TURBULENCE1

T. Kármán

1. DEFINITION OF TURBULENCE

Let us begin by considering the definition of turbulence given by Taylor: “Turbulence is irregular motion observed in liquids and gases when they flow along solid surfaces, or when two layers flow past one another.”

The emphasis in this definition lies on the word “irregular.” Taylor adds further: “The actual motion is usually so irregular that its details are little known.”

Naturally the question arises: why are we interested in so irregular a phenomenon, and what grounds have we for hoping that its study will lead to practical results?

But turbulence is by no means the only irregular phenomenon that physics attempts to analyze. Rather, one may assert that regular motion occurs in nature as an exception. Even laminar or jet motion appears regular only to an observer who views the world of molecules from so great a distance and with the aid of such crude instruments that he is able to see only the averaged motion of matter and to measure only averaged values of physical quantities. Only then, despite the irregular character of the actual motion, does he find simple relations between averaged physical quantities and simple laws of averaged motion. Instead of the words “despite the irregular character,” we ought to say: “as a consequence.” It is precisely the high degree of irregularity, the random character of molecular motions, that enables us to apply statistical methods and to obtain simple relations—not perfectly exact, but possessing a high degree of approximation and satisfactory in almost all cases, with rare exceptions.

Since the investigations of Reynolds we have known that turbulence is a statistical phenomenon. It was said above that turbulence arises when a fluid flows along solid surfaces. It is known that if a fluid flows around a body having an insufficiently “streamlined” shape, regular vortices often arise behind the body. By recording the velocity behind the body with an oscillograph, we detect periodic fluctuations of it, as is shown in

in the upper part of the oscillogram in Fig. 1. The lattice in the form of honeycombs, shown in Fig. 2, forms a multitude of vortices. As the distance from the lattice increases the vortices lose their regular character, and the oscillograms in the region of irregularity are likewise very irregular (the lower part of Fig. 1). In this region there are no longer traces of the regular systems of vortices that existed immediately behind the lattice, nor are there traces of the lattice itself. Before us is a uniformly distributed vortex motion; statistically, at every point of this region the velocity fluctuations are the same, and their components in different directions have equal values. To an observer moving together with the mean flow, all directions appear equivalent. Such a type of flow is said to have “isotropic turbulence.” In what follows we shall see that this is the simplest case of turbulent motion.

Wherever in nature we observe the motion of a liquid, we always come to the conclusion that jet-like, i.e. irregularity-free, motion is a rarity; the flow of water in rivers and the motion of air in the atmosphere are motions of a turbulent character. The motions of liquids with which the engineer has to deal are also in most cases turbulent. Examples of laminar motion are the motion of oil during lubrication, of water or steam through small openings and narrow tubes, and flows of very viscous liquids, such as molten metals and glass.

Naturally the question arises: what is the practical effect of the presence of irregularity in motion; why are we so interested in irregular motion superposed upon the mean motion? The answer is that the presence of irregular motion radically changes the order of magnitude of viscous resistance, heat transfer, and diffusion in liquids. Thus the problem of turbulence arises practically in all branches of engineering in which the motion of liquids plays a role, regardless of whether the liquid is the medium in which a body moves, or whether it itself moves within certain fixed boundaries, or whether heat transfer, diffusion, mixing, dissolution, evaporation, combustion, etc., take place in the liquid.

2. MECHANISM OF DIFFUSION

The influence of irregular motion on friction can be illustrated in the following graphic way. Let us imagine soldiers marching along a street in regular parallel columns. If each column has one and the same step, then the order of the march is not disturbed. Let us now imagine the same number of people who move irregularly among the soldiers, on average moving together with the latter. Undoubtedly, if one of the columns receives the command to outrun the others, this will be made difficult for it by the presence of the irregularly moving people. In this case there exists “internal friction between the columns.” Applying this picture to the motion of a gas consisting of molecules, we easily understand the process by which internal friction arises between two layers of gas, slid-

To the article by T. Kármán

Fig. 1

Fig. 2

...following one another, since molecules from one layer will penetrate into neighboring layers. Let us, however, consider this process of penetration in more detail. The tendency of a molecule to pass from layer to layer is due to the presence of continual molecular motion; collisions between molecules hinder this transition. Let us follow the fate of a certain group of molecules in one of the layers; for simplicity let us suppose that the molecules move only perpendicular to the layer. We shall replace the effect of collisions with other molecules by the following scheme: all molecules move by different jumps at different intervals of time, but whether a given molecule goes to the right or to the left is determined purely by chance. If, in the actual state of the gas, collisions are random, then such a scheme imitates them rather well.

Fig. 3

The diagram in Fig. 3 represents the distribution of 100 molecules after a certain number of jumps and is the result of an experiment: whether a molecule would jump to the right or to the left was decided by the tossing of a coin. As a measure of the diffusion of the molecules we shall choose the mean square of their distances from their initial position over the time \(n\Delta t\), where \(\Delta t\) is the time interval between two successive jumps, and \(n\) is the number of these jumps. Let us denote the number of molecules by \(N\); the number of molecules that have made \(p\) jumps in one direction by \(N_p\); finally, the length of one jump by \(\Delta a\). Then the mean square of the distances traversed is:

\[ \overline{S^2}=\frac{(\Delta a)^2}{N}\sum_{p=-n}^{p=+n} N_p P^2 . \tag{1} \]

The ratio \(P_p=\dfrac{N_p}{N}\) is called the probability that a certain molecule, after making \(n\) jumps, will move forward by \(p\) jumps. The theoretical values of \(P_p\), more precisely, the limiting values for very large \(N\), are given in Table 1.

TABLE 1

\(n\)	\(-6\)	\(-5\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(\sum P_p v^2\)
1						\(\dfrac{1}{2}\)		\(\dfrac{1}{2}\)						1
2					\(\dfrac{1}{4}\)		\(\dfrac{1}{2}\)		\(\dfrac{1}{4}\)					2
3				\(\dfrac{1}{8}\)		\(\dfrac{3}{8}\)		\(\dfrac{3}{8}\)		\(\dfrac{1}{8}\)				3
4			\(\dfrac{1}{16}\)		\(\dfrac{4}{16}\)		\(\dfrac{6}{16}\)		\(\dfrac{4}{16}\)		\(\dfrac{1}{16}\)			4
5		\(\dfrac{1}{32}\)		\(\dfrac{5}{32}\)		\(\dfrac{10}{32}\)		\(\dfrac{10}{32}\)		\(\dfrac{5}{32}\)		\(\dfrac{1}{32}\)		5
6	\(\dfrac{1}{64}\)		\(\dfrac{6}{64}\)		\(\dfrac{15}{64}\)		\(\dfrac{20}{64}\)		\(\dfrac{15}{64}\)		\(\dfrac{6}{64}\)		\(\dfrac{1}{64}\)	6

Fig. 3 shows what values of \(\dfrac{\overline{S^2}}{(\Delta a)^2}\) are obtained from the experiment described above. We see that for the first five jumps the experimental values of this quantity are approximately equal to 1, 2, 3, 4, 5, as indicated in the last column of our table. And indeed, in probability theory it is proved that if the number of molecules \(N\) and the ratio \(\dfrac{N}{n}\) are large, then \(\dfrac{\overline{S^2}}{(\Delta a)^2}\) is very close to \(n\), i.e., to the number of jumps. Hence \(\overline{S^2}=\dfrac{(\Delta a)^2}{\Delta t}\,t\), i.e., the increase of the mean-square displacement per unit time is a constant quantity equal to

\[ \frac{\overline{S^2}}{t}=\frac{\Delta a}{\Delta t}\,\Delta a. \tag{2} \]

In the case of the irregular motion of gas molecules this formula may be interpreted in the following way: the square of the distance traveled in a definite direction per unit time is proportional to the mean velocity of the molecules, in our case \(\dfrac{\Delta a}{\Delta t}\), and to the length of the path between two collisions, i.e. \(\Delta a\). We may also say that the mean-square distance traveled per uni-

turbulence

over time, proportional to the square of the mean free path, multiplied by the number of collisions per unit time.

The quantity \(\frac{\overline{S^2}}{t}\), proportional to the mean free path times the mean velocity, is a measure of displacement, and it can be shown that laminar friction, molecular thermal conductivity, and molecular diffusion are proportional to this quantity.

As a first illustration, let us consider two kinds of molecules, initially arranged in parallel columns and in such a way that the line \(BC\) separates the molecules of one kind from the molecules of the other (Fig. 4). Let us imagine that the molecules of each column make jumps according to the same law as in the preceding experiment, and let us count the number of molecules penetrating into alien territory during the time \(t\) (it is understood that only those molecules are counted which, after the lapse of time \(t\), find themselves in alien territory).

Fig. 4

The classification of the molecules which, after the lapse of time \(t = n\Delta t\), will be in alien territory can be made as follows: those which have made one jump, those which have made two jumps, and so on, and finally those which have made \(n\) jumps. Those molecules which have made \(p\) jumps forward and have entered alien territory were obviously located in a frontal strip of width \(p\Delta a\). If \(N\) denotes the number of molecules arranged along the front, then the total number of molecules in the strip \(ABCD\) is \(pN\), and the number of those which have advanced by \(p\) jumps is \(P_pN\). Consequently, the total number of molecules that have moved into alien territory during the time \(n\Delta t\) is

\[ N\sum_{p=1}^{n} pP_p. \]

The calculation shows that, for large values of \(N\), this quantity is proportional to \(\sqrt{n}\), i.e. to the square root of the elapsed time. It follows from this that the fraction of migrants decreases in proportion to the square root of time. It is not difficult to imagine that with the passage of time the strip of the front is populated by an increasingly mixed population, so that further migration in both directions changes its composition less and less. Speaking in scientific terms and calling the percentage ratio of one kind of molecule to the total number the concentration, we shall say that the concentration decreases with time and, consequently, diffusion slows down.

