The microstructure of turbulent flow

The microstructure of turbulent flow


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The microstructure of turbulent flow
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Physical Description:
41 p. : ; 27 cm.
Obukhoff, A. M
I︠A︡glom, A. M
United States -- National Advisory Committee for Aeronautics
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Turbulence   ( lcsh )
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An attempt is made to describe quantitatively the structure of the velocity, pressure, and acceleration fields for all scales for which the theory of Kolmogoroff is applicable.
Includes bibliographic references (p. 38-39).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by A.M. Obukhoff and A.M. Yaglom.

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University of Florida
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Full Text
V A-i/WM-I1 3 0

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By A. M. Obukhoff and A. M. Yaglom

In 1941 a general theory of locally isotropic turbulence was pro-
posed by Kolmogoroff which permitted the prediction of a number of laws
of turbulent flow for large Reynolds numbers. The most important of
these laws, the dependence of the mean square of the difference in vel-
ocities at two points on their distance and the dependence of the coef-
ficient of turbulence diffusion on the scale of the phenomenon, were
obtained by both Kolmogoroff (references 1 and 2) and Obukhoff (ref-
erence 3) in the same year. At the present time these laws have been
experimentally confirmed by direct measurements carried out in aero-
dynamic wind tunnels in the laboratory (references 4 and 5), in the
atmosphere (references 6 and 7), and also on the ocean (reference 8).1
In recent years in the Laboratory of Atmospheric Turbulence of the
Geophysics Institute of the Soviet Academy of Sciences, a number of
investigations have been conducted in which this theory was further
developed. The results of several of these investigations are pre-
sented in this paper.

The fundamental physical concepts which are the basis of
Kolmogoroff's theory may briefly by summarized as follows.3 A turbulent
flow at large Reynolds numbers is considered to be the result of the
imposing of disturbances (vortices or eddies) of all possible scales of

*"Mikrostructura turbulentnogo potoka,* Prikladnaya Matematika i
Mekhanika, Vol. XV, 1951, pp. 3-26.

'The applications of these laws to certain problems of the physics
of the atmosphere may be found in references 9 and 10.

2In addition to the results contained in the present article, ref-
erence may also be made to the theoretical investigation of the struc-
ture of the temperature field (or of the concentrations of any neutral
additive) in the turbulent flow, presented in references 11 and 12.
The applications of the latter results may be found in references 13
and 14.

3For a more detailed presentation see reference 15.

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magnitude. Only the very largest of these vortices arise directly from
the instability of the mean flow. The scale L of these large vortices
is comparable with the distance over which the velocity of the mean flow
changes (for example, in a turbulent boundary layer, with the distance
from the wall).4

The motion of the largest vortices is unstable and gives rise to
smaller vortices of the second order; vortices of the second order give
rise to still smaller vortices of the third order, and so forth, down
to the smallest vortices which are stable (i.e. the characterizing
Reynolds number is less than the critical value). Since for all vor-
tices, except the smallest ones, the characteristic Reynolds number is
large, the viscosity has no appreciable effect on their motion. The
motion of all vortices that are not too small is therefore not associ-
ated with any marked dissipation of energy; the vortices of the nth
order use practically all the energy which is received from the vor-
tices of the (n -1)th order to form the vortices of the (n+l)th order.
However, the motion of the smallest of the existing vortices is
laminarr" and depends essentially on the molecular viscosity. In these
very small vortices the entire energy that is transferred along the
vortex cascade goes over into heat energy.

The motion of all the vortices, except for the very largest, may
be assumed homogeneous and isotropic. Any directional effect of the
mean flow ceases to be appreciable for vortices of a relatively low
order. It is also of importance that this motion may be assumed quasi-
stationary, that is, a change in the statistical characteristics of the
motion of the vortices under consideration proceeds very slowly in com-
parison with the periods characteristic of these vortices. It follows
that the motion of all vortices whose scales are considerably less than
L (the microstructure or local structure of the flow) must be subject
to certain general statistical laws which do not depend on the geometry
of the flow and on the properties of the mean flow. The establishment
of these general laws, which have a wide range of applicability, con-
stitutes the theory of local isotropic turbulence.

In the investigation of the laws of the local structure, consider-
ations from the theories of similitude and dimensions are of great value.
It is only these considerations which permit obtaining a number of essen-
tial results. To apply these ideas it is necessary, first of all, to
separate out those fundamental magnitudes on which the local structure
of the flow may depend. On account of the homogeneous and isotropic
character of the motion of the vortex system under consideration, the

4The length L coincides with the length of the mixing path intro-
duced in the semiempirical theory of turbulence.

