the theoretical conditions.

On the other hand, the accuracy of measurement is low. The original
reason for it lies in the very nature of the turbulent phenomenon, and
its irregular and badly defined character. Furthermore, the anemometers
are coarse instruments, their operation not sufficiently known in the
presence of turbulence, and their interference with the fluid sometimes
a little mysterious. All this helps to lower the experimental precision.
To measure the "length of dissipation," for example, two ways are open:
one is to measure the correlations of the velocity at two infinitely close
b






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points which has no practical sense since two anemometers cannot be
brought together indefinitely; the other is to use one anemometer for
measuring the mean square value of the derivative of velocity. But the
operation of taking a derivative of a function as complicated as the
velocity is by itself inaccurate and introduces serious scattering of
the points of the derivative curve. So in both cases, the accuracy of
the results is extremely limited.

In consequence, the experimental verifications can be applied only
to the orders of magnitude. To illustrate: to check whether a parameter,
according to theory, is constant, one constructs the representative curve
of the parameter and, if this curve has a sufficiently extended maximum,
one estimates that the experiment closely confirms the theory, conceding
that, as the experimental conditions more nearly approach those stipu-
lated by theory, the maximum flattens out more and more, and one does
not appear too severe in the examination of the scatter of the test
points.

On this assumption, the verifications of the experimental laws of
turbulence, due in particular to Townsend, are encouraging. Therefore,
it is well to retain the hypotheses of chapters III, IV, and V, although
some of them obviously have their limitations, and we ought not hope for
results greater than actually can be given. Take an example drawn from
the theory of the spectrum, for instance. At the beginning of decay of
turbulence in a tunnel and at high Reynolds numbers, a certain spectral
function F(k) is of the form Ck4 for the small values of wave num-
ber k. It then passes through a maximum, then becomes proportional
to k and finally approaches zero as k m0, maybe as k-7. But
the regions of the axis of k in which the fragments of the laws to be
enumerated remain acceptable are badly defined and connected by zones
of which the structure is not known. So the future task of the theorists
will be to combine these partial results into a single acceptable law at
least to the extent that it does not become fundamentally incompatible
with the nature of turbulence, since k = 0 up to k = '. Only then
will there be a true theory of turbulence.






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CHAPTER I


GENERAL ASPECTS OF TURBULENCE

THE STATISTICAL METHOD


1. Definition of turbulence:

Our purpose is to discuss the operation which consists of measuring,
at a given point, the velocity of a flow which, to avoid every difficulty,
is assumed to be steady and uniform at the usual macroscopic scale. The
measuring instrument is an "anemometer" with approximately determined
dimensions and time constant, and of sufficiently ideal nature so as not
to disturb the flow by its presence. If this anemometer is large enough,
it measures the "velocity of the main flow" of the fluid. If it is very
small, it can be imagined that it operates in discontinuous manner, never
undergoing the influence of more than one molecule at a time. An anemom-
eter sensitive to the individual action of molecules is, of course,
unattainable, but the idea of such an instrument is convenient for repre-
senting the extreme limit of fineness of kinematic measurements in a fluid
Between these two extremes, the indications of the anemometer depend upon
the structure of the fluid. It may happen, by exception, that, when its
dimensions are progressively reduced, the velocity which it indicates
remains unchanged up to the moment where the individual influence of the
molecules starts to make itself felt and where the indications lose all
statistical significance. The flow is then said to be laminar.

But, in general, the matters are otherwise. We start with a first
anemometer which, through its dimensions, fixes a certain scale of meas-
urement. This anemometer measures the mean speed of the molecules in a
certain volume V. Then it is replaced successively by smaller anemometers
in such a way that volume V decreases progressively. It happens that,
from a certain value Vl of V, the numerical indication supplied by the
anemometer changes. If V is decreased continuously, the new indication
remains stable up to a certain value V2, then changes again and so on.
The intervals (V1, V2), (V2, V3), characterize the various scales
of turbulence, and the motion of the fluid is said to be turbulent. More
exactly, they are the conditions necessary for a fluid to be turbulent,
and which must be defined and perfected to make them sufficient and
practical.

The last value Vn of the series Vl, V2, is that from which
onward, the notion of average loses its significance, the number of mole-
cules contained in the volume V, not being large enough any longer for
statistical purposes. The series Vl, V2, .. can be discrete or






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continuous. If it is discrete, it still does not imply that the critical
values V1, V2, are mathematically defined. In the vicinity of V1,
a rapid variation of the anemometer indications occurs, which subsequently
become stabilized in the region (V1, V2), and so forth. If this stabili-
zation is not very clear, the turbulent scales are said to succeed one
another in continuous fashion.

The previous discussion ends with the notion of the turbulent fluid.
But the definition of the laminar motion given above is a little too
restrictive and the distinction between laminar and turbulent still not
precise enough, as proved by the following example:

Consider the motion of air produced by stationary waves in a sound
tube. The motion of the whole reduces, at rest, to large scale. But,
at each point there exists a speed other than zero, a periodic time func-
tion which, at a given instant, varies periodically from one point to
another. This motion, lying between the system at rest and the molecular
agitation, has not a turbulent character.

Turbulence, as shown, implies first the notion of scale. But it
should be added that, at a given scale, each component of the velocity
at a point is a function of time presenting a character of periodicity
without fundamental period. This is not a periodic function but a sum
of harmonics, the frequencies of which are not multiples of an identical
fundamental frequency. This irregularity of the turbulent agitation is
essential and distinguishes it from sound agitation, or preturbulent
vertical motions, like the cellular vortices of Benard. The mathematical
symbol for the turbulent velocity is not the ordinary Fourier series, but
Fourier's integral. It will be discussed later.

This concept of irregular agitation at a point as function of the
time is not itself sufficient. It makes it possible to differentiate the
turbulent agitation from the periodic sound agitation (musical sound),
but not from the noise, which is an agitation without definite period.
What distinguishes the noise from turbulence is the fact that it is prop-
agated by waves that exist on surfaces of equal phase, and consequently
have a regular spatial distribution, notwithstanding the irregularity in
the time of the local velocity.1

1A descriptive and purely cinematic distinction is involved here.
Its cause compressibilityy) is not discussed.

The difference between turbulence and sound agitation should become
plain from the following example: In a turbulent wind tunnel, we select
at a point the longitudinal component of the velocity with a hot wire and
send the electric current of the hot Wire to a loud speaker on the outside
of the tunnel. The atmosphere becomes the source of an irregular agi-
tation, which propagates by waves and is not turbulence, although, at
a point, the internal turbulent motion and the external sound motion
have some important cinematic elements in common.





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For a given scale, each component of the turbulent velocity is an
irregularly periodic function of both space and time.

It seems that turbulence is well defined by these kinematic condi-
tions and hence is distinguished from all other more organized fluid
motions.

2. Average values statistics:

Figure 1 represents the record of a component u of the turbulent
velocity as function of the time. (Record of the velocity of turbulent
agitation in a 20-cm by 50-cm tunnel: airspeed, 20 m/sec; intensity of
turbulence, 5-10 3; time of recording is 0.05 second.)

The most natural method of measuring the mean velocity on the graph
consists in forming the ordinary integral



u m u(t) dt (2-1)



extended over the total duration T of recording. This method is, in
general, not very satisfactory because the operation lacks precision
when the curve u(t) is complicated.

A more precise method consists in dividing the graph by parallels
to the axis of t, suitably close together in the ordinates ul, u2,
u3, ., in measuring the number ni of points where the line of the
ordinate 1(ui + ui+l) meets the curve u(t) and then in computing the
quantity


u = ui (2-2)



n = ni is the total number of points met by all parallels.

For this calculation, a profitable first stage consists in first
constructing the graph giving the corresponding statistical frequency

fi = i for each velocity ui. Crossing the limit obviously makes it
n
possible to plot a curve of frequency f = f(u) (fig. 2) such that the
proportion of the values of the velocity comprised between u and u + du






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CO
is equal to f(u)du, the integral f(u)du, which replaces n i

being equal to unity. The final expression of U is then



= uf(u)du (2-5)



It replaces (2-2) and should be compared with (2-1).

The practical operations enabling the replacement of uM by '
correspond to well-known mathematical operations. The mode of com-
puting U is that of a Lebesgue integral, and um is an integral of the
classical type of Riemann. If um is computable, both methods yield
the same result. But it may happen that Riemann's integral does not
exist because the function u(t) is too complicated mathematically.
However, in general, Lebesgue's integral exists (if u(t) is measurable,
and naturally bounded). This mathematical case corresponds to the
practical case where the curve u(t) is too complicated for an accurate
continuation of the integration. The function f(u), continuous and
differentiable in the current cases, is the medium which, determined once
for all, replaces the calculation of Lebesgue's integral by that of an
ordinary integral by means of the plotting of curve f(u).

From the function f(u), other averages can be computed. For
example, the amount of differences of the speed with respect to its
mean value can be figured by computing the mean value of (u Ui)2
Rather than defining this average by Riemann's integral



S[u(t) 2dt (2-4)



it is simpler and more precise to use the formula



(u )2 (u i) f(u)du (2-5)


which, once the curve f(u) is plotted, calls only for operations of a
simple character.





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Thus, it is seen how much the construction of the curve f(u)
simplifies the numerical calculations of turbulence. Various experi-
mental techniques make for direct attainment of this curve without passing
through the numerical analysis of a velocity record. For measuring the

averages such as (u i) 2, it is often advisable also to use specialized
equipment without first plotting the curve f(u).

3. Random variables and the laws of probability:

The theoretical significance of the function f(u) is analyzed.
It groups the statistical data contained in the initial curve u(t), with
this exception that the chronological order in which the velocities actu-
ally follow one another does no longer appear. This limitation is quite
natural, though, and it will be seen later that this order reappears, in
a certain measure, by the introduction of space and time correlations.

Obviously, only statistical data can supply stable information on
turbulence. When the same record of the velocity is begun again several
times while taking every reasonable precaution so that the conditions
are identical, it obviously results in curves u(t) which absolutely
are not superposable. The function u(t) has not, therefore, the char-
acter of permanence that is suitable for representing the laws of a
physical phenomenon. But this character is relevant to the function f(u)

or to averages such as U, (u U)2 whose values are characteristic
numbers of the investigated flow, and which are derived by simple mathe-
matical operations from the function f(u). Hence, we direct our atten-
tion to this function which can be regarded as representing the first
law of turbulence.

To say that the velocity is characterized by a curve of statistical
frequencies is to say, by comparing the frequencies with probabilities
and f(u) with a density of probability, that this velocity is a statis-
tical quantity; f(u)du is the probability that the chosen component of
the velocity is contained between u and u + du.

A priori, such a law of probability could be dependent on the time.
That would correspond to a turbulent flow for which the laws varied with
respect to time. There is no contradiction to the initial assumptions
of permanence here. It is a question of scale. In order for the experi-
mental operation by which f(u) is defined to have any meaning, it is
necessary that two conditions be realized simultaneously:

(1) The number of oscillations in the time interval T involved
must be great enough to furnish satisfactory statistics.

(2) The laws of turbulence in this time interval T must be practic-
ally permanent.







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If it is not so, it might be difficult to reconcile the statistical
theory with the experiment. Fortunately, those "ergodic" conditions are
practically always realized in the usual cases and are therefore taken
for granted in the following:

The velocity has three components ul, u2, and u3 which are treated
as three random variables, components of a random vector.

It should be noted here that the velocity is perhaps not sufficient
for characterizing the turbulence. It might appear useful to introduce
other quantities, such as the pressure, which should be treated as a
statistical quantity. But, owing to the equations of motion, this then
will be a function defined by the velocity and its derivatives. For the
present, It is assumed that the turbulent motion is sufficiently well
defined by its velocity so that the problem narrows down to the laws of
probability applied to the velocity.

The simplest of these laws, that which immediately generalizes the
experimental function f(u), is the law of probability of the system of
three components ul, u2, u3 of the velocity. This law may vary as
function of the time t (problem of spontaneous decay of turbulence) and
of the space (variation of turbulence in terms of the distance from the
walls). It is therefore a function of t and the ordinates xl, x2, x5
of the point of measurement. Its density is denoted by


f(ul, u2, u3; x1, X2, x3, t)


or, abbreviated, f(u; x, t).

The quantity f dul du2 du3 or, abbreviated, f du, represents the
probability that, at the point xl, x2, x3 (or x) and at the instant t,
the three velocity components are comprised between ul and ul + dul,
u: and u2 + du2, u3 and u3 + du3.

But a single law of probability defined in terms of four parameters
x1, x2, x and t is not adequate for characterizing turbulence. It
is necessary to introduce the more profound concept of random function
and to consider the turbulent velocity as a random function of space and
time. This is the random velocity field.