Figs. 5a, b, c, d illustrate this process of mixing. Fig. 5a shows the initial distribution of white and black molecules. In the next figure the process is shown schematically, on the assumption that each column moves across the front as a whole, but according to the law of chance defined by Table 1. In

Fig. 5a

at each jump the column advances by 2 persons. Fig. 5b shows the arrangement after three jumps, Fig. 5c—after six jumps. In Fig. 5d the columns have been rearranged so that the resulting distribution can be seen more clearly.

In order to find the differential law of diffusion, let us consider an initial state in which the concentration gradient is constant, as shown in Fig. 6. In this case it is sufficient to consider the migration of molecules located in the thre-

Fig. 5b

Fig. 5c

angular spaces indicated in the figure, since this is a stationary state in which the crossing of molecules in both directions is mutually balanced (the arrangement of molecules in this figure is ordered in the same sense as was done in Fig. 5).

The molecules passing through the boundary during the time \(t = n \Delta t\) may in this case also be classified according to the number of jumps by which they advance during the time \(n\Delta t\).

Fig. 5d

Of the molecules whose displacement during this time is \(p\Delta a\), only those can cross the boundary which at the moment \(t=0\) were nearer to it than \(p\Delta a\), i.e. those which are situated in the triangle \(ABC\). If the number \(N\) of molecules of one kind increases by \(\dfrac{dN}{da}\) from column to column, then the area \(ABC\) is equal to \(p^{2}\Delta a \Delta b \dfrac{dN}{da}\) (where \(\Delta b\) is the mean distance between molecules of one column), and the number of molecules on this area is \(p^{2}\dfrac{\Delta N}{\Delta a}\). Consequently, the total number of molecules crossing the boundary in the time \(n\Delta t\) is equal to

\[ \frac{dN}{da}\sum_{1}^{n} p^{2}P_p . \]

But we have seen that

\[ \frac{\overline{S^{2}}}{(\Delta a)^{2}} = \sum_{p=-n}^{p=n} p^{2}P_p . \]

Therefore the number of molecules crossing the boundary in the time \(t\) is equal to

\[ \frac{1}{2}\frac{\overline{S^{2}}}{(\Delta a)^{2}}\frac{dN}{da}, \]

and per unit time is equal to

\[ \frac{1}{2}\frac{\overline{S^{2}}}{t(\Delta a)^{2}}\frac{dN}{da}. \]

Fig. 6

Extending our reasoning to volume and denoting by \(N\) the volume density of some physical quantity transported by the molecules, we find that the flux of this quantity through a unit surface is equal to

\[ \frac{1}{2}\frac{\overline{S^{2}}}{t}\frac{dN}{dx}; \]

we shall call the parameter

\[ \frac{1}{2}\frac{\overline{S^{2}}}{t} \]

the coefficient of diffusion or exchange.

The laws of collision of molecules of a real gas, such as, for example, air, are much more complicated than the conditions given above. Accordingly, the various coefficients of exchange as well—for example, the coefficient of transfer of momentum (friction), thermal conductivity, and diffusion of suspended particles—are not numerically equal to

\[ \frac{\overline{S^{2}}}{2t}, \]

but nevertheless are proportional to this quantity.

In the case of molecular or laminar friction, the coefficient of exchange of momentum is nothing other than the coefficient of kinematic viscosity. We may present the results of the discussion of our two examples in the following way. Wherever the exchange process begins from a discontinuous distribution, as in the first example, the amount of diffused substance is proportional to \(\sqrt{t}\); where, as in the second example, we have a stationary state, the measure of transfer is the quantity

\[ \frac{\overline{S^{2}}}{2t}, \]

multiplied by the gradient of the transported quantity.

If thermal conductivity and diffusion of suspended particles were governed only by the molecular mechanism, then for these processes

would retain the force of similar laws. But in the case of turbulent motion the carriers of momentum or heat are not as distinctly defined as in the case of the molecular mechanism. Nevertheless, we are still able to compute the corresponding exchange coefficients from observations of the coefficients of turbulent friction, turbulent thermal conductivity, turbulent diffusion, and by proceeding from the assumption that these coefficients are a measure of the mean square distance traversed by the liquid masses—for example, by eddies—which are the carriers of the quantities of interest to us in turbulent exchange.

Let us consider, for example, the stress tensor, equal to the quantity of motion transported across a unit area in a unit time; the component of interest to us is \(\tau=\varepsilon \dfrac{\partial(\rho u)}{\partial y}\), where \(u\) is the mean velocity, \(y\) is the coordinate perpendicular to \(u\), and \(\rho\) is the density of the liquid. We shall call the quantity \(\varepsilon\) the coefficient of turbulent exchange and identify it with \(\dfrac{\overline{S^2}}{2t}\), where \(S\) is the distance traversed by the eddies, or the degree of their mutual penetration. As we shall see below, because of the large dimensions of the objects determining turbulent exchange, the laws of the latter are more complicated than the laws of molecular exchange, and, generally speaking, \(\dfrac{\overline{S^2}}{2t}\) not only changes from phenomenon to phenomenon, but even within one and the same phenomenon does not remain constant. Nevertheless, the introduction of the coefficient of turbulent exchange greatly facilitates a general understanding of the phenomenon.

3. Examples of Turbulent Exchange

The first result in this field is the fact that the coefficient of turbulent exchange is in all cases, except in the immediate vicinity of walls, of a considerably higher order than the coefficient of molecular exchange.

A few examples are sufficient to show how great the difference between these two exchange coefficients is.

As a first example, let us consider a point source of gas. Suppose, for example, that this gas carries with it heat or suspended particles—smoke. Let the amount of heat or smoke emitted per unit time be constant; let the magnitude of diffusion at some distance from the source be measured by the cross section through which, say, 90% of the issuing quantity passes; and let, finally, the outflow velocity be 30 ft. per 1 sec. In the case of laminar diffusion, the radius of the cross section of a jet containing 90% of the total amount of heat or smoke supplied by the jet (neglecting the weight of the smoke particles) will be \(1/2\) dm at a distance of 30 ft. and 5 dm at a distance of 3000 ft. But with the slightest turbulence of the jet, heat and smoke will be dispersed much more rapidly than these figures indicate. The cloud of smoke produced by the explosion of a shell is another such example. Treating the exploding shell as a point

source and measuring the diameter of the visible part of the gas cloud, we can determine the mean square distance traveled by the particles ejected from the center. Fig. 7 gives the values of \(S^2\) as a function of time, obtained experimentally. The dotted line indicates \(\overline{S^2}\) as a function of time for molecular diffusion, with the scale of ordinates increased 10,000 times.

Fig. 7a

It is clear that only the turbulent character of the motion of water in a river or of air in the wind accounts for the displacement of silt in a river and, in the air, of clouds of dust. Let \(a\) be the mean diameter of silt particles; we can estimate, starting from an assumption about the magnitude of the coefficient of turbulent exchange, how far dust particles raised by the wind will be carried [Figs. 8 and 9 present photographs of dust storms taken by the Soil Conservation Service (Organization for combating soil destruction)]. It is known that the velocity of fall of small spherical particles is well described by Stokes’ law

Fig. 7b

\[ w=\frac{2}{9}\,\eta\,\frac{a^2 g}{\nu}, \tag{3} \]

where \(\eta=\dfrac{\rho_s-\rho_a}{\rho_a}\), and \(\rho_s\) and \(\rho_a\) are, respectively, the densities of the solid particles and of the air. On the other hand, the mean square distance to which particles are carried in time \(t\) as a result of turbulence is determined by the coefficient of turbulent exchange according to the formula

\[ h=\sqrt{\overline{z^2}}=2\varepsilon t. \tag{4} \]

Putting \(h=wt\), we find

\[ t=\frac{2\varepsilon}{w^2} =\frac{81\,\varepsilon \nu^2}{2\eta^2 g^2 a^4}, \tag{5} \]

and this formula gives us an estimate of the time during which the particles remain suspended in the air. Hence, denoting by \(U\) the mean wind velocity, we also find the distance of transport

\[ l = Ut \sim \frac{40 \varepsilon^{2} U}{\gamma^{2} g^{2} l^{4}} . \tag{6} \]

Fig. 8

The coefficient of turbulent exchange varies within wide limits depending on the strength and structure of the wind. The following calculations were made under the assumption \(\varepsilon = 10^{4}\) and \(\varepsilon = 10^{5}\). The value of \(\varepsilon\) for winds of moderate strength lies precisely within these limits,

Fig. 9

with the exception of the first few feet above the surface of the earth, where \(\varepsilon\) decreases with height (Table 2).

TABLE 2

Particle diameter in mm	Settling velocity in 1 sec.	Flight time	Distance in miles with a wind of 15 m/sec.	Maximum rise
0.001	0.00824	9–90 years	\(2.5\)—\(25 \cdot 10^6\) miles	3.8–38 miles
0.01	0.824	8–80 hours	250–2,500 miles	200–2,000 ft.
0.1	82.4	0.3–3 sec.	150–1,500 ft.	2–20 ft.
1.0	8,240	0.0003–0.003 sec.	0.2–2 in.	0.3–3 in.

The figures in the last row are of only academic interest, since for such large particles Stokes’ law already loses its force. The figures in the first row are also doubtful, if only because \(v\) has been taken as independent of height. Nevertheless, Table 2 gives a vivid idea of how strongly the flight time changes as the size of the particles changes. This explains why the size of the particles found in dust storms lies within relatively narrow limits. But small particles remain suspended for a very long time; an excellent example of this is the well-known fact that ash particles ejected during the eruption of the Krakatoa volcano remained in the upper layers of the atmosphere for many years and were carried even to England and the Arctic.