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characteristics of the mean motion (of the type of length characteris-
tics, velocity characteristics, etc.) do not enter among these funda-
mental magnitudes. Therefore, only two magnitudes remain, the mean
dissipation of energy per unit time per unit mass of the fluid e,
which determines the intensity of the energy flow transferred along a
cascade of vortices of different scales, and the kinematic viscosity v,
which plays an essential role in the process of dissipation.5 These
two magnitudes thus play a fundamental part in the theory that is pre-
sented herein.

The dimensions of and v are:

(e] = L2 -3

[v] = L-1

From these two magnitudes, it is evidently possible to form a single
combination in the dimension of length
3 1/4

The length q determines an internal scale characteristic of the local
structure. By use of the previously described physical picture of tur-
bulent motion, it is possible to identify n with the scale of the
smallest vortices in which a dissipation of energy occurs (since this
picture does not contain any other characteristic length). The scale T
was first introduced in the work of Kolmogoroff (reference 1); it is
termed the internal (or local) scale of turbulence (in contrast to the
external scale L).

In the further analysis of the microstructure, two limiting cases
may be considered separately to advantage: the case of scales much
larger than and that of scales much smaller than T. First, the
system of vortices with dimensions much smaller than L but much greater
than the scale I of the smallest vortices is considered. The motion
of these vortices, as has already been pointed out, should not depend

5The fluid is assumed everywhere to be incompressible and to have
a constant density p. The magnitude p is not included herein among
the fundamental magnitudes because in the main part of the paper (sec-
tions 1 and 3), the purely kinematic characteristics of the flow, which
of course cannot depend on the density, will be considered. When, how-
ever, the structure of the pressure field (section 2) is investigated,
it is necessary to add p to e and v. Information on the fundamen-
tal magnitudes on which the local structure of the temperature field
may depend is found in references 11 and 12.

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on the viscosity v, a circumstance which immediately facilitates the
obtaining of concrete results by computation of the dimensions. In the
second extreme case, for scales of motion much less than n, the motion
may be assumed laminar. However, in the intermediate range of scales
of the order of n, the theory of dimensions gives, as a rule, less
concrete results. Thus, for example, it follows from this theory that
any nondimensional function of the distance determined by the local
structure should be a universal function of r/r. The form of this
function for values of the argument of the order of unity remains how-
ever undetermined.

In the present paper an attempt is made to describe quantitatively
the structure of the fundamental hydrodynamic fields (pressure, velocity,
and acceleration6) for all distances less than L (i.e., for the entire
range for which the theory of Kolmogoroff applies). For this purpose
some additional hypotheses are introduced which have a certain experi-
mental basis. The asymptotic formulas for r>> and for r< obtained are in agreement with known earlier results where all the
undetermined numerical coefficients that figure in these results are
expressed in terms of a single constant S (asymmetry or skewness
factor), the value of which has been experimentally determined by
Townsend (reference 4). The nondimensional magnitude S (as well as
the magnitudes e and v) enters only in the expression for the char-
acteristic scales so that with an accuracy up to the choice of units
the measurements of the structure of all the fields considered under
the assumed hypotheses are described by universal functions not depend-
ing on any experimental data (see figs. 1 to 3; the meaning of these
functions will be explained in a later discussion).

The investigation of the structure of the velocity field (section 1)
is the work of A. M. Obukhoff; the investigation of the pressure field
(section 2) was started by Obukhoff (reference 16) and continued by
A. M. Yaglom; the investigation of the acceleration field (section 3)
was carried out by Yaglom. Several results of the present work were
first published in the form of separate short communications (refer-
ences 7, 16, and 17).

1. Computation of structural functions of velocity field. In order
to be able to make use of the concepts of locally isotropic turbulence in
investigating the velocity field of a turbulent flow, it is first neces-
sary to separate out those characteristics of the field which depend
only on the local structure. The true velocity v will essentially be
determined by the mean flow. In the theory of turbulence the usual
decomposition of the true velocity v into the mean velocity v and
the fluctuating velocity v' = v v gives a component v' not depend-
ing on this mean flow; but the theory does not solve the problem pro-
posed since the value of v' will be determined mainly by the very

6The acceleration of the flow is considered herein to be the total
acceleration dv/dt of the fluid particles moving in space.