This point is now to be defined. An isolated random quantity is
defined by its law of probability. But, to define a system of coexistant
random quantities requires more than just their laws of individual prob-
abilities. The stochastic dependencies or correlations between these






NACA TM 1577


quantities must also be known. Taking a family of probability laws does
not give the right to speak of the system of corresponding random quan-
tities without completing the data by those of the correlations. Sup-
posing that the family in question depends on a continuous parameter t.
We know then an isolated random quantity U(t) for each value of t.
This immediately "suggests grouping the U(t) corresponding to the var-
ious values of t in a well-defined system of statistical quantities.
This calls for the introduction of the correlations between the U(td),
U(t2), .. corresponding to an arbitrary system E of the values tl,
tZ,, of the parameter. To proceed thus, means to define a random
function. Naturally, this also holds for laws of probabilities dependent
on several parameters. Thus, when a turbulent medium is represented as
a velocity field, the system of velocities at each point and at each
instant precisely constitutes a system of coexistant statistical quantities,
of which the physical interactions characterizing the structure of turbu-
lence have the correlations for mathematical description.

Among the systems E, the simplest are the denumerable systems and
even the finite systems and, among the latter, the simplest one which is
not trivial is that of two elements, that is, of two points of space and
time.

The concept of random function thus suggests the comparison of the
velocity vectors at two different points of their field of definition,
that is, for two different positions x and x' (x represents the
point of the coordinates xl, x2, x,) and for two instants t and t'.
It concerns a statistical comparison which makes it possible to define
the law of probability of the two systems of the velocity components at
points x, x' and instants t and t'. This law has a very clear
physical meaning and is easy to define experimentally or, at the least,
to construct the surface of (statistical) frequencies corresponding to
one velocity component at point x and a second component at point x',
the measurements being spaced at a chosen time interval T.

In practice, the question is frequently handled from a less general
point of view. One is not concerned with the laws of probability them-
selves as such but only with their most simple moments, those of the
second order which are associated, as will be shown, with certain
"physical" aspects of turbulence and, possibly, with certain moments of
the third order that play a part in modern theories. The moments of the
second order constitute the correlation tensor of the law of probability
of the velocity field at two points and at two instants. They are meas-
ured direct, without resorting to frequency curves or frequency surfaces
or to velocity recordings. A detailed study follows later.






HACA 'T3 1577


4. The concept cf random point velocity field turbulent diffusion:

The density of probability f(u; x, t) of the velocity field contains
the velocity of the whole fluid, but not its density. With p(xl, x2, x3; t)
or, abbreviated, O(x, t) denoting the quotient of the density at point x
by the fluid mass, p is a normalized function, as a density of probabil-
ity, which means that


J p dx = 1 (4-1)



where dx represents the element of volume dxL dx2 dx3 and the integral
is extended to the volume V occupied by the fluid.

In all modern studies on turbulence, the fluid is naturally assumed
incompressible, so that p is a constant, equal to 1/V in the volume V,
and zero at the outside. p could be simply replaced by a constant; but
the more general conclusions to be arrived at ultimately are more complete
if this simplification is not made. On the other hand, it is interesting
to foresee, at a certain stage of the theory, the day when it will be
possible to study the turbulent motions in conditions where the compressi-
bility is no longer negligible. For these reasons, p is treated here
as a function of x (and even of t, if necessary).

The product R(x, u; t) = pf is now formed. It obviously is
normalized with respect to the system of the six variables x, u


R dx du = 1 (4-2)


R presents thus the characters of a density of probability with respect
to these six variables. The quantities xl, x2, x3 are regarded as the
coordinates of a moving point, ul, u2, u3 as the velocity components of
this point. These are six random quantities of which the law of probability
at instant t is known. Thus a random point can be associated with the
turbulent fluid in correspondence with a given scale.

This point is now to be discussed as was the velocity field in the
preceding paragraph.

The position and the velocity of this point are random functions of
the time. The theory of random functions suggests the study of the law
of probability of the system of positions and velocities of this point
for an arbitrary combination of instants. The initial analysis was on






NACA TM 1377


the law of probability at one instant. The generalization from one to
two instants seems to us adequate for forming a physical theory of turbu-
lence and, in particular, for considering the statistical organization
in time of the velocity, and what may be called the "interactions between
turbulent particles."


The values of the positions and of the velocity must
associated to two instants t and t'. This association
relationship defined by the density of probability of the
of 12 statistical variables:


X1, x2,1 x

ul, u2, u


xL,' X2',' x'

u1', U2', UJ'


therefore be
is a stochastic
following system


position
at instant
velocity


position
Sat instant
velocity


or, abbreviated,


and it is assumed that it defines the turbulent motion. By
can it be measured?

According to the theorem of compound probabilities, G
is the product of two factors:


what stages


dx dx' du du'


(1) The probability that at the instants t and t' the statistical
image point of the fluid might have positions contained within the inter-
vals (x, x + dx) and (x', x' + dx').

(2) The probability (conditional) that, these positions being fixed,
the velocities are contained in the intervals (u, u + du) and (u', u' + du').

This last probability is designated by H(u, u'; x, x', t, t')du du'.
According to the theorem of compound probabilities, the probability (1)
is the product of the probability p(x, t)dx that the random point is
found, at instant t, in the interval x, x + dx, through the (conditional)
probability that, the position at instant t being chosen, its position
at instant t' is found in the interval x', x' + dx'.


G(x, x', u, u'; t, t')







NACA TM 1577


This last probability is designated by p(x'; x, t, t')dx'. We can
write


G(x, x', u, u', t, t') = H(u, u'; x, x', t, t')p(x'; x, t, t')p(x; t)

(4-5)

What is the significance of the three factors of which *G is the
product ?

We already know 0, which represents except for a numerical factor
the density of the fluid.

The function H is the law of a random velocity field at two points
and at two instants. It is natural to identify it with the law of the
field already discussed at the end of paragraph 5. It is seen that it
gives no complete picture of turbulence. It does not explain the func-
tion p.

The function p represents the turbulent diffusion at a chosen
scale. It is the relative density at point x' and at instant t' of
fluid elements which have passed neighboring point x at instant t.
It should be pointed out that the diffused portion of the fluid is essen-
tially compressible since the density p(x'; x, t, t') decreases in
proportion as the point x' is removed from the initial point x where
it is maximum. The spread can be materialized and p can be measured
by introducing with the necessary precautions at point x a dye that
spreads in the fluid. But it is a rather ticklish matter to separate
the effects of the various turbulent scales, especially of the molecular
diffusion. It is accomplished by adapting the particles of the "dye" to
the chosen scale. This way Kampe' de Feriet rendered the turbulent dif-
fusion of the tunnel flow visible in his experiments at the Institute of
Fluid Mechanics, at Lille, by injecting soap bubbles at a point in place
of dyed particles. These soap bubbles, because of their size, were
sensitive to the turbulent fluctuations and were used successfully for
measuring a density of turbulent diffusion.

5. Equations of development of the laws of probability:

The analysis of the turbulent velocity field made it possible to
represent a turbulent fluid by two random functions X(t) and U(t)
playing the part of the position and of the velocity for a random material
point. It was shown how this concept of random point gives a very com-
plete picture of the turbulence. But this picture is qualitative and
must be made more quantitative.





NACA DTM 1577


At the beginning, no distinction is made between the velocity
vector X(t) and the position vector U(t). The six components of these
two vectors are considered as those of a unique vector X(t) in a six-
dimensional space, by putting ul = x4, u2= x., u, = x6, and designating
the density of probability of this vector by R(x; t), this notation being
the abbreviation of R(x1, x2, x5, x4, xC, x6; t), that is, of
R(xl, x2, X3, ul, u2, u; t).

We already had applied (4-5), the theorem of compound probabilities,
to the law of probability G of the position and the velocity at two
instants t, t':


G(x, x', u, u'; t, t') = [p(x; t)p(x'; x, t, t')JH(u, u'; x, x', t, t')


The question involved essentially the separation of the velocity,
which figures in the factor H, from the position. But there is another
way of applying this theorem. It consists in separating the two
instants t and t' by writing


G(x, x', u, u'; t, t') = R(x', u'; t')K(x, u; x', u', t, t')

(5-1)

hence it results, according to the theorm of the total probabilities,
that


R(x, u; t) = fR(x', u'; t')K(x, u; x', u', t, t')dx' du'

(5-2)

K is a conditional density of probability, that of the "probability of
passage" of the state of the random point at instant t' to its state at
instant t > t'.

Abbreviated, we get


R(x; t) = [R(x'; t')K(x; x', t, t')x' (5-3)


To exploit this equation, recourse is had to a method patterned after
the concept of J. Moyal (ref. 35). This idea was to consider (5-5) as a
linear integral transformation for passing from the density of proba-
bility R at instant t' to the same density at instant t. The density







NACA TM 1377 17


of probability of the passage K is the kernel of the transformation.
With Kttt indicating the linear operation which has as kernel the
function K(x; x', t, t'), one may write symbolically:


R(x; t) = Kttr R(x'; t')


(5-4)


This transformation has special properties rather difficult to
define. Suffice it to state that it reduces to the identical transforma-
tion for t = t'. It is assumed that it has an inverse Ktt~-l = Kt't
without, however, prejudicing the relations between Ktti and Ktit.


The infinitesimal transformation applied to KttI is
To this end, R is assumed differentiable with respect to
calculated


6R(x, t)
at


lim
T ---> 0


= lim
T 0



= lim
T->0


now examined.
t and 6R/ot


R(x; t + T) R(x; t)


Kt+Tt K't R(x', t')
T



Kt+Tt' Kt't 1 R(x, t)


If, as assumed, the function R is differentiable with respect
to t, the limit of the second member exists, and it is a function of x,t
independent of t'. It can be cj'mputed by giving parameter t' (inde-
pendent of variable T) any value not exceeding t, such as t' = t, for
example. Hence


=R i
- = limn
&t T-0


Kt+Tt 1
T


(5-5)


The operator


L = limn
T -- 0


Kt+Tt 1
T





NACA TM 1577


defines the infinitesimal transformation of K, and R confirms the
fundamental functional equation


=R =LR (5-6)
at


which, theoretically, enables R(x; t) to be computed at instant t
when R(x; t') is known at an initial instant t'.

The foregoing calculation cannot be explained in a simple manner
from formula (5-5) because the limit of the function K(x, x', t, t')
does not exist when t' -- t; this is a symbolical "function of Dirac"
which expresses the identity Ktt = 1 in the functional formalism. It
is preferable to pass, as Moyal did, from the densities of probability
to characteristic functions. Moyal's calculation follows:

With the function K is associated the characteristic function of
the increment X(t) X(t'), the value of X(t') once fixed, or by
definition


0(a; x', t, t') = eia(x-x') K(x; x', t, t')dx (5-7)


One assumes likewise:


p(a, x, t) = eiaxP(x; t)dx (5-8)


the characteristic function of X(t).

According to the theorem of total probabilities, one has:


cp(a, x, t) = eiax'0(a; x', t, t')R(x'; t')dx' (5-9)


This relation replaces (5-5).

The two members of (5-9) are differentiated with respect to t


= 1m eix' O(a; x', t + t, t') (a.; x, t, t') (x' t')dx'






NACA TM 1577


Whereas the conditions of differentiability under the sign

(Permutation of signs lim and of f) cannot be verified on (5-3), they
can be here, in all current cases. Since the limit must not depend on t',
let t' = t, so that


S= /eix'R(x', t) lim g(a; x', t + T, t) 1 dx' (5-10)
6t T->O


because (a.; x, t, t) = 1, for t' = t.

The limit that figures under the f sign is a certain function

9(a; x', t'), whence follows the basic equation

aE= fea'BR(x', t)9(a; x', t)dx'


equivalent to (5-6). From that, two probability densities can be recovered
by taking the Fourier transforms of the two members. Lastly, BR/6t is
expressed in form of an integral transformation of R, equivalent to the
transformation L, which, after introducing a kernel function L(x, x', t),
gives

bR(x, t) = fR(x', t)L(x, x', t)dx' (5-11)



The second member of (5-11) is represented in integral form but, in
many cases, it can be expressed in form of a differential operator of
finite or infinite order.

Examples.- Supposing the probability of passage obeys the Laplace-
Gauss law and, to avoid any confusion of the notations, t0 now denotes
the previous instant; the differences of X(to) and X(t) are denoted
by So = S(to) and S = S(t), the correlation coefficient between X(to)
and X(t) by r = r (t, t). It is known that the law of probability
related to X(t), when the value x0 of X (to) is given, has for density

t1 x r02 (5-12)
K(x; XO to, t 1 e 2
r2xtS~l r2





20 NACA TM 1577


Using Moyal's method, it is shown that the probability density R(x, t
which assumes K as kernel of the probability of passage satisfies the
equation of partial derivatives


OR = (ri + (xR) -
ot S bdx


r'S
6x2


where r' represents the value of -a- r(tl, t) for tl = t, and S'
otl
denotes the derivative d- S(t).
dt

The equation (5-15) is, moreover, demonstrated very simply by a
direct method. By definition of K


R(x, t) = R( x0 )e 2 te (1-r2)
23S/1-2 --r


-r 2
Sdx0


this equation being, in particular, satisfied when


R(x, t) =


2
1 s2s
eS


When the two members of (5-14) are differentiated with respect to
t and x, equation (5-15) is verified. Reciprocally, the integral of
(5-15) which is reduced to R (, t0) for t = t0, can be put in the
form (5-14).

The construction of the random functions compatible with these laws
of probability is an easy matter.

To illustrate:

Let h(s) be a variable random function of the parameter s, obeying
the reduced Laplace-Gauss law h = 0, h2 = 1 and so that the increments


(5-15)


(5-14)






IACA TM 1577


h(sl) h(a2) and h(s3) h(5 s), corresponding to two separate intervals
fl, S2) and (S3, s4) are independent. The random function


t
X(t) = f (t s)dh(s)
Jo


is now considered.