4. WHEN DOES TURBULENCE HARM THE AIRCRAFT ENGINEER?

Surface friction is perhaps the most important factor limiting the speed of the modern airplane. Some types of modern wings are aerodynamically so perfect that no less than 60% of their drag is due to surface friction. The fact that air resistance is caused by surface friction and by the inertial resistance of the medium (its modern name is “form drag”) was known already to Newton. From dimensional considerations he showed that the resistance of surface friction is proportional to the first power of the velocity, whereas the inertial part of the resistance is proportional to the second power of it. Experimental investigations of surface friction date back to the eighteenth century. Bossut was the first to make measurements of friction on flat surfaces; Froude’s systematic measurements are well known. The fact that the characteristic parameter of the problem is the Reynolds number (the flow velocity at

TURBULENCE

the width of the plate divided by the kinematic coefficient of viscosity), was indicated by Lord Rayleigh. The theoretical explanation of the mechanism of surface friction belongs to a comparatively recent time. Our modern conception of surface friction is in agreement with the Newtonian law governing it: for a flow of the given form, the friction is indeed proportional to the velocity. But the very form of the flow around a body is independent of the velocity only so long as the forces of inertia are small in comparison with the forces of viscosity, i.e. only in the case of small Reynolds numbers. In this case Stokes’ law correctly determines the drag force. When, however, the forces of inertia cannot be neglected, the pattern of the flow changes sharply. Let us imagine, for example, a flat plate placed in the direction of the flow, whose velocity in the absence of the plate is assumed everywhere constant in magnitude and direction. We know that the action of the forces of viscosity is concentrated only in the region close to the plate, in the so-called boundary layer. According to the well-known law of mechanics, the loss of momentum in the boundary layer is equal to the force of viscous resistance. Denote by \(U\) the velocity of the flow relative to the plate, and by \(\delta\) the thickness of the boundary layer; then the resistance force acting on a unit width of the plate (taking the width in the direction of the flow) is equal to

\[ F = a\rho \delta U^{2}, \]

where \(\rho\) is the density of the fluid and \(a\) is a coefficient depending on the distribution of velocity in the boundary layer. If in the boundary layer the flow is laminar, then the friction force per unit length is proportional to the velocity gradient perpendicular to the plate and is equal to \(\dfrac{\beta\mu U}{\delta}\), where \(\beta\) is a new coefficient of viscosity. Consequently,

\[ F = \beta\mu U \int_{0}^{x}\frac{dx}{\delta}. \tag{7} \]

From comparison of these two expressions it follows that

\[ \alpha U^{2}\frac{\partial \delta}{\partial x}=\beta\nu\frac{U}{\delta} \]

or

\[ \delta^{2}=2\frac{\beta}{\alpha}\nu\frac{x}{U}. \tag{8} \]

With increasing velocity the flow is “pressed” against the plate. The resistance force is proportional to \(U^{\frac{3}{2}}\) and \(\sqrt{\rho\mu}\), which indicates the dependence of surface friction not on the forces of viscosity alone, or on the forces of inertia alone, but on both. Taking into account that \(\dfrac{x}{U}\) is the time \(t\) during which the flow passes along the plate from its leading edge to the point \(x\), we may write equation (8) in the form

\[ \frac{\delta^{2}}{t}=\nu\cdot \text{const.} \]

Here the analogy with the basic law of molecular diffusion considered in § 1 stands out clearly, and consequently the thickness of the boundary layer may be regarded as a measure of the penetration of the action of surface friction into the undisturbed flow.

The reader familiar with the usual theory of the boundary layer will easily recognize here the principal results of this theory. Having determined the value of the shape coefficients \(\alpha\) and \(\beta\), the theory leads to an expression for the coefficient appearing in the expression

\[ F=C_f\frac{\rho U^2}{2}, \]

in the form

\[ C_f=\frac{1.33}{\sqrt{R_x}}, \tag{9} \]

where \(R_x=\dfrac{Ux}{\nu}\). For the wings of modern airplanes the Reynolds number is about \(25\cdot 10^6\). Consequently, the coefficient of surface friction for such surfaces is

\[ C_f=0.00053, \]

provided that turbulence does not come into play. But for the same Reynolds number the actual value of \(C_f\), determined by the presence of turbulence, is \(C_f=0.0025\).

The theory that underlies our present knowledge of turbulent friction is far less developed than the theory of laminar friction.

We define the transfer of momentum through a plane parallel to the vector of the mean velocity as a component of the stress tensor caused by turbulent friction; its mathematical expression is \(-\rho \overline{uv}\), where \(\overline{uv}\) is the mean product of the velocity component in the direction of the flow and the velocity component normal to the above-mentioned plane. Hence the component of the total friction tensor is

\[ \tau=\mu\frac{d\overline{u}}{dy}-\rho\overline{uv}. \tag{10} \]

Introducing here the coefficient of turbulent exchange, we have

\[ \tau=\nu+\varepsilon\frac{\partial \overline{u}}{\partial y}. \tag{11} \]

A complete theory of turbulent friction ought to give both the distribution of the velocity fluctuations and the distribution of the mean velocity, or, what is essentially the same thing, the magnitude and distribution of the coefficient of turbulent exchange \(\varepsilon\). There is as yet no sufficiently developed theory that could give this without resorting to arbitrary and sometimes unsatisfactory hypotheses. However, the following facts appear to be sufficiently reliably established by the combined application of experimental and theoretical—

ERRATUM

through the fault of the translator

p.	line	printed	should read
33	28 from top	where $\beta$ is the new coefficient of viscosity.	where $\beta$ is the new coefficient, and $\mu$ is the coefficient of viscosity.

—mostly from dimensional theory considerations. First of all, we shall consider the case of perfectly smooth plane surfaces:

a) In the immediate vicinity of the wall the laminar friction considerably exceeds the turbulent friction; here

\[ \mu \frac{\partial \overline{u}}{\partial y} \gg -\rho \overline{uv}. \]

We shall call this layer the laminar sublayer. Its thickness \(\delta_1\) is found from the relation

\[ \frac{\delta_1}{\nu}\sqrt{\frac{\tau}{\rho}}=\mathrm{const}, \]

where \(\tau\) is the friction per unit area acting on the wall. Denoting the velocity at \(y=\delta_1\) by \(U^*\), we may approximately put

\[ \frac{\tau}{\rho}=\nu \frac{U^*}{\delta_1}, \]

and therefore the Reynolds number in the sublayer is

\[ R_1=\frac{U^*\delta_1}{\nu} \]

and is constant along the length of the plate; consequently, it is the characteristic number for the laminar sublayer. Its value is \(R\sim 200\text{--}300\).

b) Beyond the laminar sublayer there follows a transition region of some thickness, beyond the limits of which the coefficient of turbulent exchange is proportional to the distance from the wall. Consequently, the gradient of the mean velocity \(\partial \overline{u}/\partial y\) is inversely proportional to the distance from the wall. From experiment we find

\[ \frac{\partial \overline{u}}{\partial y} = \frac{1}{k} y \sqrt{\frac{\tau}{\rho}}, \tag{12} \]

where \(k\) is a universal constant, close to \(0.4\). Integrating (12), we find the distribution of the mean velocity

\[ \overline{u} = \sqrt{\frac{\tau}{\rho}} \left[ A+\frac{1}{k}\lg\frac{y}{\nu}\sqrt{\frac{\tau}{\rho}} \right]. \tag{13} \]

Comparing (12) with (11), we see that the physical meaning of (12) consists in the fact that the coefficient of turbulent exchange increases with distance from the wall. Indeed,

\[ \varepsilon = ky\sqrt{\frac{\tau}{\rho}}. \]

Equation (12) is valid in the region where the variations of \(\tau\) are small. But as \(y\) increases, \(\tau\) decreases and becomes practically zero at the outer boundary of the boundary layer. At present there is still no theory that would allow us to calculate the distribution of \(\overline{u}\) in the outer part of the boundary layer.^1

c) Fortunately, equations (12) and (13) are sufficient for computing the coefficient of surface friction, and also for computing

^1 An attempt in this direction is made by Howarth, Proc. Roy. Soc. A. 154, p. 364, 1936.

of approximate boundary-layer thickness as a function of the Reynolds number. Since the theory of turbulent skin friction, based on (12) and (13), has been developed in detail in a number of monographs in recent years, it is sufficient to set forth the results obtained. The coefficient of total skin friction is given by the formula

\[ \frac{0.242}{\sqrt{C_f}}=\lg (RC_f), \tag{14} \]

where

\[ R=\frac{Ul}{\nu} \]

(\(l\) is the extent of the flow in its direction). The coefficient of local skin friction (per unit length at a distance \(x\) from the leading edge of the plate) is obtained from (14) as

\[ c_f=\frac{C_f}{1+0.36\sqrt{C_f}}, \tag{15} \]

where \(C_f\) is the quantity obtained from (14) under the assumption

\[ R=\frac{U_x}{\nu}; \]

the boundary-layer thickness is readily computed from the value of \(c_f\)

\[ \delta=0.38\,x\sqrt{c_f}. \]

The quantity \(c_f\) decreases only slightly with \(x\); therefore the thickness of the turbulent boundary layer increases almost proportionally to \(x\), whereas the thickness of the laminar boundary layer increases proportionally to \(\sqrt{x}\).

Formula (14) was first given by the author of these lines; the constant in this formula was computed by Schönherr. Schlichting found that the empirical formula

\[ C_f=\frac{0.455}{(\lg R)^{2.58}} \tag{16} \]

is a good substitute for (14) throughout the entire range of Reynolds numbers of practical importance. Its advantage is the explicit dependence of \(C_f\) on the Reynolds number.