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large vortices, the scale of which is comparable with L. However, as
was first noted by A. N. Kolmogoroff (reference 1), the above mentioned
required that a separation of the characteristics be effected by con-
sidering the difference of the velocities at two sufficiently near
points (i.e., the relative motion of two neighboring elements of the
fluid). It is clear that this difference will not be affected by the
large vortices which transport the pair of points under consideration
as a whole. Hence, in the theory of local isotropic turbulence, the
following functions are taken as the fundamental quantitative charac-
teristics of the structure of the velocity field:

Di(M,Mt) = [vi(M') vi(M)][vj(M') vj (M)] (i,j.= 1,2,3) (1.1)

where vi(M) is the ith component of the velocity vector v(M) at the
point M, and the bar above a symbol denotes the average value. The
function Dij(M,M') is termed the structural function of the velocity
field. According to the preceding discussion, for a distance r
between the points M and M' much less than L, this function depends
only on the local structure of the flow. On account of the homogeneity
and isotropy of the motion of the vortices with scales much less than
L, the function D.i(M,M'), for r< of the vector MM' and may therefore be represented in the form

Dij(M,M') = A(r)ti + B(r)8ij (1.2)

where tl, 2,' and (3 are the components of the vector MM' (so that

2 + 2 + 2 = r) and 5ij = 1 for i = j and 5ij = 0 for

When first v1 = v. = vn where vn is the projection of the
velocity vector on a certain direction perpendicular to the vector
MM' and then vi = vj = v2 where v, is the projection of v on the
direction of the vector MM' are set into this formula, it is readily
shown that equation (1.2) may be represented in the form

Dij(M,M') = Dnn(r) qj + Dnn(r)8ij (1.3)

where the functions D (r) and Dnn(r) (the longitudinal and trans-
verse structural functions) have the simple physical meaning:

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D i(r) = [V(M') V(M)2 (1.4)

Dn(r) = [v(M,) vn(M)]2

The determination of these functions, D (r) and Dnn(r), will be
the main object of this section.7

In the theory of local isotropic turbulence it is possible to con-
sider the functions DI(r) and Dnn(r) as independent of the time.
As a matter of fact, a quasi-stationary statistical regime in a region
of sufficiently small turbulence scale is assumed. From the consider-
ations of the theory of similarity, it follows that in the range of
applicability of the theory of locally isotropic turbulence (i.e., for
r< form
D,,(r) = (1.5)

Dnn(r) = ve dnn()

where G = (v-e 1)1/4 is the internal scale of turbulence and dJJ(x)
and dnn(x) are universal functions. Formulas (1.5) may also be
represented in the form

7In the theory of isotropic turbulence, the correlation functions
(longitudinal and transverse) are usually employed.

BI (r) = v (M)v (M')

Bnn(r) = vn(M)v (M')

The structural functions in the isotropic case are connected with the
correlation functions by the following relations:

D,1(r) = 2(B(0) B,,(r))

Dnn(r) = 2(B(0) Bnn(r))

where B(0) = BI (0) = Bnn(0).

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D (r) = u12? ,,/()

Dnn(r) = u12nn l)


91 = kl V
ul = k2 -ve

The numerical factors k1 and
will always be assumed to be of the
Pnn(x) are new universal functions
the graphs of the functions dj(x)
scales along the x and y axes.

k2 can be chosen by inspection and
order of unity, and P3,(x) and
the graphs of which are obtained from
and dnn(x) by a simple change of

Since for r>>T the functions DII(r) and Dnn(r), on account
of the stated physical considerations, should not depend on the viscos-
ity v, the asymptotic equations should hold

d (x) x2/3

for x>>1


dnn(x) ~ x2/3

The same equations also hold, of course, in relation to the
functions p22(x) and Onn(x). Whence it follows that for r>>

DI (r) = Ce/3 r2/3

Dnn(r) = C'E 2/3 r2/3





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(the so-called 2/3 law). In the other extreme case, for r< difference of the velocities v(M') v(M) will be of the first order
of smallness with respect to r (for such distances the velocity at a
point of the flow is continuous and is a differentiable function of
the coordinates), so that in this case

D (r) = Ar2

Dnn(r) = A'r2

The more complete theory based on the equations of hydrodynamics
is now discussed. First use of the equation of continuity

