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 c! *! z NACA TM 1577 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 NACA TM 1577 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 NACA TM 1577 22. Transformation of the fundamental equation in spectral terms . ... 86 23. First theory of Heisenberg . .... 89 24. First theory of Heisenberg (continued). Spacetime 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 Digilized by Mhe Iniernei Archive in 2011 with lunding Irom University of Florida, George A. Smaihers Libraries wilh support Irom LYRASIS and the Sloan Foundalion htlp: www.arcliive.org details theoriesoflurbulOunil NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 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 welldefined 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 BatchelorTownsend at Cambridge. The analysis of these works became the subject of a series of lec tures at the Sorbonne in FebruaryMarch 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. NACA TM 1577 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 KarmanHowarth 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 NACA TM 1577 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 BlancLapierre. 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 NACA TM 1377 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 socalled 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 NACA TM 1577 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 k7. 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. NACA TM 1377 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 NACA 'TI 1377 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. 8 NACA TM 1577 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 20cm by 50cm tunnel: airspeed, 20 m/sec; intensity of turbulence, 510 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 (21) 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 (22) 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 NACA TM 1577 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 (25) It replaces (22) and should be compared with (21). The practical operations enabling the replacement of uM by ' correspond to wellknown 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 (24) it is simpler and more precise to use the formula (u )2 (u i) f(u)du (25) which, once the curve f(u) is plotted, calls only for operations of a simple character. NACA TM 1577 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. NACA TM 1577 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 welldefined 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 (41) 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 (42) 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) (45) 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 (45), 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') (51) 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' (52) 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' (53) To exploit this equation, recourse is had to a method patterned after the concept of J. Moyal (ref. 35). This idea was to consider (55) 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') (54) 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 T0 Kt+Tt 1 T (55) 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 (56) 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 (55) 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(xx') K(x; x', t, t')dx (57) One assumes likewise: p(a, x, t) = eiaxP(x; t)dx (58) 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' (59) This relation replaces (55). The two members of (59) 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 (53), 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' (510) 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 (56). 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' (511) The second member of (511) 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 (512) 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 (515) is, moreover, demonstrated very simply by a direct method. By definition of K R(x, t) = R( x0 )e 2 te (1r2) 23S/12 r r 2 Sdx0 this equation being, in particular, satisfied when R(x, t) = 2 1 s2s eS When the two members of (514) are differentiated with respect to t and x, equation (515) is verified. Reciprocally, the integral of (515) which is reduced to R (, t0) for t = t0, can be put in the form (514). 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 LaplaceGauss law h = 0, h2 = 1 and so that the increments (515) (514) 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 LaplaceGauss law having for typical difference S(t) = t2 and that, if T = t to V77 1 + r (t, to 2 t =1 T+ /2 3 72 8 t2 (516) 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 firstorder partial differential equation OR + 1 (xR) = 0 (517) 6t 2 t ox 2See, for instance, Bass, reference 4, or "The random functions and their mechanical interpretation." Revue Scientifique, No. 3240, 1945. (515) 22 NACA TM 1377 X(t) is a differentiable random function. Its stochastic deriva tive is deduced from the expression (515) 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 (517) 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 (518) stochastic derivative of the function which has been studied as the first illustrative example. It is shown that X(t) obeys a LaplaceGauss law having S(t) = rt for typical difference and that, if to0 t: too, r'  2t = 1 O + , 2 to (519) Therefore, equation R(x, t) verifies the secondorder partial derivative. oR l &2R dt 2 ax2 (520) 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 (xxt)2 e 2(tt 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 (520) defines here the form of'the probability of passage K(x; X0, t, to) contrary to what occurred in equation (517). This example presents an unusual peculiarity. It is easily verified that the function _1 X 2 K(x; XO, t, t0) = 1 e 2ttO) (x ) (521) satisfies the functional equation K(x2; xO, t2, to) 1 fK(x2; x1, t2, tl)K xl; x0, t1, t0)dx (522) called the ChapmanKolmogoroff equation, which characterizes the Markoff processes (or more generally, the "pseudomarkovian" processes), a functional generalization of simple Markoff chains. This equation can be written in operational notation as KtrtO = Kt2t1 Kt t0 (525) 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 T10 which must be independent of t, but it itself is not. Returning to equations (512) and (515) it now is assumed that R is not dependent on t; S is then a constant and S' = 0. If r' / 0, equation (515) is written as (xR) + S42R = 0 (524) Ox 6x2 24 NACA TM 1577 The