The results set forth in items a), b), and c) are useful in various calculations and, in particular, in extrapolating from test data in wind tunnels. If, for example, the share of skin friction in the total resistance is known, the reduction of the latter, depending on the Reynolds number, can be estimated by the formula given above. It is true that there are also factors, still unaccounted for, which act together with those already known, and in order to give a correct picture of all our knowledge about skin friction, this should be mentioned.

The total frictional resistance of smooth surfaces consists of friction on the forward part of the surface, where the boundary layer

laminar, and friction in the rear part, where we are dealing with a turbulent boundary layer. If \(R=\dfrac{Ul}{\nu}\) (where \(l\) is the full width of the surface in the direction of the flow) is large, then the friction in the laminar part of the layer may be neglected. For smaller values \((10^5<R<10^6)\), both resistances are of the same order; and, expressing the resistance as a function of the Reynolds number, we encounter the so-called transition region, in which the drag coefficient expressed by curve (9) passes into the coefficient expressed by (14). True,

Labels in the figure: \(C_f\); \(R\); “Turbulent boundary layer (Kármán 1931)”; “Wieselsberger 1921”; “Zahm 1904”; “Froude 1915”; \(R_\delta=3000\); \(R_\delta=4000\); \(R_\delta=5000\); “Baudry 1793”; “Froude 1872, Froude 1915, Gibson 1915 (Kempf 1929)”; “Gebers 1913–18”; “Laminar theory (Prandtl–Blasius 1904).”

Fig. 10

the transition from laminar to turbulent resistance does not in all cases occur in the region of the same Reynolds numbers, but depends very strongly on the shape of the leading edge, or, in other words, on the character of the oncoming flow, and especially on the degree of turbulence of the latter. Usually, experiment yields not a single transition curve, but an entire family of them (Fig. 10). It is precisely such experiments, carried out in recent years for very large Reynolds numbers, that have clarified the complete picture of this transition. In earlier experiments, however, the curves of the transition region appeared confused and contradictory.

For the aeronautical engineer an important practical conclusion follows from this: measurements of profile drag have little value if the models or the speeds in the wind tunnel are so small that the Reynolds number falls in the transition region—because in that case the magnitude of the surface friction depends strongly on the degree of turbulence of the flow in the wind tunnel.

The magnitude of turbulent friction on a smooth plate is the lower limit of the possible profile drag of an airplane and at present represents the chief obstacle to increasing flight speed. The question naturally arises whether there is not some way to “outwit nature” and, by some clever means, reduce surface friction. It is hardly feasible, however,

to stabilize the boundary layer for those Reynolds numbers which pertain to modern airplanes. Nevertheless, it may be possible to prevent the transition of the surface layer into an unstable state by removing excess air from the surface. The quantity of air to be removed is approximately determined by

\[ C_1 S \frac{U}{2}, \]

where \(C_1\) is the magnitude of the coefficient of laminar friction, which must be maintained by the appropriate device, and \(S\) is the surface on which this is to be done.

If one assumes that \(C_1\) is identical with the coefficient of local laminar friction up to the transition of the latter into turbulent friction, then \(C_1\) may be put equal to \(0.00115\)—\(0.002\). So far no practical results in this direction have yet been published.

The next question is how close we can come to the limit of resistance given by a smooth plate. Here we encounter the problem of roughness. Summarizing recent investigations on this question, we can say with certainty the following:

a) Apparently, the influence of roughness on surface friction is insignificant below a certain value of the Reynolds number. The physical reason for this behavior of a rough surface is apparently the circumstance that in this case the roughness elements are small in comparison with the thickness of the laminar sublayer1. It has already been said that the thickness of the latter is given by the relation

\[ \sqrt{\frac{\tau}{\rho}}\cdot\frac{\delta_1}{\nu}=\mathrm{const}, \]

and since friction increases with the Reynolds number, \(\delta_1\), as the Reynolds number increases, decreases. Thus, with an increase of the Reynolds number the elements of unevenness rise out of the laminar sublayer like mountains rising out of fog as it settles (Fig. 11).

Fig. 11

Reynolds number, \(\delta_1\) decreases as the Reynolds number increases. Thus, with an increase of the Reynolds number the elements of unevenness rise out of the laminar sublayer like mountains rising out of fog as it settles (Fig. 11).

b) When the height of the elements of unevenness is large in comparison with the thickness of the laminar sublayer, the surface friction, apparently—

is determined by the frontal resistance of these elements; correspondingly, the surface friction itself, at Reynolds numbers above a certain limit, becomes proportional to the square of the relative velocity of the medium and of the solid body moving in it. This limiting case has been studied by many investigators; in particular, the pressure loss was measured in aerodynamic tubes of constant cross-section and with artificially roughened walls. It was found that, for a constant degree of roughness (for the same walls), the coefficient of surface friction is proportional to the logarithm of the so-called relative roughness, i.e. the ratio of the height of the roughness elements to the radius of the tube. The coefficient of surface friction is determined by the equation

\[ \frac{2}{\sqrt{C_f}}=A+\frac{1}{k}\lg\frac{r}{h}, \tag{17} \]

whence

\[ C_f=\frac{4}{\left(A+\frac{1}{k}\lg\frac{r}{h}\right)^2} \tag{18} \]

(\(r\) is a linear dimension, for example the hydraulic radius of the tube, \(h\) is the height of the elements of irregularity). Equation (17) was first given by the author in 1929. The constant \(k\) is very close to the constant denoted by the same letter in equation (13) for the velocity distribution. Equation (17) makes it possible to establish a scale of roughness. Obviously, having assigned \(h\) arbitrarily for some rough object, for example for sandpaper of a definite grade, and having measured the coefficient of surface friction in a tube lined with this sandpaper, we can determine, from equation (17), the value of \(h\) for a surface of any kind.

From the foregoing, several important conclusions may be drawn. First, it becomes possible to estimate, for a given velocity and given dimensions, the thickness of the laminar boundary sublayer and to calculate the upper limit of permissible roughness, i.e. to determine the limits of roughness at which it does not noticeably affect the magnitude of the surface friction. It should be borne in mind that, for relative roughness, the thickness of the boundary layer plays the same role as the radius of a tube with rough walls.

It follows from this that, since the coefficient of surface friction is a function of the relative roughness, one and the same roughness will exert a greater influence near the leading edge, where the boundary layer is thin, than in the rear part of the wing, where the thickness of the boundary layer is considerable.

The theoretical prediction of the magnitude of the surface friction for a surface possessing a definite roughness is rather difficult, and the difficulty lies in the variability of the relative roughness along the surface, which in turn is caused by the variable thickness of the boundary layer. Hence, it is necessary to deter-

divide the increase of its thickness, in order to be able to use formula (17) for calculation, integrating over the surface in the direction of the flow.

In addition, there is still another uncertainty, due to the following. Equation (17) is valid for large Reynolds numbers. But in the boundary layer the Reynolds number varies from zero, beginning at the leading edge, and thus we are dealing with the whole range of Reynolds numbers as the boundary layer thickens downstream, especially in the transition region, where the height of the roughness elements and the thickness of the laminar sublayer are of the same order. Expressing the coefficient of surface friction as a function of the Reynolds number, we obtain a family of similar curves for geometrically similar forms of roughness; if, however, the geometric forms of the roughness elements are different, then the curves are different as well. For example, they depend on the ratio between the height and the spacing of the roughness elements, i.e., on the “relative wavelength” of the roughness. The curves of surface friction obtained for so-called “wavy” surfaces, i.e. those in which the height of the elements is small in comparison with their spacing, are of a completely different kind than the curves for surfaces with “sharp” roughness elements. Here a wide field opens up for new experimental and theoretical investigations.

Much promise for clarifying this question is offered by the direct measurement of drag resistance in flight, carried out by German investigators and developed to a high degree of accuracy by Jones in England. It is a useful supplement to experiments in wind tunnels, especially in tunnels with compressed air.

Fig. 12

Summing up all that has been said above, we must acknowledge that, in the present state of our knowledge, the determination from wind-tunnel experiments of the actual frictional resistance of wings and other parts of an airplane is not free from arbitrariness, especially because of the inadequacy of our knowledge concerning roughness.

Up to this point we have considered the turbulent friction of a fluid moving along a solid body. It is now necessary to dwell on the question of turbulent friction between two fluid layers. For this case Prandtl introduces the term “free turbulence.” Let us consider only the simplest case. Suppose that at the moment \(t=0\) two masses of air, in contact along the plane \(y=0\), begin to move uniformly one past the other, with the initial velocity for \(y>0\) being \(+U\), and for \(y<0\) the initial velocity being \(-U\) (Fig. 12). Let us first suppose that the flow is laminar. It is not difficult to find the exact mathematical solution for this case; but the main results can be obtained from the following simple argument.

Let us again introduce the thickness of the laminar layer, which we define by the relation

\[ \delta=\frac{1}{U}\int_0^\infty (U-\bar u)\,dy. \]

Then, as is easy to see, the momentum law gives an expression for the friction force in terms of \(U\) and \(\delta\), while the energy law determines \(\delta\) as a function of time. Denoting by \(\tau_0\) the shear stress at \(y=0\), we can find from the momentum law the expression

\[ \tau_0=\alpha\rho U\frac{\partial\delta}{dt}, \tag{19} \]

where \(\alpha\) is a numerical coefficient of form. The energy law says that the loss of kinetic energy is equal to the work of the friction forces. Obviously, this loss of energy is expressed by \(\beta\rho U^2\delta\), and the work of the friction forces is

\[ \gamma\mu\int_0^t \frac{U^2}{\delta^2}\,\delta\,dt, \]

where \(\beta\) and \(\gamma\) are dimensionless coefficients. Hence, by the energy theorem we find

\[ \frac{\beta}{\gamma}\delta=\nu\int_0^t \frac{dt}{\delta} \tag{20} \]

or

\[ \delta^2=\mathrm{const}\cdot \nu t. \tag{21} \]

We find similar results for a stationary boundary layer along a plate.