It is easily shown2 that X(t) obeys a Laplace-Gauss law having

for typical difference S(t) = t-2 and that, if T = t to
V77


1 +
r (t, to 2 t =1
T+ /2


3 72
8 t2


(5-16)


Without going into details, it is simply recalled that the demonstra-
tion utilizes as intermediary the characteristic function of X(t), of
which the logarithm, owing to the properties of independence of the
dh(s), is expressed by an elementary integral.

From these formulas, it follows that



r = 0, S
S 2 t


and that R(x, t) verifies the first-order partial differential equation



OR + 1 (xR) = 0 (5-17)
6t 2 t ox


2See, for instance, Bass, reference 4, or "The random functions and
their mechanical interpretation." Revue Scientifique, No. 3240, 1945.


(5-15)





22 NACA TM 1377


X(t) is a differentiable random function. Its stochastic deriva-
tive is deduced from the expression (5-15) by operations of classical


5tx
form and written as dh(s). It is easily proved that x
0 2 t


is the


related mean of this derivative, if the value x of X is fixed. The
equation (5-17) has many solutions which are densities of probability.
It does not determine the function K.

Next consider the elementary random function


X(t) = /dh(s)
0


(5-18)


stochastic derivative of the function which has been studied as the first
illustrative example. It is shown that X(t) obeys a Laplace-Gauss law

having S(t) = rt for typical difference and that, if to0 t:


too,


r' -
2t


= 1 O +- ,
2 to


(5-19)


Therefore,
equation


R(x, t) verifies the second-order partial derivative.


oR l &2R
dt 2 ax2


(5-20)


This is the equation of heat. The solution which reduces to a
given function Ro(x), for t = t0, is given by the classical formula


R(x, t) = f 1
-n T To


- 1 (x-xt)2
e 2(t-t 0)


R(xo)dxo


which can also be shown by using the Fourier transform of R, that is,
its charactistic function.






NACA TM 1577


Thus equation (5-20) defines here the form of'the probability of
passage K(x; X0, t, to) contrary to what occurred in equation (5-17).

This example presents an unusual peculiarity. It is easily verified
that the function


_1 X 2
K(x; XO, t, t0) = 1 e 2t-tO) (x ) (5-21)



satisfies the functional equation



K(x2; xO, t2, to) 1 fK(x2; x1, t2, tl)K xl; x0, t1, t0)dx (5-22)




called the Chapman-Kolmogoroff equation, which characterizes the
Markoff processes (or more generally, the "pseudo-markovian" processes),
a functional generalization of simple Markoff chains. This equation
can be written in operational notation as



KtrtO = Kt2t1 Kt t0 (5-25)


It expresses that the operations K form a group. The operator
Kt+Tt' Kt't-- 1 is then written simply as Kt.T 1. It is independ-
T T
ent of t'. In the general case it is its limit L only when T-10
which must be independent of t, but it itself is not.

Returning to equations (5-12) and (5-15) it now is assumed that
R is not dependent on t; S is then a constant and S' = 0. If
r' / 0, equation (5-15) is written as



(xR) + S42R = 0 (5-24)
Ox 6x2






24 NACA TM 1577


The only solution of this differential equation which defines a
law of probability is the density of the Laplace-Gauss law

x2

R(x) = 1 e 2s2 (5-25)
S|&


This example contains, as a special case, stationary random functions
for which the function r(t, t0) depends solely on the difference
t to; r' is then a constant, independent of t.

The final example deals with a vectorial random function of a type
to be utilized later. Consider simultaneously the random function


t
X(t) = (t s)dri(s) (5-26)
0


and its derivative



X'(t) = dh(s) (5-27)





which can play the part of the speed.

It is easy to form the 'functions R and K for the vector X
having X and X' as components. To find the partial differential
equation verified by the function R(x, x', t) it is not necessary to
first form the kernel K. It is simpler to begin with the expressions
of X and X', which, passing through the intermediary of the charac-
teristic function of X, X', gives




bR oR 1 2R
R+ xR (5-28)
at Ox 2 dx'2







NACA TM 1577


The operator L is thus a differential operator, of the second
order with respect to x':


L -x' + 1 62
6x 2 6x'2


(5-29)


Re-turning now to the old notations and adding the letter u to
the velocity vector, we get


L 1 62
L = -u +
ox 2 du2


(5-50)


considering
which is the
of operators
consequently


X'(t) as the velocity of point X(t). The operator L,
subject of this example, appertains to the particular class
which make it possible to define statistical kinematics and
hydirodynamics, as will be proved.


6. Random velocity. Hydrodynamic equations refss. 3, 4):

The problem involves the separation of what is position and what
is velocity in the vectorial random function with six dimensions X(t).
In classical mechanics the velocity U(t) is the derivative of the
position X(t). In the present case, the assumption is made that the
velocity is the stochastic derivative of the mean square of the position.
By theory of random functions, it follows that, if the random function
X(t) is completely or at least known locally, the function U(t) can
be defined by operations comparable to those used in classical mechanics
to deduce the velocity from the position.


The preceding paragraph contained an example
of two random functions X(t) and X'(t) = U(t)
of this kind, proving in a general way that there
satisfies the necessary relation:


(equations (26) and (27))
linked by a relation
is an operator that


/ARdu = 0


(6-1)





NACA TM 1577


and is such that the basic equation (5-6) (with original notations)
takes the form


.R+ R ukBR- = AR (6-2)
kt k 5 xk


If A is a differential operator, it can be expressed by an expan-
sion in series (infinite or finite) in terms of the symbolical powers
of the partial derivatives of which the coefficients are functions
6uk
of x and u. The vectorial operator having for components occurs
C)k
only in the first member, by its scalar product with the velocity.

The product G = RK can then also be decomposed in G = ppH (com-
pare formulas (4-5) and (5-1)). The function H defines the correlations
in the velocity field and function p is the mathematical representation
of turbulent diffusion.

Equation (6-2) must therefore play the fundamental part in the theory
of turbulence. No attempt is made here at particularization; it simply
is shown that it contains the general equations of hydrodynamics.

First, the two members of (6-2) are integrated with respect to uk,
with due regard to (6-1). We introduce the density; of the fluid


p(x, t) = R(x, u; t)du (6-3)


(or more accurately, a quantity p normalized and numerically propor-
tional to the density) and the relative mean of the velocity for a
given position, the components of which are


U = u1iR(x, u; t)du (6-4)


We end with an equation of continuity


k pu)- =0 (6-5)
t k






NACA TM 1577


which simply expresses the fact that the velocity is the (stochastic)
derivative of the position. Thus it is seen that the velocity of the
whole of the turbulent motion, defined qualitatively in paragraph 4, has
the first of the qualitative properties of the hydrodynamic velocity in
general. At the chosen turbulent scale (which is arbitrary) the mass
is conserved.

Now it will be seen that the velocity also satisfies the equations
of motion. Multiplying equation (6-2) by the velocity component ui
and integrating with respect to u, gives for the first member



-p6
6t k ui"k


Introducing the speed of fluctuation


U'i = Ui Ui


and considering the equation of continuity, we put


Tik = -pu iu'k


(6-6)


The first member becomes


dui __
at k k


1 Tik
Pk axk


the second member is written


uiAR du = pyi


7. being a certain function of x which is deduced from the operator A
and the probability density R. Hence the equations of motion





NACA TM 1577


ui + u k i + ik (6-7)
>11- u-t-
at k k"xk 1 k xk


The yi play the part of the components of the density of the field
of external forces (gravity, for example). The Tik are the components
of the stress tensor (with fixed turbulent scale)). They are, except for
the factor -p, the components of the correlation tensors of the velocity
components at a point. They can be measured directly by statistical
methods, and the equations of hydrrodynamics for a given turbulent scale
can thus be verified.

It should be noted that in this case the equations of motion of the
ensemble (average) are involved. For the present, nothing about the
behavior of the rate of fluctuation has been assumed. Later on the usual
assumption will be made that the velocity satisfies the hydrodynamic
equations and, more precisely, the Nravier equations. It should be remem-
bered that, based upon this hypothesis, Reynolds was able to establish
equations similar to equations (6-7) for the mean turbulent motion. It
is apparent that, without it being necessary to repeat Reynolds' calcu-
lations, the statistical theory in question here is entirely different
from that of Reynolds. It is prc'Lbatly more complete, but it still has
not been pushed far enough to be verified bty 'experiment, due to a lack
of suital'le hypothesis.

As simple example of the fundamental equation (6-2), the case is
chosen in which the function U(t) is itself differentiable (in mean
squares, refs. 15, 16). In this case, the statistical image point of
the fluid has an instantaneous acceleration. If rk(x, u, t) represents
the relevant mean of the acceleration, that is, of the velocity derivative,
when the position and the velocity are fixed, it proves that the density
of probability of X(t) and U(t) satisfies the equation


+R uk- kR+) = 0 (6-8)
dt k=l oxk k=i uk


The operator A = > --(kR) is therefore linear, differential
k=l auk
and of the first order. The field of the external forces to which the
fluid is subjected has, necessarily, for components, the quantities


5This expression of stresses was originally given by Reynolds (On
the dynamical of incompressible viscous fluids and the determination of
the criterion. Phil. trans. Roy. Soc. CLXXXVI, part I, 125, 1895),
proceeding from the Navier equations. However, the exact meaning of the
Reynolds stresses is different from that of Tik.






NACA TM 1577


i fklf k. .Ru = i du = Ci (6-9)


averages formed of the acceleration components when the position is fixed,
provided only that uijiR tends toward zero when the velocity increases
to infinity. If, in particular, the ri are not dependent on u, ri is
identical with yi, and R satisfies the simple equation


3R + dR + R (6-10)
6t ff 7ka
k= x k k=l uk


Once the 7k are given, this equation defines R(x, u, t) from
R(x, u, tO But the form of the linear transformation from R(x, u, t0)
to R(x, u, t), that is, the probability of transition, depends upon the
form of the original probability R'x, u, t.O ,, contrary to what happens,
say for the equation of heat (type (5-19)); this illustration does
not appear to rest on hypotheses sufficiently inspired by reality to
serve as basis of a turbulence theory. First of all, the starting point
must be modified, as will be done in the following paragraph:


7. Systems of molecules refss. 15 and 14):

The statistical quantities to which the analysis of velocity records
leads, represent only certain scales of turbulence, those which correspond
to the ensemble of "vortices" whose dimensions are superior to a limit
approximately fixed by the employed anemometer. Can the theory be changed
so that all the possible scales can be represented simultaneously? It
seems that it suffices for this purpose to start from the finest scale,
that is, the molecular scale. The gas is therefore considered as a system
of N molecules, and, to explain the method with as much simplicity as
possible, the assumption, which is not verified for air, is made, that
the molecules are identical, monatomic, comparable to material points
subjected to central interactions. V(r) denotes the potential of the
force of interaction of two molecules separated by the distance r. If
r is great (with respect to the diameter of the molecules, which will
not be introduced explicitly), V(r) is negligible. For the small values
of r, V(r) expresses the repulsion of the molecules, generalized form
of shocks. No other information about V(r) is needed beforehand, at
least in a general theory.

The motion of this system of molecules is controlled by the equations
of dynamics. But the extreme complication of the trajectories of the






NACA TM 1577


molecules prompted the replacement of rational mechanics by statistical
mechanics. We prefer to introduce random mechanics where each molecule
is a random point. By means of an ergodic hypothesis the random motion
of a molecule can be considered as being statistically equivalent to the
motion of the ensemble of the fluid, in quasi-steady conditions, the
ensemble of successive states of the visualized molecule replacing the
ensemble of the simultaneous states of all the molecules. The molecule
would therefore be the concrete image of the abstract random point serving
up to now for representing the fluid. The molecular scale is a true ulti-
mate scale of turbulence, separated, however, from the actual turbulent
scales by a poorly defined but finite interval. While for experimental
turbulence the concept of a random image point is a mathematical abstrac-
tion, it is a natural idea and a starting point for the molecules.

This idea is now explored but by a method slightly different from
that discussed in the preceding paragraphs. The probability density
fN(xl, ., x1, U, ., u ; t) of the positions and the velocities
of N molecules simultaneously, rather than singly, is introduced. The
gas appears then as a random point with three N dimensions, in a space
of configuration, and no longer as a random point of ordinary and physi-
cal space. For the time being, the notations xl, x2, ., xN shall
have a vectorial character and represent the system of the three coordi-
nates of the molecules4 numbered 1, 2, ., IN.

The potential of interaction of the molecules of rank i and j
is indicated by Vij, and the external force to which the molecule of
rank i is subjected, by 7i. If m is the mass of a molecule, the
equations of motion of the molecules read


dxi u mdui 7i I Vij (7-1)
dt dt 7 3x,


the summation applying to all values of j from 1 to N, when Vii is
assumed to be zero. Naturally Vij = Vji-

When these equations are written, the hypothesis is made that the
velocity of each molecule is differentiable. It follows, that the random

In the preceding paragraphs the notation f(x, t) represented
already the probability density of a vector x, of components xl, x2, x2 .
What is used only temporarily is the meaning of the subscripts, particular
to this paragraph. A change in notation could be avoided only by compli-
cations in writing which would be more harmful than useful.