only solution of this differential equation which defines a law of probability is the density of the LaplaceGauss law x2 R(x) = 1 e 2s2 (525) 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) (526) 0 and its derivative X'(t) = dh(s) (527) 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 (528) 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 (529) Returning now to the old notations and adding the letter u to the velocity vector, we get L 1 62 L = u + ox 2 du2 (550) 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 (61) NACA TM 1577 and is such that the basic equation (56) (with original notations) takes the form .R+ R ukBR = AR (62) 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 (45) and (51)). The function H defines the correlations in the velocity field and function p is the mathematical representation of turbulent diffusion. Equation (62) 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 (62) are integrated with respect to uk, with due regard to (61). We introduce the density; of the fluid p(x, t) = R(x, u; t)du (63) (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 (64) We end with an equation of continuity k pu) =0 (65) 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 (62) 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 (66) 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 (67) >11 ut 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 (67) 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 (62), 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 (68) 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 (69) 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 (610) 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 (519)); 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 quasisteady 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 (71) 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 (72) 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 (62), 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 (72) 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 (74) 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 (74) becomes a functional equation in f . NACA TM 1577 For s = 1, (74) is, in a certain measure, comparable to (62) and reads +f f1 1 oifl 1 /6V12 + u + 7   2 dx, du2 3t Ox1 m oul m dul ox2 Equation (75) differs from (62) by the presence of the terms in f.. The presence of the second member relates (75) to (62), but its form makes it different from it. Equation (75) 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 (62) 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 (74) 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') (81) The scalar is introduced also: R = R aR = jux)ucx') (82) 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 secondrank tensor RP. They are "secondrank properties" of the statistical vector U%. They must be extended to certain thirdrank 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) (85) 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 (84) 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 (85) 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 (86) It is seen that u12 = u22 = u 2 2 u = u = ulu2 = 0 (87) 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) (88) 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 (89) 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 (91) 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 \ (92) 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) + . (93) 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 (94) g"(0) is called length of dissipation. The triplecorrelation 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) (95) NACA TM 1577 It follows from formula (89) 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 r40 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 (96) This equation is general. In the particular case of isotropic turbulence it leads to KhArman's equation g = f + (97) 2 2r NACA TM 1577 In the same manner, one then finds that: c = 2b a = b 0 (98) 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 (99) 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 (910) 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 (911) 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 (912) where U is the mean velocity. The total intensity of turbulence is the ratio u 2 U .5 U (915) 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 6x2  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 t240 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 (101) 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 An = 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 (102) 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 (105) u(xt) = fz(A,12,'5,tt)ei(Xlxl+2x2+Y x5)d. dX2 d? (104.) n1, n2, and n3 are three integers that vary independently from (nIn*2,n5) w to +mn in formula (105), and on which the quantities Z , S("1 n2) n"5) j1 32 are dependent. In formula (104), 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) (105) 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 (106) 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 (107) Now the spectrum of turbulence in the different cases (105), (104), and (105) 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 (103) or (104) 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 (105). 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 (111) 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 (112) 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 + ) = (115) 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 (114) Suppose Z (A) satisfies "Dirichlet's conditions" or, more gener ally, is a limited variation; then the integral which figures in (114) has a finite limit which, when Z,(A) is continuous, has for value 3 8 A (z)Zp(A)e P= dA (115) 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 a4m, 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 socalled 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 (114), 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? (116) 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 (117) (118) 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 (118) 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 (118). But it is much preferred to turn elsewhere. and have recourse to the representation of the velocity by Fourier's sto chastic integral (105). The product u (x)u (x + E) is a double sto chastic integral 5 Si 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 (1110) 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 (119) Z ( \hit 