Let us now turn to the case of turbulence. Except for the very beginning of the motion, the forces of laminar friction are small in comparison with the forces of turbulent friction. This means that the loss of kinetic energy of the mean flow occurs at the expense of an increase in the kinetic energy of the fluctuations. Thus, introducing the notation

\[ \frac{1}{2}q^2=\frac{1}{2}(u'^2+v'^2+w'^2) \]

for the mean velocity of the turbulent fluctuations, we obtain

\[ \int_0^\delta \frac{\rho}{2}(U^2-\bar u^2)\,dy = \frac{\rho}{2}\int_0^\delta q^2\,dy. \tag{22} \]

On the other hand, the loss of momentum during the time \(t\) is equal to the impulse of the friction force \(\tau_0\) acting at \(y=0\). Hence

\[ \rho\int_0^\delta (U-\bar u)\,dy = \int_0^t \tau_0\,dt. \tag{23} \]

Substituting here also the numerical coefficients of the form, we may write equation (22) as

\[ \alpha \rho U^2 = \beta \rho q_0^2, \tag{24} \]

where \(q_0\) is the value of \(q\) at \(y = 0\), and equation (23) takes the form

\[ \gamma \rho U \delta = \int_0^t \tau_0 \, dt . \tag{25} \]

Hence, by differentiation, we find

\[ \tau_0 = \gamma \rho U \frac{\partial \delta}{\partial t}. \tag{26} \]

But \(\tau_0\) is a component of the turbulent-stress tensor and is equal to \(-\rho \overline{uv}\). Denoting by \(K_1\) the mean value of \(\dfrac{-\overline{uv}}{q^2}\) (the so-called correlation factor of the components \(u\) and \(v\)), we may write

\[ \tau_0 = K_1 \rho q_0^2 . \]

It may be assumed that \(K_1\) does not depend on time. Then, from equation (24), we obtain

\[ \tau_0 = \frac{\alpha}{\beta} K_1 \rho U^2, \tag{27} \]

and, using (26), find

\[ \frac{\partial \delta}{\partial t} = \frac{\alpha \gamma}{\beta} K_1 U. \tag{28} \]

The physical interpretation of these equations is that the transition region spreads with a constant velocity proportional to the initial velocity of the relative motion of the upper and lower masses of fluid. The turbulent friction between these masses, in turn, is proportional to the square of their relative velocity. It is remarkable that the coefficient of friction of fluid on fluid is a definite constant quantity, whereas the coefficient of friction of a fluid on a solid smooth wall is a function of the Reynolds number and tends to zero when \(R \to \infty\). The numerical value of this fluid friction is of considerable interest. For the nonstationary case considered by us up to now, we do not yet have sufficient experimental data. In any case, these results may be transferred to the case of stationary friction of two masses of fluid having a finite difference of velocities along the surface of their contact. If the process of turbulent mixing begins at \(x = 0\), then we find that the length of the mixing region is proportional to \(x\) (just as in the nonstationary case we have proportionality to \(t\)). This case has been studied experimentally; the coefficient of friction at \(\rho \dfrac{U^2}{2}\) (\(U\) being the relative velocity of both po-

currents) is of the order of \(C_f=0.02\). A very rough surface would have the same coefficient of friction. Thus we see that, generally speaking, the friction of liquid on liquid is greater than the friction of liquid on a solid body (Fig. 13). This explains the well-known paradox that a cylindrical solid body, floating in a river so that its axis is parallel to the direction of the flow, has a speed greater than that of the flow itself. The explanation is that, if the solid cylinder were replaced by a liquid one, the turbulent friction would increase.

Fig. 13

We encounter friction of liquid on liquid in many practically important cases. The question of the gradual weakening of the flow produced by a propeller, the spreading of a jet of liquid injected into a liquid, and the decay of vortices formed behind a moving body are phenomena of this type. Prandtl, Tollmien, Schlichting, and others have given a semi-empirical theory of these cases, based on the transfer of momentum. Taylor and Matteoli also considered certain similar cases, proceeding from other views on the problem of turbulence. In the opinion of the author of this article, a satisfactory solution can be obtained only by considering the exchange of all the mechanical quantities involved: momentum, energy, and vortices. The simple dimensional consideration mentioned above may serve as the basis for a more complete theory.

5. WHEN TURBULENCE IS USEFUL TO THE AERONAUTICAL ENGINEER

Although turbulence represents an undesirable addition to surface friction, it should not be forgotten that streamlined flow is practically impossible without instability of laminar motion. The concept of the boundary layer brought clarity to two problems. It made understandable the mechanism of surface friction, the mechanism of separation of vortices, and the formation of a vortex region behind a body. Lord Rayleigh considered the form resistance of plates—

Proceeding from the theory of discontinuous potential motion developed by Kirchhoff and Helmholtz. In the few cases that it was possible to investigate with the help of this theory, the formation of a vortex region behind a body was explained by the presence of sharp corners. However, a circular or elliptical cylinder has no sharp corners, and nevertheless separation exists. The boundary-layer theory explained this on the basis of the presence of an adverse pressure gradient. But it is known that a laminar boundary layer has so little resistance in comparison with the adverse pressure gradient that only for sections of very small relative thickness can the advantages of streamlined flow be used. The favorable influence of turbulence in the boundary layer on pressure drag is well known. The matter began with a discrepancy in the measurements of the drag coefficient of a sphere in Eiffel’s laboratory and in Prandtl’s laboratory. The measurements were made for various Reynolds numbers in two different wind tunnels. Eiffel then extended his measurements to a broader range of Reynolds numbers and found a sudden decrease in the drag coefficient, observed, as Lord Rayleigh pointed out, always at the same Reynolds numbers. Finally, Prandtl explained this sudden drop by the transition from a laminar to a turbulent boundary layer. Later it was found that the critical value of the Reynolds number depends strongly on the degree of turbulence of the air stream, which also served as an indicator of the degree of turbulence of the air in wind tunnels.

A similar influence of the turbulence of the air stream on the resistance of airplane wings was discovered several years ago. The investigation of this problem was also prompted by a discrepancy in experimental results, this time between measurements in the N.A.C.A. variable-density tunnel and in the tunnel with ordinary atmospheric pressure at the California Institute of Technology. The latter tunnel has a very low level of flow turbulence (about 0.4% velocity fluctuation); the transition from laminar to turbulent state is delayed, and velocity loss is observed at smaller values of the lift coefficient than in a tunnel with a higher level of turbulence. By introducing artificial turbulence it proved possible to bring the measurement results in the two tunnels into agreement. Consequently, the maximum lift of a wing turns out to depend on the turbulence of the oncoming flow; it is a function both of the Reynolds number and of the turbulence level. This gave Dryden grounds for calling turbulence “a companion of the Reynolds number.”

This phenomenon is very complex. Turbulence, as we shall see in the following paragraph, is not yet fully characterized by the level of turbulence, i.e., by the relative magnitude of the velocity fluctuations. At the very least, one must also take into account some length corresponding to the extent of individual vortices; otherwise it will be impossible to describe the action of external turbulence on

the place of transition of the boundary layer from laminar to turbulent. It was clearly shown in the above-mentioned investigations that, in experiments conducted in the free atmosphere, a gusty flow does not produce the same effect as increased turbulence in wind tunnels. In other words, large-scale turbulence acts differently from small-scale turbulence. Taylor recently found that the critical Reynolds number \(R_C\) must satisfy the relation

\[ R_C = F\left(\frac{u}{U}\right)\left(\frac{D}{L}\right)^{\frac{1}{5}}, \]

where \(u\) is the magnitude of the velocity fluctuations, \(D\) is the diameter of the sphere, and \(L\) is the length characterizing the “coarseness” of the turbulence.

From this brief review of experimentally established facts it follows that both the transition from the laminar to the turbulent state and the separation of the turbulent boundary layer influence one another in many very important phenomena. Recently, velocity stall (tip stalling), caused by the increased tapering of wings, has drawn new attention to the problem of separation. The most important questions to which we still cannot answer appear to the author to be the following:

a) the influence of curvature and of the pressure gradient on the position of the place of transition of the layer from laminar to turbulent;

b) the influence of external turbulence on the position of the place of transition;

c) the influence of curvature and of the pressure gradient on the separation of the turbulent boundary layer;

d) the influence of roughness on the same phenomena.

The attempt by Buri and Gruschwitz to solve the problem of separation by a semi-empirical method does not seem fruitful to us. The author is convinced that clarification of this question is possible only through the establishment of the fundamental laws of turbulent exchange.

In one respect we are in a better position than in the case of the phenomena of molecular exchange: owing to the macroscopic nature of the phenomena that interest us, their direct observation is possible. Nevertheless, as has already been said, the theoretical formulation of the problem is much more difficult than in the case of molecular phenomena; atoms and molecules are stable systems that can be destroyed only by enormous forces. Vortices, on the other hand, are easily born and easily destroyed; precisely for this reason the application of statistical theory to them requires more subtle reasoning than does the kinetic theory of gases.

At the present time we still have only the childhood years of the statistical theory of turbulence, its first attempts to walk. Nevertheless, I am deeply convinced that the methods and results of this theory will have the same influence on practical aerodynamics as the Lanchester–Prandtl wing theory once had on it, and, to a certain extent, the theory of the boundary layer, which is an inseparable part of modern practical aerodynamics. That is why

we have considered it useful to give a brief survey of the basic principles of the statistical method.