6 vi
i- = 0
i=l 1


shows with little difficulty that

Dnn(r) = D ,(r

r dD1 (r)
+ + dr

[v(M') v(M)H[p(MI) p(M)J = 0


where p(M) is the pressure at point M (see, for exmaple, refer-
ences 2 and 15 and compare also references 18 and 19). Now with the
aid of equations (1.12) and (1.13) and the equations of motion

3 6v
+ v x_= -1 p +VAvi
j=l 1xj P

(i = 1,2,3)

and that




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it may be shown that the function DI (r) is connected with the struc-
tural function of the third order

D (r ).= [v (M') v (M)]3 (1.15)

by the known relation of Kolmogoroff (reference 2).8

dDi (r) 4
D (r) 6V dr = -r

In the case of homogeneous and isotropic turbulence, the equation
relative to the correlation functions (references 18 and 19) is easily
derived from equation (1.14):

6B / B 2 4 2B
"=. + r B3 +2v -" + 4


BZ= v 2(M)V(M')

When the correlation functions are replaced by the structural
functions given by the formulas in the previous footnote (and by an
analogous formula for B11), the following is obtained:

-4 = + fD1 6v -3dB

from which equation (1.16) is obtained after a single integration with
respect to r. In a similar manner, equations (1.12) and (1.13) may
be obtained from known results relative to isotropic turbulence. It
may likewise be shown that equations (1.12), (1.13), and (1.16) are
also valid within the framework of the theory of a locally isotropic

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For r<<, the term D--(r) may be neglected in this relation
(since for these values of r the function DIlI(r) will be of
third-order smallness with respect to r) and therefore, equations (1.16)
and (1.12) give the solutions

D(r) = I r2

for r<< (1.17)

Dnn(r) 2 r2
15 v

This is an improvement in the accuracy of relations (1.10). On
the other hand, for r>> the term with the viscosity may be rejected

D2(r) = er for r>> (1.18)

The nondimensional magnitude, the asymmetry of distribution of the
probabilities for the longitudinal component of the velocity difference
is now introduced

S = (1.19)
[D (r)1312

From the considerations of the theory of dimensions, it follows
that for r>> the magnitude S should have a constant value (it
can depend only on r and on e, but from these two magnitudes it is
not possible to obtain any nondimensional combination). From equa-
tions (1.19), (1.18), and (1.12) it follows that for r>>

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) = 4 2/3 2/3 2/3
Dn (r) = 'S e r

Dnn(r) 4 2/3 2/3 r2/3

The coefficients C and C' of formulas (1.9) are thus connected
with the asymmetry S by the following simple relations:

4) 2/3
C' =4C

It follows that S is always negative: S = I S. Formulas
(1.17), (1.9), and (1.21) were obtained by A. N. Kolmogoroff in 1941
(references 1 and 2). Up to that time, the results obtained from the
equations of hydrodynamics only slightly improved the accuracy of the
results obtained previously from a dimensional analysis and they
referred only to the two extreme cases: r>> and r<< T. In the
matter of the computation of D (r) for the intermediate values of r,
the single relation (1.16) is of course not sufficient. In this rela-
tion are two unknown functions D (r) and DI11(r), and therefore
still another relation between them is required for their determination.
The theory does not give this needed relation, but an attempt may be
made to derive it from experimental data.

At the present time, results are known of the direct measurements
of the magnitude S for various distances, conducted by Townsend
(reference 4) in wind-tunnel tests at very high Reynolds numbers for
the purpose of checking the theory of Kolmogoroff. These measurements
have shown that the asymmetry S may, with a sufficient degree of
accuracy, be assumed as constant not only for r>> but in general
for all values of r lying within the range of applicability of the
theory of locally isotropic turbulence. The experimental value of S

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for all values of r is approximately -0.4.9 This experimental fact
provides the additional relation between D1(r) and D111(r), which
permits the determination of these functions uniquely for all values of

Thus the asymmetry S is assumed constant. From equations (1.16)
and (1.19)
dD 4
6v -d + IS [D (r)] /2 (1.22)

where IS is constant. This equation in the function D1-(r) with
coefficients depending on v, e, and IS| is considerably simplified
if transfer is made to nondimensional magnitudes and the as yet unde-
termined numerical factors kI and k2 are in the expressions for the
scales (i.e., use is made of formulas (1.6) and (1.7)). Then for

6 k22 + IS L k23 (x = klx (1.23)

The magnitudes e and v no longer enter into this equation.
For a corresponding choice of the constants k, and k2, it is also
possible to eliminate the experimental constant ISI and obtain for
031(x) an equation with numerical coefficients. It is convenient to
choose k1 and k2 such that

IS kk2 \3/2
6 (1.24)
2 k2
2 -
15 k2 2

9The experimentally determined values of S fluctuate between
the limits -0.36 and -0.42. This scatter lies within the limits of
accuracy of the measurements. As the most probable value of S
Townsend gives the value -0.38. However, this value may not be assumed
reliable for purposes of this report.