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 (121) 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 (97) 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 (122) NACA TM 1577 51 we must have identically Sdh = 0 (125) hence, with p being fixed When this condition is applied to the components (121) of the isotropic spectral tensor, it is seen that Ak + B = 0 We put A F B F 4ick4 4nk2 hence F(k) (124) 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 (125) 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 (123). X is (126) 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 (127) 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 (127), 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 (128) NACA TM 1577 a being the function of A. This form of (p n, similar to (118), is too restrictive; (b) S = 0 i aA = o Z b7. = 0 * = SilZ aX 2 + 2 IL b.aX12 (129) We write rather Sl a *acXa*X + S2 b fb XX and identify with the original form (125) of r. It is seen that 9aQ = Slaa*ap + S2ba b (1210) 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 (1211) NACA IM 1577 By Pythagorean theorem 'ZM2 VE 12 +1 \ a 2 + 1'IZ6b^ 2 X ( 212) (1212) 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) (1215) 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 (1214) (p (7) = C *( ) c( N) + with cM(A)h, = 0  L2 NACA TM 1577 The isotropy stipulates that c = 0, so that formula (124) 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 (151) But IcL(X) = F (132) 2itk Hence E = 1 F(k dA (155) IT; k2 In passing to the polar coordinates in the space of the wave numbers one finds that E = LF(k)dk (154) 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' (155) 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 + (156) 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} Th am+ + + ah) + 1P2 ] (138) Consequently, if E is the mean dissipation of energy per unit mass e= I cp=f Vp2p + d( iA a ~ pap~ (157) (159) 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 (1310) 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 (141) 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 (142) 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 (143) Which, after a short calculation, yields R(r) = 2]' s r F(k)dk (144) Conversely F(k) 1 r sin rkR(r)dr (145) k n 10 (b) Relations between f(r). g(r), and R(r): The integration of the differential equation yields rf' + 5f = R (146) uo or dr O= 2 dr\ /' 2 NACA TM 1577 with the condition f(0) = 0. It is uO2f(r) = rS 2R(r')dr r31 Jo (147) 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 (97) 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 (148) (149) NACA TM 1577 18 VUI f(r) = 1 r2 _E + (1410) 50 vuo2 g(r) = 1 1 r2  + 15 2 Comparison with (99) 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) (1411) 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 (151) 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)] (152) and at the same time that R(r) = 2 sin rkF(k)dk = cos ar[A(w) + 2B(u)] dao JoC rk Jo (153) 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 LdkJ O k (154) (155) NACA TM 1577 The result is A(w) = F( k )dk E(c) =1 +F(k J2 k( k2 kkdk (156) 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, (171) 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 (172) ,)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 (171) by u' and summation with respect to a, gives CL t bxl 6 X0 a p u a3 2u :au' + I c u'02 (175) 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 overall 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 + ca2uI P x % %a/ a, \ cxp2 / i 2 (174) 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 overall 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%xu 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 (175) The viscosity term reads 2vEau u' = 2vAZ7Ra 2VAR P `3 a a CLC (176) where A is the Laplacian symbol in threedimensional 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 ilzrr. f f ;...,   a;. a 'n D7 Z~ir *ts if 2 r.z , ess 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 = 301 trIi_ 8a fJ. 4;L . LSi a' "In Y .i f'IW ri * Ziff ~r ar = a __ azia NACA "Y 157 AcccrdIc to the properties of T (section 9): =a k(c + 2a) = UO c' + S The iivergence of this xec tr is equs. to i r c' + 72 ;!L c\I The f.naental e aiorn (11Z) is now transcribed: r r * r2 orI t r r r2 LC (1i1) (152) r;] In fact. it is rere=tered tnat the Laplacian of a function ndinensicnal space is ()= + r r1 w(r) in (185) 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, " (185) NACA TM 1577 This is the KarmianHowarth equation (ref. 30) in its classical form. The general equation (1710) has the form of the equation of heat propagation in ordinary threedimensional 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 (1710) 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 (185) 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) (186) c' + 4 1 a (r4c r r4 Tr\ / then one may write: t u2f) 2v 5uo2) r (187)i(rc 6t ((0) r ( T (187) It is an "equation of heat," isotropic naturally, but in a fictitious fivedimensional 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 dr = 0 provided that r4f and r4c tend toward zero when reasonable, and difficult to check. Consequently rPm, which seems uL2(t) r4f(r,t)dr (189) 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 (1710) is 24 R + T, its integral, on a sphere of center is zero. Consequently, the integral of dxl dx2 dx5 = Cte (1810) the triple integral being extended over the entire space. If the turbu lence is isotropic, it leaves r2R(rt)dr = Cte JO (1811) 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) (188) 72 NACA TM 1577 If rf f'40, r5f tends toward zero also, and this limit is zero r2R dr = 0 (1812) 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 (1812) is approximated to formula (145) 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 (1815) with C =  r4R(r)dr The constant C can be expressed by f, because, according to (147) 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 rm 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 rf 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 (189). 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 (181) is radial, that is to say, that the Tn are proportional to the E. In effect: UT5(c + 2a) T r (1815) Hence one may put: 4T = T grad r T = T grad r (1816) T(r) = uO (c + 2a) being a scalar. 19. Local form of the fundamental equation: The problem is to ascertain what the equation (1710) becomes when r40. 