6. THE BIRTH, LIFE AND DEATH OF TURBULENCE. BASIC PRINCIPLES OF THE STATISTICAL THEORY

As was already said in the introduction to this report, turbulence is characterized by the mixing and entangling of a large number of individual vortices. In recent investigations of turbulence, carried out both in England and in the USA, grids have been used in order to obtain a homogeneous turbulent field in a flow. In this case the energy of the turbulent fluctuations is obtained through the action of the individual rods or wires of which the grid consists. Directly behind the rods, individual regular systems of vortices are observed; but farther from the grid they soon disappear, and the wakes from the rods consist of an irregularly swirled fluid; still farther on, all traces of the grid disappear, and the turbulent fluctuations become uniformly distributed over the whole cross-section of the flow.

If a fluid slides along a solid surface, then turbulence is continuously created from vortices produced by the irregularities of the surface, or arises as a result of the instability of the laminar sublayer. It is remarkable how rapidly, in this case, all traces of the origin of the turbulence disappear—already at a small distance from the place where the vortices are born. For example, in the case of a pipe with rough walls, already at a distance twice as large as the height of the irregularities, we see that the level of turbulence becomes constant in the mean direction of the flow, independently of whether the place of observation is opposite a valley or opposite a crest of an irregularity.

The measurement of turbulent fluctuations can be carried out in various ways—by anemometers of the hot-wire-anemometer type, especially suitable for recording velocity oscillations, or else by visual observation. To carry out the latter, some particles are suspended in the fluid, or the difference in refractive indices of heated and cold fluid is used. In the free atmosphere and in a river, vortices are usually so large, and consequently the rate of change of the fluctuation is so small, that it becomes possible to use ordinary anemometers, which, generally speaking, have considerable inertia, for recording them.

Let us denote by \(u_1,\ u_2,\ u_3\) the velocity fluctuations in three mutually perpendicular directions, \(x_1,\ x_2,\ x_3\), and their mean squares by \(\overline{u_1^2},\ \overline{u_2^2},\ \overline{u_3^2}\); by the mean kinetic energy per unit mass we shall understand the expression

\[ \frac{1}{2}\left(\overline{u_1^2}+\overline{u_2^2}+\overline{u_3^2}\right), \]

Reynolds indicated that the quantities

\[ \frac{1}{2}\rho \overline{u_1^2};\quad \frac{1}{2}\rho \overline{u_2^2};\quad \frac{1}{2}\rho \overline{u_3^2}; \]

represent the components of pressure per unit area, perpendicular respectively to \(x_1, x_2, x_3\), while the quantities

\[ -\rho \overline{u_1u_2};\quad -\rho \overline{u_2u_3};\quad -\rho \overline{u_3u_1} \]

represent shearing stresses; we call them the components of turbulent friction. These six components define the tensor of turbulent stresses. The “level of turbulence” is characterized by the mean value of the square of the velocity fluctuations.

The first quantity that we can determine from records of fluctuations is the frequency with which a given magnitude of velocity occurs. Apparently, experiment always shows that the distribution of velocities of different magnitude conforms rather accurately to the law of errors (Fig. 14).

The simplest case of a turbulent field is so-called isotropic turbulence, first studied in detail by Taylor. By definition, this is a type of turbulence in which there is no difference in the velocity fluctuations in different directions, so that an observer moving together with the mean flow does not find this difference. Such, for example, is the case of a uniform artificial wind with a constant level of turbulence over the cross-section of the flow, and, to a known degree, a natural uniform wind at a sufficient height above the ground.

Fig. 14

Isotropic turbulence is to a certain extent analogous to the molecular state of a gas at rest. In such a gas all directions are equivalent, and as a result of molecular collisions we have three equal components of pressure; the shearing stresses are equal to zero. Likewise, in the case of isotropic turbulence the three normal stresses are equal, and the three shearing ones, determined through mean values of products of the form \(\overline{u_1u_2}\), are zero. Indeed, if \(\overline{u_1u_2}\) were different from zero, the condition of isotropy would not be satisfied. If, for example, we reverse the direction of the \(x_2\) axis, then the sign of the product \(\overline{u_1u_2}\) will also change to the opposite one, and consequently there would be a difference between the directions \(+x_2\) and \(-x_2\), which would contradict the definition of isotropy.

Nevertheless, there is an essential difference between the molecular field of a gas at rest and the field of isotropic turbulence. If radiation and heat transfer from the walls are neglected, then the molecular system is not dissipative, i.e. the mean kinetic energy of the molecules (which is nothing other than the store of thermal energy in the gas) remains unchanged. But in a turbulized liqu-

the velocity, which is constantly subject to the action of frictional forces; turbulence therefore decreases, and the macroscopic kinetic energy of the turbulent fluctuations is transformed into the kinetic energy of molecular motion, i.e. into heat.

The establishment of the laws governing the decay of turbulence due to friction is interesting from the standpoint both of theory and of practice for many applications in engineering and meteorology. The components of the friction tensor are proportional to the spatial derivatives of the velocities, and therefore dissipation depends not only on the level of turbulence, but also on the degree of graininess of the turbulent field, i.e. on the sizes of the vortices.

In studying the structure of turbulence one may use either a descriptive or a statistical method. The former was used by Schmidt and, recently, by Scherhag in their interesting investigations of the structure of strong winds (gales) (Fig. 15, lines of equal velocity as a function of height and time).

Fig. 15

In the statistical method we study mean values and try to determine the sizes of vortices from the correlation between the components of the velocities. The advantage of the statistical method is that all characteristic quantities can be determined mathematically; but, of course, individual events and configurations are erased in the process of averaging.

What, then, is correlation? Let us mark the occurrence of a certain event at two points \(A\) and \(B\) of space; suppose also that the occurrence of the event and its non-occurrence have equal probability. If the event occurs, we mark this in our records as \(+1\); otherwise we write \(-1\). For example, \(a=1\) means that the event occurs at point \(A\); \(b=1\) means that the same event occurs at point \(b\). Obviously, if these events are independent, then the mean value of the product \(\overline{ab}\) is equal to zero; if the events at \(A\) and \(B\) invariably both occur or both fail to occur, then \(\overline{ab}=1\); if, finally, the occurrence of the event at \(A\)

implies its non-occurrence in \(B\), and conversely, then \(\overline{ab}=-1\). But if the connection between events in \(A\) and \(B\) is incomplete, i.e., there is only a “correlation,” then \(0<\overline{ab}<1\). It is clear that in a similar way the correlation between any two different events in \(A\) and \(B\) is measured.

Let us now imagine that we measure two arbitrary components of the velocity \(u_i, u_k\) at two different points of an isotropic turbulent field. In order to establish whether there is a correlation between the fluctuations at \(A\) and \(B\), we form the mean values of the products \(\overline{u_{iA}u_{kB}}\); we shall denote them by \(q_{ik}\). Obviously, there are nine such quantities; three of them are pairwise equal owing to the equivalence of all directions, for example, \(u_{iA}u_{kB}=u_{kA}u_{iB}\), or \(q_{ik}=q_{ki}\). As for the remaining six, for a homogeneous unbounded field we may choose \(A\) arbitrarily, and then \(q_{ik}\) are functions of \(B\). Taking \(A\) as the origin of coordinates \(x_1,x_2,x_3\), we see that \(q_{ik}\) are functions of \(x_1,x_2,x_3\). At a large distance between \(A\) and \(B\), i.e., for a large radius

\[ r=\sqrt{x_1^2+x_2^2+x_3^2} \]

the mean values \(q_{ik}\) are small; usually they are practically zero. If \(A\) and \(B\) coincide (\(r=0\)), the mean values \(q_{11}, q_{22}, q_{33}\) are equal to \(\overline{u^2}\), where \(\overline{u^2}\) is the common value of the quantities \(\overline{u_1^2}, \overline{u_2^2}, \overline{u_3^2}\); \(q_{12}=q_{23}=q_{31}=0\), because, as was said above, the shear stresses vanish.

Dividing the six quantities \(q_{ik}\) by \(\overline{u^2}\), we shall call the resulting quotients the correlation functions \(R_{ik}\). In the case of isotropy these six functions can be expressed in terms of two functions \(R_1(r)\) and \(R_2(r)\). The first of these functions is the correlation between the components of velocity along the straight line \(AB\) joining the observation points \(A\) and \(B\); the second function is the correlation between components perpendicular to \(AB\) and parallel to each other. Therefore we call \(R_1\) the longitudinal correlation, and \(R_2\) the transverse correlation. If the fluid may be regarded as incompressible, then these quantities are related by

\[ \frac{dR_1}{dr}+2(R_1-R_2)=0, \tag{29} \]

which is nothing other than the equation of continuity. The existence of this relation may be regarded as a criterion of the isotropy of the turbulent field. Measurements by Simmons, in the aerodynamic wind tunnel of the National Physical Laboratory, showed that this criterion is very well satisfied (Fig. 16).

We imagine turbulent motion as a random motion of a large number of vortex clusters or clumps; the character of the correlation curve gives us an idea of the size of these clumps. Let us look, for example, at the curve \(R_1\) of the longitudinal correlation in Fig. 16. For \(r=1\) (in inches) we find \(R_1=0.36\). Thus,

there is a comparatively high probability that two fluid particles located at a distance of 1 dm move in the same direction. Hence we conclude that in the flow studied by Simmons there exists a considerable number of vortices of similar dimensions; indeed, if most of the vortices were smaller than 1 dm, then the motions at \(A\) and \(B\) would depend very little on one another, and the correlation would be close to zero. Thus, consideration of the correlation curves obtained under different conditions allows us to draw conclusions about the “physical dimensions” (we shall speak of the “grain size”) of a turbulent flow.