NACA TM 1377


point of 5N dimensions having for coordinates the ensemble of the coordi-
nates of the molecules is represented by a random vector function of the
configuration space of 5N dimensions doubly differentiable. This prop-
erty which, as stated before, is not necessarily true for the three-
dimensional random point used previously, is the reason, according to the
theory of statistical functions, why fN verifies the partial derivative
equation

ffN / N oVj))fN
afN r af N 1 yi N
t +_ Ui + u- i (7-2)
it i1 xi m 1i=1 J=1xj i


which expresses that fN is an integral of the equations of motion. It
has the same form as Liouville's equation of statistical mechanics. But
it should be remembered that its original meaning is a little different.
This equation replaces and defines, in the space of 3N dimensions, the
equation (6-2), and must now be exploited.

The turbulence involves "particles" or "eddies" formed, at a given
scale, by groups of s molecules, s being a very large number, but
at the same time very small with respect to N. Let us investigate what
functional equation is satisfied by the probability density f. relative
to the molecules of rank 1, 2, ., fs is defined by



s -J dxs+l dus+1 dx d uN (-53)


The fundamental equation (7-2) must be integrated with respect to
Xs+1, us+1, *, xN, uN. The operation is obvious, except for the
avi
terms in ---i; Vij being function solely of ui and x3, the integra-
ox.
tion gives a zero result if i and j are both superior to s.

1 oVij sf,
If i and j are both inferior to s, then .


If j s, i > s, the integration with respect to ui shows that
the result still is zero.

If, finally, i S s, j > s, we first can integrate with respect
to Xk, uk, for k i.





NACA TM 1577


1 Vij afs+1
The result is where fs+i represents the probability
m dx. du.
J i
density relative to s + 1 molecules of rank 1, 2, s, j. The
last integration


1 f V f..f 1 a OV
J /^ dx du = -- i .s5l dx. du.
mj xj x ui m Ou Xj S+ j J


cannot be extended farther, because both x- and fs+1 are functions

of xj.

Now it will be noted that in the sum


i / l dut
fij dx du
m j=s+l ui s+1 J J


the terms are identical because the molecule j plays an anonymous
part.

The final result is

6f S 6f 1 -fs N s s
t-- + ui 2 i- 7i- is+1 s +1 dus+1 d+1
dt i=1 dxi m i=1 6ui m i=1 ui oxs+1

(7-4)


This equation seems capable of serving as basis for a theory of
turbulence. The scale there appears explicitly for the numbers I and
s
- and the problem (not taken up here) consists in formulating a reasonable
N
hypothesis which enables fs+1 to be expressed with the aid of f., so
that (7-4) becomes a functional equation in f .







NACA TM 1577


For s = 1, (7-4) is, in a certain measure, comparable to (6-2)
and reads



+f f1 1 oifl 1 /6V12
+ u-- + -7 ---- --- --2 dx, du2
3t Ox1 m oul m dul ox2


Equation (7-5) differs from (6-2) by the presence of the terms in
f.. The presence of the second member relates (7-5) to (6-2), but its
form makes it different from it. Equation (7-5) reduces to a functional
equation in fl only if f2 is tied to fl by a suitable assumption.
Such an assumption has been made by J. Yvon who demonstrated with its
aid Boltzmann's fundamental equation on which the kinetic theory of gases
is based. More recently it was taken up again Lty Born and Green, who
also studied the case of s = 2, and applied it to the case of liquids on
the basis of an assumption by Kirkwood binding f to f 2

It seems that the case of large values of s has never been studied.

Equation (7->) like (6-2) and despite the not necessarily linear
character of the second member with respect to fg, involves equations
of hydrodynamics which are easy to write, if the method indicated in
paragraph 6 is applied. Since these equations have so far not been used
in concrete applications to turbulence, only these summary indications
are given here. The application of equation (7-4) to turbulence raises
difficulties which are far from being solved, or even stated but it is
important to know that this equation exists.


CHAPTER II


CORRELATIONS AND SPECTRAL FUNCTIONS


8. Introduction correlations in space homogeneity, isotropy:

In this chapter we deal no longer with the general laws of proba-
bility of turbulence. We limit ourselves to the study of the correla-
tions of the law of the field H(u,u'; x,x',t,t'). Following are some
preliminary remarks on the subject:

Experiments furnish the correlations in form of temporal averages.
Chapter I shows how it was possible to change their interpretation for
converting them into stochastic averages. The assumption is made that





NACA TM 1577


this operation is always possible. But for starting, there is no neces-
sity for knowing whether it has been effected, since onl, properties of
symmetry, of tensional character, are involved here. On the other hand,
arriving at the dynamics of turbulence we shall see that the use of tem-
poral averages leads to serious difficulties, which disappear when they
are transformed into stochastic averages, and our calculations will deal
only with stochastic averages. Temporal averages are resorted to only
when the calculations have reached the laws which must be compared with
physical reality.

The components of the speed fluctuations5 at point xj,x2,x3 are
designated by ul, u2, and u3. They are statistical quantities whose
mean value is zero. Time plays no part at present and the parameter t
in the formulas is disregarded. The correlations of the velocity between
the two points x and x' = x + 5 are defined by a tensor having for
components


R (x,x') = u (x)u (x') (8-1)


The scalar is introduced also:


R = R aR = jux)ucx') (8-2)
a


The turbulence is said to be homogeneous (within a certain field of
space) when the Rp are not separately dependent on the two points x,x',
but only on their relative position, that is, on the vector 5 = x' x.
Then Rap(t) replaces R0,1(x,x').

The turbulence is said to be isotropic at point x if it is homo-
geneous in the vicinity of x and the tensor R o is invariant to any
rotation of the axes and any reflection. The form which isotropy imposes
on the components of this tensor is discussed later.

These two definitions concern only the second-rank tensor RP.
They are "second-rank properties" of the statistical vector U%. They
must be extended to certain third-rank tensors. Homogeneity and isotropy
can be given a more complete and also more restrictive definition by
extension to all possible moments of velocity components taken at any
number of points. It is then more convenient to define the homogeneity

5They were called u'1,u'2,u'5 in chapter I. But it is advantageous
to use thereafter the notation u' for other purposes.






NACA TM 1577


as the invariance to translations of the law of probability of the
ensemble of velocity vectors having for origin a certain number of
arbitrarily distributed points, and the isotropy, once homogeneity is
achieved, as invariance to rotations and reflections.

Without restrictive hypotheses the tensor R.B has nine distinct
components, functions of the seven variables xl,x2,x3,x'l,x'2,x'3,t.
If the turbulence is homogeneous, the nine functions remain, but of four
variables only 1',2't ,t. The isotropy is now written.

To the tensor Rap(w) we associate the scalar bilinear form


Z R a3X Y-


where X and Y are two arbitrary vectors. This form is the mean
value of the product of two scalar products I u,(x)X, and
Su (x' )Yc. If it is invariant to rotations, it is an algebraic com-
bimation, separately linear with respect to X, and Ya,, of invariants of
vectors XaC,Ya,,m with respect to rotations. These invariants are



SX Y V X Y r2s2
x- Ya 5- E y r2 = tZ 2


There are therefore two scalars A(r), B(r), functions of r, so that



S Rm XaYp = A Y7 XYa. + B ( Xa) (Zip3y) (8-5)


The identification shows that, if 6ap represents the classical
symbol of Kronecker, zero if m3 a, equal to unity when m = a:


Rp = A(8 + B aSp


(8-4)





NACA TM 1377


The tensor R ., in the isotropy hypothesis, depends therefore solely
on two distinct functions of two variables r and t. The notations are
changed as usual and one puts with Karman and Howarth (reference 50):



R LO2 g + g6 j (8-5)


f and g being functions of r, t becoming unity for r = 0, and
uO2 a simple function of t.

If r = 0, RaO is reduced to


Ra(0) = u02g(0) 6 (8-6)


It is seen that



u12 = u22 = u 2 2 u = u = ulu2 = 0 (8-7)


Also, owing to the isotropy, the tensor R., is symmetrical.

Lastly, it should be noted that the quadratic form E_ R aX Xa is
the first member of the equation of an ellipsoid of revolution. This
shows that, if one takes, no matter how, a symmetrical table of numbers
R n, they are not, in general, components of a correlation tensor. These
numbers must verify the inequalities which state that they are coefficients
of the first member of the equation of an ellipsoid. In other words, the
roots of the equation of the third degree RHO S -j5C = 0 (equation in
S classical) must be positive.

A similar method permits the reducing of the components of the tensor


TaY = ua(x)uO(x)uY(x + 0)


(8-8)







NACA TM 1)7?


which will be needed later. This tensor is symmetrical with respect to
a and j, and the trilinear form





is invariant to rotations. By applying the classical notations of Karman
and Howarth, it is found, after a few calculations, that


(8-9)


TuO5 a6v + a c '2a
10[2,67 + 2A 373ra + E) 3'a~


a, b, and c being three functions of r and t.


9. Properties of the functions f, g, a, b, c. Incompressibilityb:

Other forms of symmetr; less particular than isotropy can be vis-
ualized, as for instance, axial symmetry, or invariance to rotations about
a given axis, instead of about a point. The case of isotropy is that in
which the RF, depend on the smallest number of separate functions.

The functions f and g are correlation coefficients. For example,
f can be defined by taking two velocity components along the axis of xl
(direction of velocity of the ensemble) at two points situated on a
parallel to this axis (fig. 5)


f(r) =


ul(X1>,x2,x)ul(xl + r,x2,x)

"o2


(9-1)


g is defined by taking two components still parallel to the axis of
the xl, but at two points located on a line perpendicular to this axis.


0The authors
But the letter k
recent fashion to
with this meaning


(reference 30) use q, i, k where we use a, b, c.
is also used in a just as classical although more
designate the spectral frequency and will be used
in the present report.





NACA TM 1577


For example



g(r)= ul(x1x2,x )ul(xlx2 + r,x \ (9-2)
2
uo


Experience indicates that near r = 0, the functions f and g are
continuous and twice differentiable and allow a tangent to the "horizontal"
origin (fig. 4).

Theory confirms experience, which unfortunately lacks precision when
the distance r becomes small. Much greater accuracy is obtained by
correlation measurements with difference in time (experiments by Favre,
Report to the VIIth Intern. Cong. of Appl. Mech., London, 1948), because
only one anemometer is used and the timing can be reduced as much as
desired.

Hence, one may write developments of the form



f(r) = 1 + "(0) + .

(9-3)

g(r) = 1 + (0) + .
2


f"(0) and g"(0) are negative quantities, possibly time functions,
having the dimensions of the inverse of the square of a length. The
length


2 (9-4)
g"(0)


is called length of dissipation.

The triple-correlation functions a,b,c have interpretations simi-
lar to those of f and g. For example:


c(r) = l2 u12(xx2xul x + r,x2,x5) (9-5)






NACA TM 1577


It follows from formula (8-9) that, if T., has a well defineri
limit when --40, this limit can only be zero. Hence, a(0),b(0),c(0)
are zero. In a general way, Ta4y is an odd function of Stagjpy-
Consequently, considered as function of r, c(r) is an odd function.
Its development has no term in r2. Lastly, it is shown that it has no
term in r, either. The coefficient of this term is the mean of


S2 1 Aim u1i(xl + rx2,xj) ul (x,x2,x5)
Ul 1 = lim
dx1 5 axl 5 r->0 r


or


rlin 1 3xl + lx2,x3) u~l$ "l 2x
3 r-40 r + 125) -


But on account of the homogeneity extended to the averages of the

third order, ul 5(x + r,x2,xj) is equal to u15 (x1,x2,x5). Hence the
limit is zero. Lastly, c(r) is an infinitesimal of the third order
with respect to r. The same holds for b(r) and a(r).

If the flow is incompressible, the various functions that charac-
terize the correlations of isotropic turbulence are not independent. From

the relation -ua- 0 it is, in fact, immediately deduced that




Y_ -0 (9-6)



This equation is general. In the particular case of isotropic
turbulence it leads to KhArman's equation



g = f + (9-7)
2 2r





NACA TM 1577


In the same manner, one then finds that:


c = -2b a = -b 0 (9-8)
2 or


So, of the five functions f,g,a,b,c which define the double and
triple correlations of isotropic turbulence, only two remain distinct if
the flow is incompressible.

The incompressibility results, in particular, in the
relation g"(0) = 2f"(0). Therefore, in terms of dissipation length A:

r2
f(r) = 1 +
2?,2
(9-9)

g(r) = 1- 2



Together with the length h, two other numerical parameters (time
functions) are frequently used to give an idea of the turbulence.

First, the correlation length


L = j f(r) dr (9-10)



is introduced; it depends upon all values of f(r) for 0 < r < m, while
'A depends only on the form of the function f(r) at the origin. It may
be pointed out that L is expressed by means of the scalar

R = u (x)u%(x + g) = u02(f + 2g). Owing to the incompressibility,

f + 2g = 5f + r = 2f + -- (rf). If rf(r) approaches zero when r-->o:
or +r


(f + 2g) dr = 2L (9-11)
UO







NACA TM 1577


The intensity of turbulence in one direction, that of the axis of xl,
for example, is the dimensionless quantity


U
U


(9-12)


where U is the mean velocity. The total intensity of turbulence is the
ratio


u 2
U .5 U


(9-15)


If the turbulence is isotropic, its intensity is the same in every

direction. It is equal to the total intensity and to -U-.
U

In the case of isotropic turbulence the mean square values of the
rotational components are directly associated with the quantities uO
and .