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that is, to set
k 4V-- 1 5.035
k 4^F2 1 1.838

The equation for Z11(x) is then

dx (x+ I (x32= x (1.26)

Equation (1.26) together with the initial condition P3,(0) = 0
uniquely determines the nondimensional longitudinal structural function
hP7(x) which describes the structure of the velocity field.10

10The structure of a turbulent flow may likewise be described with
the aid of the spectral energy distribution. In this case, E(p)
denotes the energy of the system of disturbances the wave number of,
which is larger than p (the scale of disturbance is inversely propor-
tional to the wave number). In the statistical theory of homogeneous
(stationary) processes and fields, it is shown that there exists a one
to one correspondence between the correlational (structural) functions
and the functions E(p); the formulas that permit expressing one of
these functions in terms of the other approximate in type the Fourier
transformation (cf. references 20 and 31). The 2/3 law for the struc-
tural functions, equations (1.9), is equivalent to the ratiq of the
spectral function E(p) for p< the ratio of the spectral density dE(p)/dp = E'(p) to the magnitude
p-5/3). The scale rT corresponds in the spectral theory to the critical
wave number pl = l/ The 2/3 law was first obtained in this form by
A. M. Obukhoff (reference 3) in 1941. The complete description given
in the text of the structural function Dn(r) is equivalent to the
determination of the spectral function not only for p< in general, for all values of p. There are a number of attempts
(references 3, 21, 22 and 5) at a direct theoretical computation of the
function E(p) for all p. The results thereby obtained are however
difficult to compare with experimental data.

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The corresponding nondimensional transverse structural function pnn(x)
is determined from the relation (1.12) which, after substitution from
equation (1.6), may be represented in the form

Inn(x) = 1(x) + x d -(x) (1.27)

Figure 1 shows the graphs of the graphs of the functions 0 ,(x)
and nn(x), where 3,,(x) was determined with the aid of numerical
integration11 of equation (1.26) for the conditions P 1(0) = 0, and
Pnn(x) was computed with the aid of p22(x) from relation (1.27).
The dotted curves denote the asymptotic values of these functions for
small and large values of x:

pI(x) x2

for x<
Onn(x) x2

~ (x) = 2/3
22 4
for x>>l (1.29)

Pnn(X) = x2/3

These formulas correspond to the asymptotic equations (1.17) and (1.20)
for the structural functions. The particularly simple form of the
asymptotic formulas for the function Pnn(x) permits a very simple
determination of the magnitudes of ql and ul of equation (1.6) from
the transverse structural function nn(x) which was obtained from

11For large values of x (for x>8), it is convenient to make
use of the asymptotic expansion for 131(x):

S3 2/3( 1 -4/3 5 -8/3
W x 1 3x + .

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experiment.12 It is for this reason that the previously mentioned values
for the coefficients k1 and k2 were chosen.

A direct comparison of the computed curves with the experimental
curves obtained in wind-tunnel 'measurements is technically difficult
to make because of the smallness of the scale T. In wind-tunnel
measurements it is thus usually possible only to check the agreement
with the 2/3 law (see for example references 4 and 5). With relation
to the results which refer to the trend of the curve for r -~ 1'
it is necessary to be satisfied with an indirect check of the type used
in checking the accuracy of the constancy of the asymmetry factor.
From this point of view measurements in the free atmosphere are evi-
dently more convenient because here the scale Tl is somewhat larger
(of the order of several mm). Nevertheless, such experiments are very
complicated and up to this time only one investigation containing data
referring to scales of the order of nl is known. This is the investi-
gation of G6decke (reference 23) in which the mean absolute differences
in velocity in a direction perpendicular to the base (which corresponds
to the transverse structural function) is measured for distances of r
varying from 0.1 to 80 centimeters at an altitude of 1.15 meters [?].
The evaluation of these data (reference 7) has shown that they are in
good agreement with the theoretical curve obtained herein for pnn(x)
where Tl = 0.54 centimeter and ul = 2.02 centimeters per second.