Fig. 16

The curvature of the correlation curves at the point \(r=0\), where they have their maximum value, is of special interest. Indeed, the squares and pairwise products of the derivatives of the velocity fluctuations depend only on

\[ \frac{d^2 R_1}{d r^2} \quad \text{and} \quad \frac{d^2 R_2}{d r^2} \quad \text{for} \quad r=0. \]

Let us consider, for example,

\[ \left(\frac{d u_1}{d x_2}\right)^2 . \]

This is of interest because the sum

\[ \sum \mu \left(\frac{d u_i}{d x_k}\right)^2 \]

represents the work of the frictional forces per unit time, i.e. a measure of the dissipation of kinetic energy in turbulent motion. But

\[ \overline{\left(\frac{\partial u_1}{\partial x_2}\right)^2} = \lim \frac{(u_1' - u_1)^2}{(x_2' - x_2)^2}, \]

where \(\overline{u_1}\) is the magnitude of the velocity fluctuation at the point \(x_2\), \(\overline{u_1'}\) at the point \(x_2'\), and the limit is taken as \(x_2' \to x_2\). Hence we find

\[ \overline{\left(\frac{u_1}{\partial x_2}\right)^2} = \lim \frac{u_1'^2 + u_1^2 - 2u_1' u_1}{(x_2' - x_2)^2}. \]

But \(\overline{u'^2}=\overline{u_1^2}\), \(\overline{u_1 u_1'}=R_2 \overline{u^2}\), and therefore

\[ \overline{\left(\frac{u_1}{\partial x_2}\right)^2} = 2\overline{u^2}\, \frac{1-R_2}{(x_2' - x_2)^2}. \]

Let us now expand the correlation function in a series in powers of \(x_2' - x_2 = r\)

\[ R_2 = 1 + \frac{1}{2} r^2 \left(\frac{d^2 R_2}{dr^2}\right)_0 + \cdots . \]

Obviously

\[ 2(1 - R_2) = - \left(\frac{d^2 R_2}{dr^2}\right) r^2 - \cdots , \]

and, passing to the limit, \(r = 0,\ x_2' \to x_2\), we find

\[ \overline{\left(\frac{\partial u_1}{\partial x_1}\right)^2} = -\overline{u^2}\left(\frac{d^2 R_2}{dr^2}\right)_0 . \]

Taylor gave a simple geometrical representation for

\[ \left(\frac{d^2 R_2}{dr^2}\right)_{r=0}. \]

In the region of the maximum value of the correlation curve \(R_2(r)\), it may approximately be regarded as a parabola and written

\[ R_2 = 1 - \frac{r^2}{\lambda^2}, \]

where \(\lambda\) is the distance from the origin to the intersection of the parabola with the axis of abscissae. Obviously

\[ \frac{1}{2}\left(\frac{d^2 R_2}{dr^2}\right)_0 = -\frac{1}{\lambda^2}, \]

and we have

\[ \overline{\left(\frac{\partial u_1}{\partial x_2}\right)^2} = \frac{\overline{u^2}}{\lambda^2}. \]

Carrying out similar calculations for the remaining terms of the sum

\[ \sum \mu \overline{\left(\frac{d u_i}{d x_k}\right)^2}, \]

we find an important equation for the decay of turbulence

\[ \frac{d^2 g}{dt} = -10 \nu^2 \frac{g^2}{\lambda^2}, \tag{30} \]

where

\[ g^2 = \overline{u_1^2} + \overline{u_2^2} + \overline{u_3^2}, \]

i.e. the doubled mean value of the kinetic energy of turbulence referred to unit mass; \(\nu\) is the coefficient of kinematic viscosity, and \(x\) is a length characterizing the steepness of the velocity gradient in a turbulent flow. Taylor calls \(\lambda\) the size of the “smallest vortices”; the exact definition of \(\lambda\) was given above.

The attention of researchers presently working on the statistical theory of turbulence is focused precisely on equation (30). The length \(\lambda\), as has been said, is the “linear size of small vortices,” whereas the width of the correlation curves shown in Fig. 16 gives the mean size of the “large” vortices. A very essential question is what determines the further development of the vortex sizes \(\lambda\)? If we imagine such a turbulent field, for

for which \(\lambda\) is constant, then in such a field the energy will decrease rather rapidly and proportionally to an exponential function of time. One should expect that \(\lambda\) increases with time, since small vortices perish sooner than large ones. As a consequence of this process, \(\lambda^2\) increases, in all probability, proportionally to time, so that the energy itself is proportional to a certain negative power of time. But at present it is not yet possible to give a definite answer to this question; it still requires further discussion.

A similar reasoning may be applied to the dissipation of the mean vorticity of a fluid. Denoting the three components of the vorticity vector by \(\omega_1, \omega_2, \omega_3\), we can show that \(\omega^2=\omega_1^2+\omega_2^2+\omega_3^2\) satisfies the following equation:

\[ \frac{d\omega^2}{dt}=-10\nu\,\frac{\omega^2}{\lambda_\omega^2}+\omega_i\omega_k\overline{\frac{\partial u_i}{\partial x_k}}, \tag{31} \]

where \(\lambda_\omega\) is a length, analogous to \(\lambda\), characterizing the spatial variation of the fluctuations of vorticity. The first term on the right represents the dissipation of vorticity caused by friction; the second term represents the change of the mean vorticity caused by a change in the dimensions of the vortices. Consider, for example, a finite part of a vortex tube: if the diameter of the tubes decreases, then the mean vorticity also increases; if the length of the cylinder shortens, then the mean vorticity also decreases. In what direction does this phenomenon develop in turbulent motion? The answer to this and similar questions will help in the search for the final form of the fundamental equations of turbulent flow.

Fig. 17

The experimental investigation of correlation functions is of enormous significance not only for the development of theory, but also for elucidating a number of applied problems in meteorology, oceanography, hydraulics, and other fields.

Figs. 17–21 give examples of the distribution of the magnitudes of velocity fluctuations and of the correlation function, according to measurements in a wind tunnel, in a river (the Mississippi), and in the free atmosphere.

An example of a direct engineering application of correlation measurements is the estimation of the magnitude of vortex gusts acting—

Fig. 18

Fig. 19

—on an airplane or on a part of it. If the size of this part is small in comparison with the size of the gusts, then it must be assumed that the relative velocity and the angle of attack vary with time, but are practically constant over the wingspan or the length of the airplane. Such a conclusion would, however, already be incorrect for airships and, perhaps, also for the giant craft currently under construction—

-speed aircraft. It is therefore necessary to develop instructions for the probable distribution of pressures; they would be useful not only in strength calculations, but also in studying the dynamic effect of vortex wakes. Correlation measurements and statistical analysis of vortex wakes can provide a firm basis for investigations of this kind, just as in ship calculations one has to specify the form of the waves.

Fig. 20

The concept of correlation thus plays an essential role in understanding the process of turbulent exchange, which, one may say, is the principal object in the theoretical and experimental study of turbulence. In our preliminary reasoning we proceeded from the assumption that the law governing molecular diffusion remains valid also for the diffusion of vortices, but with increased coefficients. But the law of molecular diffusion is based on the fact that the mean square of the distance traversed by particles increases linearly with time. Applying this law to the case of so-called Brownian motion, Einstein was able to show that the diffusion coefficient of particles is equal to

\[ \frac{\overline{S^{2}}}{2t}, \]

where \(\overline{S^{2}}\) is the mean square of the distance traversed by the particles in time \(t\).

Fig. 21

However, on more careful consideration of the problem we become convinced that this law is only approximate and is inapplicable for small observation times. The following arguments belong to Langevin; they are set forth here because, in my opinion, they will be of substantial help in our problem.

Let us consider the mean motion of a large number of particles and calculate the value \(\overline{S^{2}}\), i.e., the mean square of the path traversed

TURBULENCE

of particles in some direction over some interval of time \(t\). The differential equation of motion is

\[ \frac{d^{2}S}{dt^{2}}=\frac{P}{m}-\frac{K}{m}\frac{dS}{dt}, \tag{32} \]

where \(P\) is a random force, and \(K\dfrac{dS}{dt}\) is the frictional resistance of the particle against the medium. Let us multiply both sides of this equation by \(S\) and perform an averaging over a large number of particles; since the magnitude and direction of \(P\) vary according to the law of chance, there is no correlation between \(P\) and \(S\), and the mean \(\overline{PS}\) is equal to zero. On the other hand, we have

\[ S\frac{dS^{2}}{dt^{2}} = \frac{d^{2}}{dt^{2}}\left(\frac{S^{2}}{2}\right) - \left(\frac{dS}{dt}\right)^{2}, \]

and therefore

\[ \frac{d^{2}}{dt^{2}}\frac{\overline{S^{2}}}{2} = \overline{\left(\frac{dS}{dt}\right)^{2}} - \frac{K}{m}\,\overline{S\frac{dS}{dt}} \tag{33} \]

or

\[ \frac{d^{2}}{dt^{2}}\frac{\overline{S^{2}}}{2} = \overline{\left(\frac{dS}{dt}\right)^{2}} - \frac{K}{m}\frac{d}{dt}\frac{\overline{S^{2}}}{2}. \tag{34} \]

Let us now suppose that the mean \(\overline{\left(\dfrac{dS}{dt}\right)^{2}}\), i.e. the mean square of the velocity component in the direction \(S\), is constant and equal to \(\overline{u^{2}}\). Then our equation is soluble with respect to \(\dfrac{\overline{S^{2}}}{2}\), and we find

\[ \frac{\overline{S^{2}}}{2} = Ae^{-\frac{Kt}{m}}+B+\frac{\overline{m u^{2}}}{K}\,t \tag{35} \]

or, since at \(t=0\) we have \(\overline{S^{2}}\) and \(\dfrac{\overline{dS^{2}}}{dt}=0\),

\[ \frac{\overline{S^{2}}}{2} = \frac{m^{2}\overline{u^{2}}}{K^{2}} \left(e^{-\frac{Kt}{m}}-1\right) + \frac{\overline{m u^{2}}\,t}{k}. \tag{36} \]

Let us now introduce the relaxation time \(t_{0}=\dfrac{m}{K}\), equal to the time during which the frictional resistance reduces the velocity of the particle to \(\dfrac{1}{2.73}\) of its initial value.