Intro.iucing the components


S1 2 0- 1 o0U
ux2 Ox, ox5 6xl


of the rotational, the averages (2, 2 22
and isotropic turtwulence are computed:


__, '2 '2

S\ .\ 3)


6u2 bul
x1 6--x2
- Ox, Oxp


)2 for homogeneous and




2 6u2
2--
Oxg OX





NACA 'M 1577


We note that, for example:


S. im j u)xlx2 + a2,x5) u5(xl,x2,x)
Ox2 t2-40 2


() u= 21[u32Qxlc,x2 + x + u 2 (xx2,x) -



2u (xlx2,x) u (x12+ x 2,x


The first two bracketed terms are equal to u02. The third is equal

to R (O,g2,0O) = uO2g, or, except for the third order, uO2 ( 2).
Lastly

(u 2 2u02
^x2 2

Likewise


(3u2 = 2uo2







NACA TM 1577


Lastly:


3x2 63
5u, <^i
15gd~


u5(X1,X2 + 62,x3) ul(x1,x2,x5) u2(x1x2,x3 + 3) u(x1x2,)x3


= lii 2(o,2, E) R(12(,0,9) R203(o2,0) + R12 0,00)]


5253


u2
2),


Finally


2
uO


Naturally,


2 2
U0 UO
7? h ?


2 2 2
(i2L and ate are equal to h .


10. Spectral decomposition of the velocity:

We know that the representative curve of a turbulent velocity compo-
nent u(t) in terms of time suggests the idea of an irregular periodic
phenomenon. The analytical representation of u(t) is not a periodic
Fourier series, but rather an almost periodic series or a Fourier integral.

In the first case, u(t) is a sum of harmonics without common base
period; by adopting the complex notation


(10-1)


u(t) = Ae A t
n=-oo


the pulsations ah forming a succession of real numbers increasing
with n. It can always be supposed that n_n = -%h. u(t) is a real
quantity if A-n = An*, the notation An* designating the conjugate
imaginary of An. The series An must be convergent.


1 m
S,2 -0


2 2
A2
2 uO
tol = 2 +


r2 2





44 NACA TM 1577


In the second case:


u(t) = A()eiCtdau (10-2)


the function A(u) being absolutely summable and such that A(-cu) = A*(ai).

The numbers An or the function A(w) depend upon the position of
the anemometer.

This representation is practical when the turbulence is steady in
time, which enables a vigorous application of the ergodic principle and
the calculation of the averages connected with u(t) in a time interval
as great as desired. But, among the simplest problems involved, there
is, first of all, that of a turbulence homogeneous in space (at least in
a sufficiently restricted range) and developing in time. The pseudo-
periodic character of the speed is then manifested in space rather than
in time. (A detailed study follows.)

The component ua of the velocity of fluctuation at point x1,x2,x3
at a given instant can be developed in series or by Fourier integral,
depending upon whether the spectrum is discrete or continuous



U (xt) = nn2,n3) (t )ei 1,(n ) 2"2) n
nl,n2,n5

(10-5)

u(xt) = fz(A,12,'5,tt)ei(Xlxl+2x2+Y x5)d. dX2 d?

(10-4.)


n1, n2, and n3 are three integers that vary independently from
(nIn*2,n5)
-w to +mn in formula (10-5), and on which the quantities Z ,

S("1 n2) n"5)
j1 32 are dependent. In formula (10-4), uu is a
triple integral extended over the entire space A of wave numbers (The
wave number, inverse of a length, is the equivalent in space to the fre-
quency, inverse of time. It is a vector). ua is a real quantity if
1 -,1.-.2-.2 ),t = Z (l2' ").






NACA TM 1577


These notations are of classical form. But, since the u are
steady, random functions of space, it is preferable to represent them as
stochastic Fourier integrals



uu.(x,t) = lei xl*+2x2+x)dh(,t) (10-5)



In this formula, the ha(?,t) are random functions of XlN2 53
with orthogonal increments (or noncorrelated). Or, in other words, if
A and A' are two different points in the space A, the mean of the pro-
duct of the two increments dha(/,,t) and dhp(A',t) is zero. On account
of the complex notations it naturally is a question of a product of
hermitian symmetry. Hence


dh M*(A,t) dh (X',t) = 0 if 7' A (10-6)


If A' = A at the same point, this average is, in general, no longer
zero. It is infinitely small of the order of dX. We put


dhb*(\,t) dh (A,t) = rpc1(,t) d- (10-7)


Now the spectrum of turbulence in the different cases (10-5), (10-4),
and (10-5) is defined.


11. Spectral tensor and correlation tensor:

Assuming spatial homogeneity it is now attempted to form, from the
spectral decomposition of the velocity, the expression of the compo-
nents RO(k1, 2,15t,t) or abbreviated, %Rx(g) of the correlation tensor
of the velocity at a given instant.

If the formulas (10-3) or (10-4) are utilized, the spatial averages
in a very great volume, physically limited but practically infinite with
respect to microscopic turbulent lengths, must be used. It involves a
cube with edges parallel to the axes, of arbitrarily large length 2a, of
which the center, which can be placed anywhere because of the homogeneity,
is placed in the origin of the axes of the coordinates.





NACA TM 1577


First, take the case of the Fourier series (10-5). To form R.,
involves, first, the product u (x)up(x + t), by associating an arbitrary
term of the series which represents uL(x), or rather u *(x), to any
one term of the series that represents up(x + E). To form the average
in the volume V = 8a5, involves division by V and integration of x.
in space. The exponentials are restricted, and likewise their integrals
in V, so that the quotients by V approach zero when a-->4. There
is an exception for the terms of the series uj(x) and u,(x + 5) which
correspond to the same wave numbers, and which give the products


-i x zc.*
Z.*nlyn2,n5) P=1
Z e


l (xtp)+ (n
p=l


i I1 %% p)
Z C l* (n",n Zn (nln2ns)e p=l (11-1)
a P


independent of xa. They are therefore equal to their mean value and
consequently


o p()= u(x)u (x + ) =


n2
nl,n2,n,


Z "ln2,n3)z (nl,n2,n5)e p=l


(11-2)


This is none other than the Parseval formula for the almost periodic
Fourier series of three variables. It naturally assumes conditions of
uniform convergence which need not be defined here.


SZ(ninn) e






NACA TM 1577


In the case of the Fourier integral, um(x)up(x + t) is a sextuple
integral


3 3
-i ) ?#iV+i > M"P(Xp+tp)
(N~z IX7 e P=l p=l dX d?
fd?\ dM


ua(x)up(x + ) =


(11-5)


extended over all values of AW,


72', 2 5?l3 '2, and A' .


Integrating under the sign fin the finite volume V and dividing

by V, we find that, mathematically speaking, the mean value which is
being sought is the limit for an infinite of


S Z (A)Z (XI)e


J


i


p


3

p=l




p=1


p
d?


AZ (A)Z (V)e
AA *


5
p=l dx=
v


sin (''l 1) a
A'1 1


sin (2 ) a sin (' )15 a
X 2 7 2 N 11


(11-4)


Suppose Z (A) satisfies "Dirichlet's conditions" or, more gener-
ally, is a limited variation; then the integral which figures in (11-4)
has a finite limit which, when Z,(A) is continuous, has for value

3

8 A (z)Zp(A)e P= dA (11-5)






NACA TM 1577


This is Dirichlet's theorem.

The mean value of ua(x)up(x + E), the quotient of this integral by
V = 8a5, thus approaches zero when a-4-m, and the formalism of the
Fourier integral does not furnish the spectrum. This conclusion is
paradoxical.

But every difficulty disappears when it is assumed that the turbu-
lence might have only a line spectrum, incompatible with the so-called
Fourier integral. But such an assumption does not seem reasonable.

It seems more satisfactory to concede that the functions representing
the turbulent velocity are too complicated for applying Dirichlet's theorem
and, more precisely, that their oscillations are too crowded to be repre-
sented by functions with bounded variation. A mathematical process
avoiding this difficulty is to follow.

But an approximate argument can also be made. In reality, a is
very great (with respect to turbulent wave lengths), but finite. Hence
the mean value of Lu(x)ug(x + t) is given by formula (11-4), without
transition to the limit. However, a transition to the approximate limit
can be made and assumed that, with suitable accuracy


uj(x)uo(x + 0)


= 8 jZ (A)Z((; )ez
8n3 CL


p=l


I? (11-6)


So, if a spectral function cpqO(X) is defined by


RBp(O) = ua(x)uo(x + t)


3
S i
= Pa2(.)e p=l
J^


CPcL(A) = 8n3 *(-)Z )
V


we get


(11-7)


(11-8)






NACA TM 1577


In a more accurate way, the Tcra(') constitute the components of a
tensor, the spectral tensor, transformed from Fourier's correlation tensor.
But the approximate formula (11-8) has the drawback of yielding a spectral
tensor of a too restrictive form. In fact, there is no reason that the
cpa be the product of a function of subscript a by a function of sub-


script 0, that is, the general product of the vector


itself. This new defect can be corrected by superposing a temporal average
on the spatial average (11-8). But it is much preferred to turn elsewhere.
and have recourse to the representation of the velocity by Fourier's sto-
chastic integral (10-5). The product u (x)u (x + E) is a double sto-
chastic integral


5
S-i pxp+i
u (x)u,(x + ( = e
aL V


5
> N'p(xp+tp)
p=l


On averaging,
involved which is,
form q4j(' )dN if


it is seen that the quantity dha. *(.)dh (,') is
as explained earlier, zero when ?.' / ?, and of the
*' = t. Hence simply


R"O(O) = ua(x)u(x + F) =


5

p=l
e C
JA


(11-10)


The correlation tensor RaO(U) is the Fourier transform of the
spectral tensor pa3(?'). The qrp(A) here have the generally desirable.
They are subject to the two following conditions:

(a) q, ('A) has hermitian symmetry


PaOP = q a


(11-9)


Z (
\h-it





NACA TM 1577


(b) Whatever the complex numbers X the hermitian form

3




which is real according to the first condition, cannot be negative.

Any system of numbers Pa4p constituting a tensor and satisfying
these two conditions can, a priori, be used as spectral tensor.


12. Spectral tensor of isotropic, incompressible turbulence:

I. Suppose, with Heisenberg (reference 25), that the turbulence is
isotropic. Then, as supplementary condition, the flow is incompressible.


The method used to express the
is applied to all tensors, whatever
isotropic when two functions A(k),
t such that


isotropy of the correlation tensor
its meaning. The spatial tensor is
B(k) exist depending, moreover, on


(Pap(5) = A(k)VO? + B(k)65


(12-1)


S2 2 2
k2 denoting the quantity A\ + A2 + ?3

If the flow is incompressible, the functions A and B are formed
by a relation equivalent to Krmain's relation (9-7) between the functions

of the correlation f and g. We write, in fact, that -a = 0.
Sax
Since


5
p iZ22~~px,
le =l
/e
JA


Z -Acdh (


;~OL


(12-2)






NACA TM 1577 51


we must have identically


Sdh = 0 (12-5)


hence, with p being fixed






When this condition is applied to the components (12-1) of the
isotropic spectral tensor, it is seen that


Ak + B = 0


We put



A F B F
4ick4 4nk2

hence



F(k) (12-4)
ok2\ k2


F(k) is the spectral function introduced by Heisenberg. It will be
interpreted later.

II. With Kampe de Feriet (reference 26) we first express incompressi-
bility without disturbing the isotropy.

The rp are subject to the condition that the hermitian form


= XX X* aP (12-5)





NACA TM 1577


is positive or zero. It is positive, except when the vector
proportional to vector \h, since



>_ "&p'aa = 0



on account of the incompressibility condition (12-3).


X is




(12-6)


The hermitian form
and the equation in S,
forms


Y1 = a -a


4r is now reduced.
three real numbers


2 = bLXC


According
Sl,52,5


to classical theory
and three linear


5 = caXa


(12-7)


can be found with which q can be expressed in the form


= SJYL1 2 + S21Y2 2 + S3 Y2 2


the three vectors of components aa,baca being two by two orthogonal.
If y 0, the numbers Sl,S2,S5 are positive or zero.