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 (191) dt .4 Recalling that, according to paragraph 14: u 2 = 2E 2 = 10v 5 t The formula (191) becomes therefore9 = E (192) 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 (192) is valid in more general cases. In fact, (192) can be derived from (1710). 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 (159) 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 (194) 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 (192), limit of (1710) 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 (195) 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, (196) 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 (1710) becomes the equation of heat aR = 2v AR (201) 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 (201) must thus be resolved knowing R for t = 0, and being aware that, if r40, 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  +  (202) at \,2 r or To satisfy the limiting conditions, elementary solutions of the form R(r,t) = R1(t)R2(s) s = r (205) S8bvt are necessary. Hence t'(t) 2(s) i tH(t=  L + s + )R'2(s) (205) 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 (204) or, reduced to classical form, by the transformation, sR2 = es H (205) Quantity H confirms in fact the Herrritian differential equation H" 2sH' + (Q 4)H = 0 (200) 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 (207 ds and R2(s) s2y(s) (208) 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 tlH2p1 r (209) The general solution is a superposition of elementary solutions: r2 R(r,t) = e r A Hrpe1 r p=l tp H / t (2010) 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 r4o, whatever the nonnegative 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 (2012) f would not converge toward zero when r)4o, which proves that this example cannot be suitable for a real motion. 2. (KarmanHowarth (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 (2011) NACA TM 1577 it then yields R = u02(t)[3f(s) + sf'(s) (2015) 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 H2pl. The function f(s) satisfies a differential equation similar to (204) written by Karman and Howarth. But a great advantage accrues from the use of function R and reduction to Hermitian polynomials. Remark. According to (2010), 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 (1810) 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 (211) 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(*) (212) 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 (1814), defines the triple correlations. Considering the relation (191) 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 (215) or, schematically arranged: 1a1(t)0P(') + a2(t)2() + a.5(t)8P(*) + c4(t)P4(4) = 0 (214) Equation (215) 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 (214) be resolved? NACA TM 1577 The mi(t) are considered as the coordinates of a point A in fourdimensional 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 fourdimensional space. Hence there are, a priori, three possible cases: 1. (A) is a line passing through the origin. (B) is in the com plementary threedimriensional space orthogonal to this line. 2. (A) and (B) are in two completely orthogonal planes passing through the origin. 35. (A) is in a threedimensional 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 (215), 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 (191), which we recall here t2 = _lOvu (191) 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 (216) 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 (215) becomes S(22)'a'(0) + a"(;.) + 2a '(.) + 5a(') + 1;jp, '(l) + 0() = 0 (217) and is represented schematically by (t)p( ) + a2(t)02() + a(t),() = 0 (217) The discussion then deals with threedimensional 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 tiree constants ml, 2,m5, so that 84 [1ACA TM 1577 + 2 W) + 5a(b) 3'() 0) 4vul m2 2m S2 (218) 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 (191) J2 = 4vt 2t/2 = Cte Rt5/4 = Cte which is incompatible with the second equation (218). Following this, uO and ?h are solutions of the system of differ ential equations: mlX 2 2 + "0'= 0 / ,2 (219) "O =lov  a({.) verifies the secondorder differential equation: O" + n 2 + 5a = 0 (2110) iq 4vml, NACA TI1 1577 after which [ is determined by the firstorder differential equation of + = AVa' S2vml (2111) 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 (219) is set in the form DR lk vk BQ 10v (2112) 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 (2115), R and U = v=. The equation (2110) generalizes equation (204) 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 (2115) (2114) 12k 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 (2115) 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 fundamental 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 (1710) in spectral terms. (Compare spectral intermedium, section 19.) By (1110): 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, (224) 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. (225) 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 (224) reads Equation (224) reads 6F + 2vk2F = 2nk2 r at (227) 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 (225) 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 (22t) is not .ssentially different from (1710). 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 (228); 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() (251) 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 (251) must be introduced in the equations (252) + U oxl P CJX(L u where U is the constant overall 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() (254) 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 mrIanrs 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 (255) is then resumed. The first equation (255) 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 () (256) a fSm*( '1( widl> considering the equation of continuity, gives 7, d O 2 + 2vk2 j Wt7) = R(Z) (257) ot R(Z) denotes tie real part of 2. After averaging, it is known that when the turbulence is isotropic: 2nk 2 2nkr 1IACA TM 1577 Consequently d2 F2iF + 2vkF) = R(Z) (258) 2rk2 /t What was called I in paragraph 22 is given by the equation 2nk2R(Z) = d (259) 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). Spacetime 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 (241) 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 (242) 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) (245) 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,, (244) 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') + (245) 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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