Then equation (36) reads

\[ \overline{S^2}=2\overline{u^2}\,t_0^2\left(e^{-\frac{t}{t_0}}-1\right)+2\overline{u^2}tt_0 . \tag{37} \]

Suppose, further, that \(t \ll t_0\). Then

\[ \overline{S^2}=\overline{u^2}t^2+\text{terms of higher orders}, \tag{38} \]

i.e. \(\dfrac{\overline{S^2}}{t}\) increases proportionally to time. On the other hand, if \(t \gg t_0\), then

\[ \overline{S^2}\sim 2\overline{u^2}t_0t \tag{39} \]

or

\[ \frac{\overline{S^2}}{t}\sim 2\overline{u^2}t_0 . \tag{40} \]

In Brownian motion the viscous resistance is so great that \(t_0\) is always less than \(10^{-5}\) sec., and we are dealing only with the second case.

Transferring the idea of this reasoning to the case that interests us—the Brownian motion of large eddies in a turbulent field—we must regard the lifetime \(t_0\) of the eddies as large (for example, 100–1000 sec. for turbulence in the free atmosphere); and indeed, we observe both phases of the diffusion process.

Taylor was the first to show, on the basis of experiments carried out at the Bureau of Standards, that the temperature distribution behind a heat source placed in a turbulent air stream does not follow the law \(S^2\) proportional to \(t\); but when the correction for molecular thermal conductivity is introduced, the law \(S\) proportional to \(t\) holds. Thus the region behind the source in which some part of the heat obtained from the source is found has the form not of a paraboloid, but of a cone. This is easy to imagine if one assumes that the heat is transported by eddies whose dimensions are large in comparison with the dimensions of the heat source; these eddies carry the heat received by them at practically constant velocity. The classical diffusion considered by us in the first part of this article begins when the first eddy disappears as a result of friction and the velocities of the air particles carrying the heat undergo a large number of changes both in magnitude and in direction.

Diffusion of the second type appears at a large distance from the source, and also in cases when heat or momentum is transmitted in a parallel or approximately parallel flow of fluid layers and its mean flow is constant or changes little.

The author of the present article believes that the phenomena of so-called free turbulence, for example, the formation of regions through

which accomplish the transfer of the quantities of interest to us between masses of liquid having different velocities—correspond to the first type of turbulent diffusion, whereas the cases of stationary flow between solid walls and flow in the boundary layer correspond to the second type.

The following remarks pertain to steady flow in the vicinity of a solid surface.

Equation (40) shows that \(\dfrac{\overline{S^2}}{t}\) is determined through \(\overline{u^2}\) and \(t\). The quantity \(\overline{u^2}\) is given by the level of turbulence; the principal difficulty lies in the interpretation of \(t_0\). There are grounds for supposing that \(t_0\)—the lifetime of an eddy—must in some way be connected with the characteristic properties of the turbulence itself. If it is assumed that this duration is determined by the forces of viscosity, then the most probable assumption is that \(t_0\) is proportional to \(\dfrac{\lambda^2}{\nu}\), where \(\lambda\) is the size of small eddies and \(\nu\) is the kinematic viscosity.

Indeed, the viscous forces acting on an element of liquid whose linear dimension is \(\lambda\) are proportional to \(\mu u \lambda\) (Stokes’ law); the mass of the same element is proportional to \(\rho \lambda^3\). Therefore the deceleration of the velocity is proportional to \(\dfrac{\nu u}{\lambda^2}\), and the lifetime is proportional to \(\dfrac{\lambda^2}{\nu}\). Hence we see that the coefficient of diffusion is proportional to \(\dfrac{\lambda^2}{\nu}\cdot u^2\). The same result is obtained when calculating the time of weakening of an eddy by viscosity.

Let us compare this reasoning with the so-called “mixing length” theory, proposed and defended chiefly by Prandtl. We may introduce into our consideration the concept of mixing length by identifying \(u t_0\) with \(l\) (the mixing length). Then from the relation \(t_0=\dfrac{\lambda^2}{\nu}\) we find that the mixing length is proportional to \(\dfrac{\lambda^2}{\nu u}\). This result is supported by the following simple argument. Imagine an ordinary shear motion of a liquid, in which a constant amount of motion is transferred from layer to layer per unit time. In this case it is natural to assume that in each layer of unit thickness a constant amount of kinetic energy is dissipated owing to viscosity. Taylor showed that it is proportional to \(\dfrac{\overline{u^2}}{\lambda^2}\). Hence we obtain the correct energy balance by putting

\[ \tau \frac{\partial U}{\partial y}=\text{const}\cdot \rho \nu \frac{\overline{u^2}}{\lambda^2}, \tag{41} \]

where \(\tau\) is the shear stress, and \(\dfrac{\partial U}{\partial y}\) is the gradient of the mean velocity. On the other hand, since the exchange coefficient is proportional to \(lu\),

\[ \tau=\text{const}\cdot lu\rho \frac{\partial U}{\partial y}. \tag{42} \]

Eliminating \(\dfrac{\partial U}{\partial y}\) and taking into account that \(\tau\) is constant, we find the relation \(l=\mathrm{const}\,\dfrac{\lambda^{2}}{\nu}\,u\).

Let us now apply these considerations to the flow at a solid wall, as is observed in the boundary layer. In this case experiment tells us that
\[ \frac{\partial U}{\partial y}=\frac{1}{k}\sqrt{\frac{\tau}{\rho}}\cdot\frac{1}{y}, \]
where \(y\) is the distance from the wall, and \(k\) is a universal constant. We assume that this general law is due to the circumstance that the mechanism of turbulent exchange is the same for different \(y\). In the language of statistical theory this means that the correlations between the velocity components are the same everywhere. Since the shear stress \(\tau\) is equal to \(\rho\,\overline{uv}\) (\(u, v\) are the components of the velocity fluctuation in the directions \(x, y\)), it follows from the principle of similarity that
\[ \frac{\overline{uv}}{\overline{u^{2}}}=\mathrm{const} \]
and
\[ \tau=\mathrm{const}\,\rho\,\overline{u^{2}}. \]
Therefore we find from (41) that
\[ \frac{\partial U}{\partial y}\frac{\lambda^{2}}{\nu}=\mathrm{const}, \]
or
\[ \lambda^{2}=\mathrm{const}\,\frac{\nu y}{\sqrt{\dfrac{\tau}{\rho}}}. \]

We obtain the following picture of the phenomenon. The dimensions of the small vortices grow proportionally to the square root of the distance from the wall. The mixing path is proportional to this distance; it does not depend on the Reynolds number, whereas \(\lambda\) depends on \(\nu\) and \(\sqrt{\dfrac{\tau}{\rho}}\) and decreases as the Reynolds number increases. In other words, the grain of the turbulent field becomes finer with increasing velocity.

The value of the theory would be enhanced if it could predict the degree of turbulence and the distribution of the mean velocity in the motion of a fluid in a tube of constant cross-section. Furthermore, such phenomena as the influence of curvature, expansion and contraction of the tube, variable density, etc., would have to be included in the general equations.

None of the theories of turbulence has been developed to such an extent that it could give the distribution of the mean velocity over the cross-section or the level of turbulence. Practically every investigator concerned with the theory of turbulence has constructed for himself a theory of turbulent exchange; the best-known examples are Prandtl’s theory of momentum transfer and Taylor’s theory of vorticity transfer. Matteoli proposed a theory that is a combination of these two, and Gebelein attempted to find a solution by adopting the high standpoint of the general theory of probabilities; I fear that this standpoint is situated too high for it to be possible from it to discern the simple facts at issue. Wehrle and his collaborators made a very interesting attempt to apply the principle of maximum probabil-

...to the derivation of general equations for a single type of turbulent flow. Since I myself am trying to construct a theory of turbulence, it is better for me to refrain from criticizing the theories that have been proposed. Of course, I cherish the hope that experimental and theoretical investigations will advance so far in the coming years that sufficiently reliable connections will be established among the various basic relations; as for a complete mathematical justification of the theory and the inclusion in it of the main particular cases, that will fall to the lot of those who come after us.

Many contemporary engineers regard the problem of turbulence as merely an interesting chapter of mathematical physics. Perhaps they are right. But they should remember that if we encounter some practical question of aerodynamic calculation to which we cannot give an answer, then this inability almost certainly stems from our insufficient knowledge of turbulence. And therefore I believe that, despite the mathematical and physical complexity of this problem, the scientist has the right to say to the engineer: Tua res agitur (this is being done for you).

The main task of my lecture was to show this.

1. This is confirmed by Goldstein’s investigation, Rep. a. Mem. 1763. He found that the relation
   \[ \frac{vh\sqrt{\frac{\tau}{\rho}}}{\nu}>\sqrt{R_c} \]
   is the condition that an element of unevenness of height \(h\) does not affect the main flow. Here \(R_c\) is of the order of \(30\)—\(50\). ↩↩