But, there exists a particular vector 7y, transformed from vector ?7
by (12-7), so that *4 = 0. This can happen only; when the three terms of
which is the sum, are zero at the same time. Hence two separate
hypotheses


(a) S2 = 0


S = 0


' = Sl aaXJ 2


9apj = Sla,*a


(12-8)






NACA TM 1577


a being the function of A. This form of (p n, similar to (11-8), is
too restrictive;


(b) S = 0


i aA = o


Z b7. = 0


* = SilZ aX 2 + 2 IL b.aX12


(12-9)


We write rather



Sl a *acXa*X + S2 b fb XX



and identify with the original form (12-5) of r. It is seen that


9aQ = Slaa*ap + S2ba b


(12-10)


This is the most general form of a spectral tensor in an incompress-
ible medium. In more simple form


A a = a'


There are two vectors a'.,
tangular trihedral and such that


b = vector ba trirec-


b' forming with vector X, a trirec-


cpa = a'*a'p + b'a b'p


(12-11)





NACA IM 1577


By Pythagorean theorem



'ZM2 VE 12 +1 \ a 2 + 1'IZ6b^ 2
X ( 2-12)

(12-12)


with


a2 = a' 2


By this formula, the b'
of cpMa


2= b 2


can be eliminated from the expression


'Pa= p )2\ a' a + b '2


?~2r3)


(12-15)


Finally we put


b2 F(k)
47k


1 b,2 CCL
a2


This is the canonical form which Kampe de Fe'riet has given for the
spectral tensor of an incompressible fluid


(12-14)


(p (7) = C *( ) c( N) +


with


cM(A)h, = 0


- L2







NACA TM 1577


The isotropy stipulates that c = 0, so that formula (12-4) is
obtained again.


15. Energy interpretation of the spectral function F(k):

Except when stated otherwise, the isotropic turbulence is assumed
to be in the incompressible state.

Hence

2 2 2 2
ul = u2 2 u = U0


The total energy of the turbulent fluctuations per unit mass is



E = 1 u2 =1 Raa() = R(0) = 3 2 1 c Z ()P d
2 2 2 2 2


(15-1)


But


IcL(X) = F- (13-2)
2itk


Hence


E = 1 F(k dA (15-5)
IT; k2


In passing to the polar coordinates in the space of the wave numbers
one finds that


E = LF(k)dk (15-4)





NACA TM 1577


Which, after minor changes, gives the amount of the energy of the
turbulent fluctuations per unit mass in the "sphere of the wave numbers"

> v2 < k2 as


Ek f= F(k')dk' (15-5)



k' being an integration variable.

The "small values of k" correspond, of course, to the large scales
(large vortices).

The next problem is to find the expression of the energy E dissi-
pated as heat per unit mass. While the preceding calculations were by
nature strictly kinematic, the dissipative function here must be so intro-
duced that it yields equations of motion, that is, Navier equations;
hence, an assumption associated with the dynamics of turbulence, which
is discussed in chapter III.

With v denoting the coefficient of kinematic viscosity, the energy
dissipated in heat by the molecular motion is



v2 i + + 2 + (15-6)
oxJ 2 Ox 3


This is to be expressed in spectral terms and averaged. If

i >1 NPxp
u= e p=l dh, ()



we get




__ -i e P= dah)






NACA TM 1577


and


5

> p x p
p=1
e p ( dh^ + 1 dhp
A.'


ax


which, squared, re


+ -


ads





Ledh


A Sdh


p=



p* (hy


Lp dh-*(A) +



)]['P dh,(X') + A' dho(v'


To obtain the averages, involves operation under the f sign. The
result is zero, except when the points A and h' coincide in the
space A. of the wave numbers. It leaves


o0,u+ u
( 2


i= {/2 2doM\+ V f h + dho. dh] + a2 I h12}


T-h am+ + + ah) + 1P2 ]

(13-8)

Consequently, if E is the mean dissipation of energy per unit mass


e= I cp=f Vp2p + d(
iA a ~ pap~


(15-7)


(15-9)





NACA TM 1577


This formula supposes neither incompressibility nor isotropy. If,

now, the turbulence is incompressible, T Nmiopco = 0. Owing to the

isotropy, Pa = F
2n k

Hence, by integrating in polar coordinates in space A


E = 2v k2F(k)dk
Jo


(13-10)


E being a physical quantity, hence finite, it is seen that the
preceding integral is finite, which gives a first limitation of the
possible forms of the functions F(k).


14. Relations between spectral function F(k) and correlation func-
tions f(r) and g(r):

The spectral tensor cp O was defined by the formulas


R (h) = u (x)u (x + h) = aCp (')e d


which assume homogeneity only. Incompressibility and isotropy are to be
added.

According to section 8


2 [f -2g
R (S) = uo 2 + 15 p



and according to section 12



C )p 4k2


r rf
2 3r






NACA TM 1577


By Fourier's reciprocity formulas




(D) =W / a R(E)e p=1 (p (14-1)



the integral being extended over the entire space. The general formulas
for defining F in terms of f or g, and f, g in terms of F must be
particularized.

It is pointed out that F q is the scalar invariant of the
2xk2
spectral tensor co ("trace of tensor" or contracted tensor). Hence it
seems logical to compare with F the analogous invariant of the correla-
tion vector Rc defined by


R(r) = R Ba = uO2(f + 2g) = uO2(5f + rf')


Thus the following problems must be solved:

(a) express F(k) by means of R(r), and conversely

(b) express f(r) and g(r) by means of R(r)

(c) express f(r) and g(r) by means of F(k)

Also to be defined in precise manner is what is called the trans-
versal (lateral) and the longitudinal spectrum of turbulence.

(a) Relation between F(k) and R(r):



Vp
R(r)= e p=1 Qad = Fe F(k) d (14-2)
JA JA 2k2






NACA TM 1577


To reduce the triple integral to a simple integral, simply select

a new axis of the X, perpendicular to the plane 7 \% = 0; it
results in the integral




R(r) = firxe Si1 d = feirk e 0os 6F(k)sin 0 dk d q dp
fA 2%2k2 2x fA

(14-3)

Which, after a short calculation, yields



R(r) = 2]' s r F(k)dk (14-4)


Conversely


F(k) 1 r sin rkR(r)dr (14-5)
k n 10


(b) Relations between f(r). g(r), and R(r):

The integration of the differential equation yields



rf' + 5f = R (14-6)
uo

or



d-r O= 2
dr\ /' 2






NACA TM 1577


with the condition f(0) = 0. It is


uO2f(r) = rS 2R(r')dr
r31 Jo


(14-7)


R(r) having for limit 5u 2, when r tends toward zero, it con-

firms that the integral is equivalent to u02r3, and that f(r) tends
rather toward unity.

From that, it is deduced, by formula (9-7) that


u02g(r) = 1R(r) 1
2 2r5


(c) Relations between f(r), g(r)

A simple calculation gives




uO f(r) = 2 fF(k)


u02g(r) =


r'2R(r')dr'



, and F(k):


cos rk\dk
r2k2 dk
r2k2 /


f F(k) [cos rk
0 r2k2


From that the first terms of the limited developments of R(r), f(r),
and g(r) for small values of r are derived in terms of the dissipated
energy e


(14-8)


(14-9)






NACA TM 1577


18 VUI



f(r) = 1 --r2 _E + (14-10)
50 vuo2



g(r) = 1 1 r2 -- +
15 2



Comparison with (9-9) indicates that the dissipation length A is
related to the dissipated energy E, the viscosity v and the kinetic
energy of the turbulent fluctuations E = ug through the formula



F(k)dk
=2 E= 10 = k) (14-11)

S k2F(k)dk




15. Lateral and longitudinal spectrum:

The spectral measurements do not provide the spectral function F
directly, but merely the spectral terms corresponding to certain simple
associations of velocity components, for the observers placed in the
particular relative positions. It concerns longitudinal components of
the velocity (parallel to the velocity of the main flow), at points
placed either on a parallel to the axis or on a perpendicular to the
axis. The corresponding correlations are uO2f(r) and u02g(r).






NACA TM 1577


According to the original notations of G. I. Taylor


u0 f(r) = cos wrA(u)dma


(15-1)


uO g(r) = cos orEB(w)dw


where A(w) and B(w) are termed the longitudinal and lateral spectral
functions. They now must be expressed by means of F(k).

Since g = f + -f', elementary calculation shows that
2


B(w) = A(A1 ) GA' (w)]


(15-2)


and at the same time that


R(r) = 2 sin rkF(k)dk = cos ar[A(w) + 2B(u)] dao
JoC rk Jo


(15-3)


Elementary Fourier transformations yield


F(k)
A(w) + 2B(W) = 2 k L0 k
k


A and E are determined by the two equations


2B = A aA 1


A + 2B = 2 mJc Ldk|J
O k


(15-4)


(15-5)






NACA TM 1577


The result is



A(w) -= F( k )dk E(c) =1 +F(k
J2 k( k2 kkdk


(15-6)


CHAPTER III


DYNAMICS OF TURBULENCE


16. Introduction:

To construct a dynamics of turbulence, that is, to set up the dif-
ferential or finite laws which govern the development of the statistical
quantities characterizing the turbulence in time and space, it is neces-
sary to start from elementary laws and apply the statistical methods to
them. The most natural idea, the only one which actually produces con-
crete results, consists in utilizing the IHavier equations. To what
extent are they applicable to turbulence? The turbulent motion is always
a macroscopic motion with respect to a finer scale motion, and, at the
limit, with respect to the molecular disturbance. Therefore it is rea-
sonaule to believe that it satisfies the equations of the mechanics of
fluids. The next step is to find the solution of the Navier equations
which, for certain limiting conditions, have the turbulent aspect, and
calculate the particular averages from these solutions. Unfortunately,
it is rather difficult to define these limiting conditions. So, the
remaining resource is to examine whether, among all the possible solutions
of the Navier equations, there exist any sufficiently complicated for rep-
resenting the turbulence, without attempting to determine them logically
by the limiting conditions. But then a new difficulty arises. Solutions
for the Navier equations are known only for simple conditions which are
far from resembling turbulence. In other words, while conceding their
validity, we practically do not know how to solve them.

Since it is not acceptable to take the averages on the solutions of
the Navier equations, it is attempted to take the averages on the differ-
ential equations themselves and to write the differential equations which
verify the statistical quantities. Since this method has produced results
it is set forth in the simplest case, that of the Karman correlation
tensor.







NACA TM 1577


17. Fundamental equation of turbulent dynamics:

It is assumed that the total velocity U is uniform and along the
axis of xl. The Navier equations read then



+ U + up + v Au, (17-1)
6t ox p dxP P dxQ



in the absence of external forces. p is the density, usually assumed
constant, and p the pressure. These equations must be supplemented
by the equation of continuity



S -X O (17-2)
,)ax,


The notations are abbreviated by representing the velocity? at
point x'a = x, + Pa by u'a = u (x') = u (x + t). Multiplication of

(17-1) by u' and summation with respect to a, gives




CL t bxl 6 X0



a p u a3 2u
:au' + I c u'02 (17-5)
oxi ap oxp






7.In chapter I, section 6, ul,u2i,U represented the velocity compo-
nents of the whole and u'l,u'2,u'5, those of the velocity of fluctuations.
The over-all velocity having then U,o,o, as components, the notations are
changed; ui represents the velocity fluctuations at point x, and u'i
the velocity fluctuations at point x'.





bb NACA TM1 1377

Permutation of the points x and x' yields a similar equation,
which is added to the preceding one



u uu' + U + u au + U + UMuU =




-I u p + u+ v u 'a- + ca2u-I
P x % %a/ a, \ cxp2 / i 2


(17-4)

Taking the averages of the two members it is then assumed that the
turbulence is homogeneous, nothing more. The scalar product T UMU'O
is a function of tg, and is not individually dependent on x. and x',.
Consequently


Z u T U' a= Z uau'
0x1 a 6, z '


2 u u'a, a = u
ax', 1S,


The term containing the over-all velocity U disappears. The
following term can be written



ZI U'u0 +cu-- U 'u' -
Z ., UU -

af 3 ap d 0X,







NACA TM 1577


The last two terms disappear as a result of the equation of continuity.

I U%-x-u up is written:8
Uax


Z- 1V u'_.
P 'Xf a


Likewise, as a result of the



ci >I y" u u' u' ,
t TX Ua a
Pox 5 a


--iLCL^k)
P3 2rp a
s
~ at 0


homogeneity, u %jU --_U I becomes




t- aSp a


Since T7 (-t) = -TBy (t), the system of these two terms is equal to


-2 '- L T Q
P 0^ a
2- 1 4c


(17-5)


The viscosity term reads


2vE-au u' = 2vAZ7Ra 2VAR
P `3 a a
CLC


(17-6)


where A is the Laplacian symbol in three-dimensional space.

Lastly, it is shown that the pressure terms disappear. This is done
by extending the definition of the homogeneity to the averages containing
the pressure, with due regard to the incompressibility. The averages such
as pu'a are not functions separately of x and x' but merely of the

difference & = x' x, so that, on these functions, -

8
See the definition and the properties of the tensors Rap and
T in chapter II, sections 8 and 9.
u437







"rE. m 57


Tm --'f. r =sIg ilzr-r-. f f


-;...-,- -- -

a;. a


'n- D7


Z~ir *-ts if -2 r-.z ,- -e-ss zex. "b tsae.". r*Es:' is e*:E
- ] '--." 0- j -Z a 'm-- ,rlt -


- LT


- :-zif -a a I


-I I


F~_ ii 2 -C= --r -



-r-


-- "s .t -I. s -n.: ~ .' .. t- gz:.