2. Computation of structural function of pressure field. The
study of the local structure of the field of pressures in a turbulent

Technically, the measurement of Dnn(r) can be affected much
more simply than the measurement of D11(r). For this reason Dnn(r)
is generally measured in experimental work. Approximation of the curve
obtained for D nn(r) to a parabola for small values of r to a parab-
ola and to the 2/3 law for large values of r gives precisely the mag-
nitudes of Tl and u 2, the coordinates of the point of intersection
of these two asymptotic expressions. The above construction is con-
veniently carried out on logarithmic scale; the parabola and the 2/3 law
are thereby represented by two straight lines (cf. reference 7).

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flow is considered
of this structure,
ponding structural

in this section.13 As a quantitative characteristic
as in the case of the velocity field, the corres-
function is chosen

I(M,M') = [p(M) p(M)]2


In the case of a locally isotropic flow, the
for a distance r between the points M and M'
external scale of turbulence L, will depend only

function IT(M,M'),
much less than the
on r:

II(M,M') = II(r)


and will be entirely determined by the local structure of the flow.

From considerations of the theory of dimensions it follows that

tI(r) = q12,L r


1 = k
i1 = k 1 T1

qg = pul2 = k22 ? pe


the numerical coefficients kI and k2 being assumed to coincide with
the coefficients in equation (1.25) and t(x) being a universal func-
tion. Further, since for r>>1 the structural formula II(r) should
not depend on the viscosity v, the asymptotic equation is

lt(x) ~ x4/3

for x>>l


and therefore

TI(r) 02F4/3r4/3 o2 D2(r)]2 for


13From the fact that when deriving the fundamental equation connect-
ing the second and third moments of the velocity field of an isotropic
(locally isotropic) incompressible flow, the pressure is excluded (see
references 18 and 19 and also equation (1.13)), it does not follow that
in an isotropic (locally isotropic) turbulent flow fluctuations of the
pressure are absent. Such an erroneous conclusion has been drawn by
M. D. Millionshtchikov (reference 24).


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It will now be shown how the numerical coefficient in this formula
and, in general, the entire trend of the function i(x) may be approx-
imately computed.

For this purpose use is made of equations (1.14). If the ith
equation is differentiated with respect to xi and summed over i, then
on account of relation (1.11) the terms with v/vipt and with Avi
drop out and

x( Ap (2.7)

3 ov. ov.
Ap = -p (2.8)
i,j=l 7x_.xj

(the equation of continuity is again applied).

From equation (2.8) it is not difficult to derive the differential
equation for the function TI(r). It is simplest to proceed as follows.
At first the assumption is made that the velocity field and pressure
field are statistically homogeneous and isotropic (and not only locally
homogeneous and locally isotropic). In this case, the left and right
sides of equation (2.8), written out for the point M with coordinates
xl, x2, x3, are multiplied correspondingly by the left and right sides
of the analogous equation for the point M' with coordinates xj, x2,
x', and the result is averaged and after taking into account the fact
that in the case of a homogeneous and isotropic pressure field

Ap(M)Ap(M') = 2[p(M) p (M' )]

where when differentiation is carried out on the right side with
respect to the components ti = x' xi of the vector MM'

2 2 E v.(M) )v (M) 6vk(M') bv (M')
A2p(M)p(M') ='p2 C1 -v- ()- (2.9)
i,j,k,3 -xj xi dx dx

It should now be noted that in the case of a. homogeneous and iso-
tropic flow the correlation function p(M)p(M') is connected with the
structural function (2.1) by the relation (see previous footnote):

NACA TM 1350

II(r) = 2[p2 p(M)p(M')] (2.10)

Equation (2.9) may therefore be rewritten in the form

A2(r) d4I(r)+ 4 d-IT(r)
dr4 r dr3

S 2 v i (M) v (M) vk(M') 6v (M')
-2p ioj-k,- I xx (2.11)
ijk,l -x 7-7i x xT

This is the required equation. It also has a meaning in the case
of locally homogeneous and locally isotropic (but not homogeneous and
isotropic) flow, and with the aid of more complicated considerations
it may also be derived without the assumption of homogeneity and iso-