MC = :5 -D = 30-1









trIi_ -8a- fJ-. 4;L -. LSi a'-- "In Y --.-i f'IW ri *- Ziff


~-r
ar


= a


__ azia







NACA "Y 157


AcccrdI-c to the properties of T (section 9):



=a k(c + 2a) = UO c' + S



The iivergence of this x-ec tr is equs. to


i r- c' +
72 ;!L


c\I


The f-.na-en-tal e a-iorn (1-1Z) is now transcribed:


r r *
r2 orI t r r r2 LC


(1i-1)


(15-2)


r;]


In fact. it is rere=tered tnat the Laplacian of a function
n-dinensicnal space is



()= +
r r1


w(r) in




(18-5)


Multiplication uy r2 is followed by integration vith respect to r.
It is easily- verified that all1 the terms cancel out for r = 0. Bearing
in mind tLhat


.22 = 22_ r5f' = -r25 f" + if
ar Q drr r3 1 r )


(1l-)


one arrives at the equation


+ f') +- 5(c' + -)
g ] r


-u 2f" = 2vuO2, "


(18-5)






NACA TM 1577


This is the Karmian-Howarth equation (ref. 30) in its classical form.

The general equation (17-10) has the form of the equation of heat
propagation in ordinary three-dimensional space, the quantity being
propagated is the scalar R, the invariant of the double correlation
-4
tensor. The "second member" div T is tied to the triple correlations
as R is to the double correlations. The result is an equation of
-4
partial derivatives between two unknown functions R and div T. For
reasons which appear later on, R is usually considered as the principal
unknown. The problem of solving (17-10) consists in defining R by means
of suitable physical hypotheses on the triple correlations. The equation
with two unknowns finally arrived at is the result of the nonlinear char-
acter of the Navier equations. Its mode of operation will be explained
later.

The isotropicc" equation (18-5) leaves no trace of the original
properties of tensorial symmetry. Its purpose is to connect the func-
tions f and c, which can be measured directly. If it is borne in mind
that


f"(r) + -f'(r) = %f(r)

(18-6)

c' + 4 1 a (r4c
r r4 Tr\ /


then one may write:



t u2f) 2v 5uo2) r (18-7)i(rc
6t ((0) r ( T (18-7)


It is an "equation of heat," isotropic naturally, but in a fictitious
five-dimensional space. This remark is interesting, but it seems arti-
ficial and without real physical significance.

Nevertheless, an unusual property of the functions ug(t) and f(r,t)
can be derived from it. Multiplying by r4dr followed by integration
from 0 to yields






NACA TM 1577


0- 2 Jo r4f d-r = 0


provided that r4f and r4c tend toward zero when
reasonable, and difficult to check. Consequently


r-P-m, which seems


uL2(t) r4f(r,t)dr


(18-9)


is a numerical constant, independent of t, during the development of the
turbulence. It is the Loitsiansky invariant, which connects the energy
of fluctuation E = u 2 to the correlations f.
2


It also should be noted
the divergence of 2v grad
and infinitely large radius,
i- s zero, and
at


fR


that, since the second term of (17-10) is
24
R + T, its integral, on a sphere of center
is zero. Consequently, the integral of


dxl dx2 dx5 = Cte


(18-10)


the triple integral being extended over the entire space. If the turbu-
lence is isotropic, it leaves


r2R(rt)dr = Cte
JO


(18-11)


But, according to the expression of
beginning of this paragraph:


-r2R dr = uO2[5]O = u02 lim
0o 0 r --


R(r), remembered at the


(rEf)


(18-8)





72 NACA TM 1577


If rf f'-40, r5f tends toward zero also, and this limit is zero


r2R dr = 0 (18-12)


This proves that the function R, which is equal to 5u02 for
r = 0, and to zero for r = -, takes negative values for sufficiently
high values of r. If formula (18-12) is approximated to formula (14-5)
which expresses F(h) by means of R(r), it will be found that, pro-
viding R(r) tends sufficiently fast toward zero at infinity, the
development of F(k) in powers of k starts with a term in k4; near
k = 0 (large size eddies)



F(k) = Ck4 (18-15)


with


C = -- r4R(r)dr


The constant C can be expressed by f, because, according to (14-7)


UO2 r f(r)dr = f r dr O 2R(r')dr



In order to change the order of integration, the integrals of the
second member are written in the form (fig. 5)

S r" r r'
rim r" dr" r'2R(r')dr' = Is r'2R(r')dr' J r dr



pr
= lim r2 r'2R(r)dr' -
2 r--m Jo


1 fr4R(r')dr'
2 Jo






NACA TM 1577


Now:


r
r2 r'2R(r')dr' = u 2r5f(r)
Jo


So, if f(r) tends toward zero at infinity fast enough so that
not only r-f but even r f- approaches zero, then


2 f(
C = rf dr (18-
5" o


14)


To a factor --, C is therefore identical with Loitsiansk&'s
invariant (18-9).

These results could be extended to include the nonisotropic
turbulence.

Lastly it should be noted (ref. 2) that, in the case of isotropic
turbulence, the vector T defined by (18-1) is radial, that is to say,
that the Tn are proportional to the E-. In effect:


UT5(c + 2a)
T r


(18-15)


Hence one may put:


-4T = T grad r
T = T grad r


(18-16)


T(r) = uO (c + 2a) being a scalar.


19. Local form of the fundamental equation:

The problem is to ascertain what the equation (17-10) becomes when
r--40.





74 NACA TM 1577


It is known that f(r)->l. As f = 1 -- + f" +f
2\2 r
approaches


Lastly, as c is of the third order with respect to r, c' + L
r
tends toward zero.

The final equation, due to G. I. Taylor, reads


2
S0 -10V- (19-1)
dt .4



Recalling that, according to paragraph 14:



u 2 = 2E 2 = 10v
5 t


The formula (19-1) becomes therefore9



= -E (19-2)
dt


It establishes an elementary relationship between the dissipation
of energy E by viscosity and the decay as a function of time of the
energy of fluctuation E; the turbulent energy of fluctuation is totally
dissipated as heat by viscosity, this dissipation following a process to
be analyzed in chapter V.

The foregoing demonstration stipulates isotropic turbulence. But
the result (19-2) is valid in more general cases. In fact, (19-2) can be
derived from (17-10). First it is known that R(0) = 2E. The connection
between A R and E is established by the intermediary of the spectral
functions q ,(AM). If the flow is incompressible, formula (15-9) reads


E being function of the sole variable t, can be replaced by
dt
-, a notation which represents an ordinary derivative with respect to
dt
time (not a derivative with respect to motion which would be without sense
here).







NACA TM 1577


E = V r2z (P()d (19-)0


but on the other hand,




R() = e10Je) = Je pq1 2- ()d?



consequently:




AR = k2e p (p()dh (19-4)



and in particular, when p approaches zero, vAR is reduced to


r? -4

To prove that div T = 0, at the limit, the simplest way is to revert
to the Navier equations.

It is plain that equation (19-2), limit of (17-10) when r->0 can
be obtained without going by way of the finite values of r. Simply mul-
tiply the Navier equations by ul,u2,u5, add them up and average. Assuming
the disappearance of the pressure terms to have been proved, it is



Sd + > US = v > U 2 (19-5)
2 dt ax, dx 2






NACA TM 1577


The term Id- u 2 is written dE. The second member,
2 at a dt
according to the foregoing proof, is equal to -E. Hence it must be

shown that 2 u u -- 0. This quantity reads




6 u 2 1T 6 2 1 Z u 2 L au,
2 V> U0.Z a aZ 3 ax -2u 2 M Zua,2 6
pap ^p c a,


(19-6)


By virtue of the homogeneity, u 2u is not dependent on x, and
the first term is zero.

Owing to the incompressibility, the second term is zero. The first
member is therefore zero, which provides the proof.


20. Solution of the fundamental equation, when the triple correlations
are disregarded:

When the triple correlations are discounted, equation (17-10) becomes
the equation of heat


aR = 2v AR (20-1)



This hypothesis, which mathematically is convenient, is physically
quite difficult to justify. Its accuracy increases as the viscous
effects become greater (prevalence of term 2v AR over the term div T ),
or as the Reynolds numbers decrease (whatever their definition). It also
would be verified if the velocity components followed the Laplace-
Gaussian law. And, what is more interesting, it may be added that it
corresponds to a state of turbulence in which the forces of inertia are
negligible against the forces of viscosity.

Equation (20-1) must thus be resolved knowing R for t = 0, and
being aware that, if r-40, R has for limit a finite quantity 5uo (t)
temporarily considered as known.






BIACA TM 1377 77


In isotropic turbulence, R depends only on r, t, and (20j-) reads


6R /o2R 2 ':\'
= 2 -- + -- (20-2)
at \,2 r or


To satisfy the limiting conditions, elementary solutions of the form


R(r,t) = R1(t)R2(s) s = r (20-5)
S8bvt


are necessary. Hence


t'(t) 2(s) i
tH(t= ---- -L--- + s + -)R'2(s) (20-5)
R (t) 2R2(s) 2 s


The two terms of this equation have a constant value --. There-
4
fore, R2 satisfies the linear differential equation



sF"2 + 2(s2 + 1)R'2 + QsR2 = 0 (20-4)

or, reduced to classical form, by the transformation,



sR2 = e-s H (20-5)


Quantity H confirms in fact the Herrritian differential equation


H" 2sH' + (Q 4)H = 0


(20-0)






HACA TM 1377


Since R(r,t) must cancel out for r infinite, hence also H(s)
for s infinite, Q 4 is necessarily an even integer 2n. In that
case H is the nth Hermitian polynomial:



Ha(s) = es2dn e2 (20-7
ds


and


R2(s) -s2y(s)


(20-8)


But R2
1H(s) be an
Q = 4p + 2.


mentary solution:


must have a finite limit when s-->0, which requires that
odd polynomial, hence that n be an odd integer: n = 2p 1
The function I(t) is then equal to 1. Hence the ele-
pt-
t 2


r2

R(r,t) = -e tlH2p-1 r-


(20-9)


The general solution is a superposition of elementary solutions:


r2
R(r,t) = e r A Hrpe-1 r
p=l t-p H / t


(20-10)


the constants Ap being so chosen that the series converges and R(0,t)
is equal to 5u022

Examples: 1. Limited to the term in p = 2 it is found that, by
interpreting the Hermitian polynomial, H3(s), and denoting an inde-
terminate constant by A:







NACA TM 1577


R(r,t) = -t

t2


r2
- -if


hence


2
r
f(r,t) = e bvt


5
O2(t) = At 2


This example is rational. It is compatible with the hypotheses
made at that time. In particular, rmf approaches zero when r--4o,
whatever the non-negative number m may be. However, f = 0 for
t = 0, except when r = 0. At the initial instant the turbulence is
concentrated at one point from which it ultimately spreads throughout
the entire fluid. Such a turbulent structure seems rather difficult
to conceive when assuming it to start from t = 0. An interpretation
will be given later (section 34) in connection with the final phase of
decay of turbulence behind a grid.

For p = 1, one would have


r2
R(r,t) = Ae 8vt

t2


r '2
r =tD

) Jo


(20-12)


f would not converge toward zero when r-)4o, which proves that this
example cannot be suitable for a real motion.

2. (Karman-Howarth (ref. )0). Suppose the correlation function f
does not depend separately on r and t but only on the vari-
able s = r
Sincvt

Since


R = u02 ( + r
Tr-


(20-11)





NACA TM 1577


it then yields



R = u02(t)[3f(s) + sf'(s) (20-15)


so that R is the product of a function of t by a function of s.
Thus R is an elementary solution of the problem treated in the first
example, corresponding to a chosen Hermitian function H2p-l.

The function f(s) satisfies a differential equation similar to
(20-4) written by Karman and Howarth. But a great advantage accrues
from the use of function R and reduction to Hermitian polynomials.

Remark. According to (20-10), R(r,t) is a sum of "elementary
solutions" corresponding to various values of the integer p, starting
from p = 2. For each one of these solutions, except for p = 2, it
is verified that the invariant of Lbitsiansky (18-10) is zero. For
p = 2 it has a finite value.


21. Solutions involving a similarity hypothesis (see ref. 7):

If the triple correlations are no longer negligible, the fundamental
equation cannot be solved completely. It is assumed that the correlation
functions f(r,t) and c(r,t) are not separately dependent on the two
variables r,t, but solely on ;i r where i(t) is a length in
Z(t)
terms of time. That is to say, that at each instant t the correlation
curves are superposable, by means of a unit change on the axis of r.
However, it is recognized that this hypothesis, called "total similarity"
is often too specific, and it is therefore replaced by a "partial
similarity" which is verified only in a finite interval 11 < r < 22.

For total similarity,, the Loitsiansky invariant reads



UO2(t) j rf(rt)ir = uO2(t)O(t) 4f(')d4 (21-1)


Consequently, uc2(t)Z7(t) is a constant independent of t in the
course of the motion. This is no longer true in the case of partial
similarity.






NACA TM 1577


To transform the fundamental equation, it is possible to put


R(r,t) = 5uo2(t)a(*6)


T(r,t) = 5uo5(t)o(*) (21-2)


12


a(*) is a function that takes the value unity for r = 0; p(4*) is
a function that cancels out with *. Rj is a Reynolds number in terms
of the time, and T is the scalar which, according to (18-14), defines
the triple correlations.

Considering the relation (19-1) which affords d u2 in terms of

202 and of the length ?., the fundamental equation

= 2v AR + div T
6t

is easily transformed to



Z2)'a.') + -a_ (.p ) + a."() + 2'(9) + ) + =
'N2 1L1

(21-5)

or, schematically arranged:



1a1(t)0P(') + a2(t)2() + a.5(t)8P(*) + c4(t)P4(4) = 0 (21-4)


Equation (21-5) is a sum of four terms each of which is the product
of a function of t by a function of *r. How can an equation such as
(21-4) be resolved?






NACA TM 1577


The mi(t) are considered as the coordinates of a point A in
four-dimensional space, this point describing a curve (A) in terms of
the parameter t. The same applies to the pi(*). It results in two
curves (fig. 6) such that the straight lines joining the origin to any
point A of the first curve and to any point B of the second are
always perpendicular. These curves must fit two orthogonal complemen-
tary subspaces in four-dimensional space. Hence there are, a priori,
three possible cases:

1. (A) is a line passing through the origin. (B) is in the com-
plementary three-dimriensional space orthogonal to this line.

2. (A) and (B) are in two completely orthogonal planes passing
through the origin.

35. (A) is in a three-dimensional linear space. (B) is the orthog-
onal line.

In the first case, the ai(t) are proportional to constant num-
bers mi.

In returning to the notations of equation (21-5), the quantities


(12) 1

2
are proportional to constant numbers. In other words, 12 1-, and R
X2
are constant. Following this, the functions a(*) and 03(') are
connected by a unique differential equation. Obviously one may assume
1 = h. Then X2 is a linear function of time and ugQ. is constant.
Considering equation (19-1), which we recall here


t2 = -_lOvu (19-1)


we obtain of necessity:



2 2 = 10 vt u02 =Ct R = Cte (215)







NACA TM 1577


The Loitsiansky relation is not verified, so that no total sinLi-
larity can prevail; a(i) and p(w) are joined by the relation


a" + ( + 2' + 5a + .(, + =P o (21-6)
2 2- \


which can be subjected to experimental check.

A complete discussion of the second and third possibility is fore-
gone, in favor of the case where I = A is prescribed. Then (21-5)
becomes


S(22)'a'(0) + a"(;.) + 2-a '(.) + 5a(') + 1;jp, '(l) + 0() = 0


(21-7)

and is represented schematically by



(t)p( ) + a2(t)02() + a(t),() = 0 (21-7)


The discussion then deals with three-dimensional rather than four-
dimensional space.

Two cases are possible:

(1) The point ai(t) describes a straight line (issuing from 0),
and point 3i(2) remains in the plane perpendicular to this line
(passing through 0).

(2) The roles of the two points are permutated.

The first case is not distinct from that which we have studied.
In the second case, there exist ti-ree constants ml, 2,m5, so that






84 [1ACA TM 1577



+ 2 W) + 5a(b) 3'() 0)

4vul m2 2m
S2 (21-8)


m + m2 + mR = 0


They are the conditions proposed by Sedov. They can be checked
directly by experiment; m, cannot be zero, because a; would be = Cte
and at the same time a(*) = Cte, which is not possible; m2 and m3
cannot be zero simultaneously.

If m3 / 0, the similarity cannot be total, because one would have
u02?5 = Cte and by (19-1)



J2 = 4vt 2t/2 = Cte Rt5/4 = Cte


which is incompatible with the second equation (21-8).

Following this, uO and ?h are solutions of the system of differ-
ential equations:


mlX 2 2 + "0'= 0

/ ,-2 (21-9)
"O -=lov- --


a({.) verifies the second-order differential equation:



O" + n--- 2 + 5a = 0 (21-10)
iq 4vml,






NACA TI1 1577


after which [ is determined by the first-order differential equation


of + = -A-V-a'
S2vml


(21-11)


The system (21-,)) can be reduced to quadratures. If the solution
cannot be written in finite terms, it still is desirable to integrate
as far as possible.

When the first equation (21-9) is set in the form


DR l-k vk
BQ 10v


(21-12)


where RO and
found that


k are constants replacing mi,


2)i and m3, it is


\ = cte- n RO) k



Whence it is deduced that


dR Cte kRo
dt R (1 k)O


The problem
follows from R,


is thus reduced to a quadrature. By (21-15),
R
and U = v=.


The equation (21-10) generalizes equation (20-4)
but it is less simple and will not be discussed.


to a certain extent,


Other types of solutions satisfying a similarity hypothesis can be
examined, such as the case where hN is very small, that is, where the
dissipation due to viscosity is low. In that case a length other than

A must be chosen for length 1, such as L = f(r)dr. The discussion


(21-15)


(21-14)


1-2k

R






do NACA TM 1577


proceeds on the assumption that the viscosity terms of the fundamental
equation are negligible against the other two. It leaves then


R = div T
dt


(21-15)


The calculations are not developed (for further details, consult
Latchelor's report (ref. 7)). The sole purpose in this paragraph was to
give an idea of the various methods that can be applied to solve the
funda-mental equation on the basis of the similarity hypothesis. The
physical study of the problem is deferred until later.


22. Transformation of the fundamental equation in spectral terms:

The formulas of reciprocity between the correlations and the spec-
tral functions made it seem interesting to transcribe the fundamental
equation (17-10) in spectral terms. (Compare spectral intermedium,
section 19.)

By (11-10):


R = R. =


Si

tiA


S'PIP
p=
p=i


Scp )(d


le


5

pz


with


'. 2 2 ) = Zi a


Likewise


CR = fe pl k2qdX\






IJACA TM 1577


Hence


5

S- 2v R = J e + 2vk2c),d
t .t


This quantity is equal to div T,
for components T Wap vie put:
Qra


div T '(.)e Pi
JA


where T is the vector having
where T is the vector having


p,p


The fundamental equation becomes


'- + 2vk24 =
ot


In this equation q, and i are Lunkjiown functions of 7,


(22-4)



t, and


2 ^ 2
k2 = > ;?p2.
p=l

The total energy of the fluctuation is



E -= 1 .,7




(it was shown in section 9 that f dA = 0).



For isotropic turbul.ences, the spectral function F(k) = 24k2cp,
which does not depend separately on 's in the general case
for cp, but solely on k, is introduced.


(22-5)






NACA TM 1577


The factor can also be expressed by means of the scalar


T = uJ(c + 2a) = r'c0 + c) = (cr4



It is actually known that

1 'r2
div T = T' + -T = (r2T)'
r r2


/which, after elementary calculations, gives


p(k) r sin rk r2T)I dr
2Equation2 (22-4) reads


Equation (22-4) reads


6F + 2vk2F = 2nk2 r
at


(22-7)


Putting: 2jrk2,; = ,(k,t), where' is a function connected with
triple correlations and such that



l dk = 0
Jo


yields


6 + 2vk2F = (
at


By the use of the formula (22-5) and the limited
sin rk, it is readily proved that the function I(k)
small as k0, provided that the correlation function


development of
is infinitely
c approaches







I1ACA TM 1577


1
zero faster at infinity tnan -1. But, with the relation k = (0,


nothing more is known a priori about its form.

Equation (22-t) is not -.ssentially different from (17-10). It
raises the same difficulties and is always a single equation between
two unknowns F and T. However, it is much easier to make mathemati-
cally or physically reasonable assumptions about the form of than
about the form of the triple correlations, as will be shown in the fol-
lowing two methods, both due to Heisenberg, which take advantage of
(22-8); the first involves the introduction of the correlations in time
and will go beyond the purpose originally assigned to it, the second
leads effectively to certain possible forms of the spectral function F.


25. First theory of Heisenberg (ref. 25):

The hypotheses to be examined primarily are those which form the
subject of the last part of Heisenberg's report. We shall utilize the
ideas and follow the calculations as closely as possible, but change
the notations and substitute st'ochIastic averages0 for the spatial and
temporal averages.

The expressions giving the statistical velocity and pressure of
homogeneous turbulence read:


5 5
i ^xp i. 3 >'x
u e d ) p = Je = dq() (25-1)



The equations of motion are first written in spectral terms. It
can be assumed that ha(\) is derivable on each test with respect to
ou
the time, and its derivative, which corresponds to --, is designated
6t






10Bass, reference 5. The calculations are rather difficult. The
reader who wants to avoid reading them will find a summary of the
results at the end of paragraph 24.





NACA TM 1577


The expressions (25-1) must be introduced in the equations (25-2)


+ U-
oxl


P CJX(L


-u-


where U is the constant over-all velocity.


The only
cussion is


term of wlich the equation merits a more detailed dis-


u = -i
U -a -


f 1 i xpX dhh (ip)
JAXAe


Putting up 7'p = Xp and considering the equation of continuity
yields


- ?a. dha(t) = 0


which had already oeen used in the form


C)U Xp i X
I3--= i /e (c Thp(Ci)
2Xf3 J C()


(25-4)


the subscript (uL) specifying that. the integral
of p. This quantity is the Fourier transform


is extended over the values
with respect to ", of


i: f(i -I )h(


-+ T -
3t p 6ox


+ v
P oxp







NACA T1Il 1577


y mrI-anrs of somE ele .r:.nttr,, transformations two equations are
obtained:




Mdha() + i *f l 2)hp() + PiU 't ('1 =



dq(',) V Ckd (.) I






P a(



whitre h(*,) designates the derivative of ha(') with respect to ?.


The demonstration of paragraph 17 from equations (25-5) is then
resumed. The first equation (25-5) is multiplied by dha*(N), added up
with respect to a, i is then changed to -i and the two equations
obtained are added member by member.

P-%tting


S= -2i *' tih d(') p )dh () (25-6)
a fSm*( '1(



widl>- considering the equation of continuity, gives



7,- d O 2 + 2vk2 j Wt7) = R(Z) (25-7)
ot


R(Z) denotes t-ie real part of 2.

After averaging, it is known that when the turbulence is isotropic:



2nk 2
2nkr





1IACA TM 1577


Consequently


d2 F2-iF + 2vkF) = R(Z) (25-8)
2rk2 /t


What was called I in paragraph 22 is given by the equation



2nk2R(Z) = d (25-9)


In order to find a useful expression of Z several hypotheses
originating with Heisenberg are made.

First of all, it is known that the statistical functions dh (X)
are of orthogonal increments, that is, that the averages of the pro-
ducts dh (?)dh,*(A') are zero, if the two points X,M' of the wave
number space are distinct. The first hypothesis is that the statistical
variables dh (A), dhc*(?' ) are not only orthogonal but independent
(in probability).

Recalling that the average values dh (A?) are zero and that
dha*(A) = dh (-?), it is seern that the average value of


dh*(7,)dh(p )d&hp(G)


is zero, except when at the same time





that is, A = 0 = 0. If it is assumed that the probabilities of sudden
jumps of h (A) are zero, then Z = 0.

Paragraph 20 dealt with the case where Z = 0. It is certain that,
in general, Z is not rigorously zero, but it is likely that Z is small
compared to the viscosity term, or the term of development -. In other
at
words, the hypothesis of orthogonality is too general, and that of the






NACA TM 1577


independence too restrictive, yet it is difficult to formulate an inter-
mediary hypothesis. But, starting from the independence it is possible
to obtain, by means of certain interesting transformations, a significant
expression for Z. This expression is defective since there is contra-
diction. Put it seems likely that it represents a good approximation
of Z. We shall therefore study it.


24. First theory of Heisenberg (continued). Space-time correlations:

If the average of R(Z) disappears, it is because Z contains an
odd number of factors dha.(X); the equations of motion are not linear.
Replace dhj(p) by the time integral of its derivative dhi (p) and
express ih-() by means of the equations of motion. Certain terms
linear in dii are replaced by quadratic terms; products of four fac-
tors dh appear in Z, and Z is not zero. A kind of technique of
solving the equations of motion is involved here, which reduces them
to integral equations (bearing on simple averages).

If T is a very great positive number

t Pt+T
dh (4,t) P (p,-T) = dyip(p,T)dT= O dh (pt T)dT



(24-1)


If T is sufficiently large it may be conceded that hp(4i,-T)
tends toward zero, that is, that the fluid has "started from rest."
Hence


dh (p,t) = d (d,t T)dT (24-2)



To lighten the notations, h' (i) is to represent the func-
tion hg(p,t T). Then


Z =-2i ) dh *(A) dT dh *( A)di'p(P) (24-5)
2- Jo JCL)






NACA TM 1577


and di (4) is eliminated by means of the equations of motion from
which we removed previously the term of the pressure in dq:



P+
dh pu) = -i d h',*(4' -^dh (p') +




i^o a I h'L,*(p' W)dh n,(.') (Vi2 + ipU)dh' (p)
0. f3 ( j,,


(24-4)


with V2 2.


Now a new assumption is made according to which the original assump-
tion of independence (for T = 0), which is certainly; verified for the
great values of T, remains valid for the whole time interval 7, leaving
only the averages with four factors



Z= -2 -S d A (\)dY( ?)dhl*(1.L P)dh',(C') +


(24-5)


2a3 T W' 2 ,dh' *( ,dhO(M )
COpa- 0 2 foJw |


The dh being independent in distinct points of the frequency spaceJ
it is necessary to associate the four points which correspond to the
four dh by pairs, not forgetting that dha*(\) = dh0(-?.). Considering
the transformation of the first integral, the only admissible combinations
are


(1)st


? = L


P' P = P,


hence ut = 0. 1o contribution to the integral.

(2)nd


S= p 1 4 A = P
P c




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