The structural function l1(r) is thus seen to be a solution of
equation (2.11), in the right side of which appears a combination of
four moments of the derivatives of the velocity field. Unfortunately
these moments are not known, and in order that any use may be derived
from equation (2.11), it is necessary to make an additional assumption
which will permit computing these moments. The assumption adopted herein
is that proposed by M. D. Millionshtchikov (reference 24) which states
that the fourth moments of the velocity field are expressed in terms
of the second moments in the same manner as in the case of the normal
Gaussian distribution.14 As a first approximation this assumption
appears to be an entirely natural one. This assumption finds a certain
justification in the measurements of Townsend (reference 4) which show
that the experimental value of the fourth moment for the velocity deriv-
ative 6vl/xl1 differs by no more than 15 percent from the value com-
puted by the measured value of the second moment on the assumption of
normal distribution.

For any four chance magnitudes w1, w2, w3, and w4 subject to a
four-dimensional normal-distribution law, the equation holds (see for
example, reference 25):

lw234 2 w 34 13 24 + 4 w 23

141t is noted that in the recent work of Heisenberg (reference 21)
a hypothesis with regard to the spectral functions of an isotropic tur-
bulent flow precisely equivalent to that proposed by M. D. Millionsctchikov
was used.

NACA TM 1350

When this formula is applied to the product of the four derivatives
of the velocity field which enter into the right side of equation (2.11),
the following equation is obtained:

avY(M) avj(M) 6vk(M')
x. x. i



avi(M) Vk(M ) Ir. +
3x 1x xi ;


vi(M) xvj(M)
6x 3x i

6vk(M') 6avZ(M')
8xy kx



The first term on the right-hand side of the equation is propor-
tional to Ap(M) Ap(M'. In the case of a locally isotropic flow, it
is easily verified that this term becomes zero, as can be derived, for
example, from equations (1.3) and (1.12). The last two terms of equa-
tion (2.12) are equal to each other. It is further noted that in the
case of a locally isotropic flow

2 Di(MM')



3vi(M) 6vk(M')
Oxj x -

where Dik is the structural function in equation (1.1) and
C = x! x.. From this it follows that for the assumption made about
the relation of the second and fourth moments, equation (2.11) may be
represented in the form

d41(r) 4 d3I(r) 2
4 + -r -p i
dr4 r dr3 i,j,k,l

2 Dik
ae ai&

2 Dj,

Ci Ek


The function on the right side of this equation depends, of course,
only on r

P(M,M') =


o2 Dik(M,M')

Ii t

o2 D I(M,M')
-- i = 4(r)
b^i ^ vlk


vi(M) v1z (M') v (M) avk (M)
6x. 6x 3x. x
j k

NACA TM 1350

With the aid of equations (1.3) and (1.12), equation (2.15) may be
reduced, after rather long transformations, to the form

6 2 d2 D d 2
6 (dD12 21 + 20 + d(d2D 2 dDl1 D
$(r) 6--D1) +=20 dr d_2 + 41D1 + 4 d d (2.16)
r2 dr r dr dr2 dr2 dr dr3

In accordance with the definition (2.1), the function Xl(r) is
even and assumes the value zero for r = 0. Therefore, the two bound-
ary conditions which result are:

n(0) = 0 (2.17)
JII'() = 0 (2.18)

As a third boundary condition use is made of15

11(r) -0 for r-*w (2.19)

Equation (2.14), for the conditions of equations (2.17), (2.18),
and (2.19), has a unique solution which is the required structural

Since all linearly independent solutions of the homogeneous equa-
tion corresponding to equation (2.14) are found without difficulty
(they are 1, r, r and r-1), the required solution of the nonhomo-
geneous equation can be constructed with the aid of Green's functions.
It is easily verified that in the case of the boundary conditions
expressed in equations (2.17), (2.18), and (2.19), this function for
equation (2.14) has the form

151t may be shown that this condition is required so that the
correlation between the differences in the values of the pressures at
two pairs of points will approach zero as one pair of points recedes
infinitely from the other (the distance between the points for each
pair is assumed to be fixed).

161t may appear strange that only three boundary conditions are
used, whereas equation (2.14) is of the fourth order. The fact is,
however, that equation (2.17) is a double condition: Zero is a singu-
lar point of equation (2.14) and therefore one boundary condition will
be the requirement that the function have regularity at zero.

NACA TM 1350

G(r,e) =
3 2 4
LrL -
2 2 6r

for r

for r at

The required solution for 11(r) may be represented in the form

(r) = -p2 G(r,)@(-g)dg
IT~r = J-


The function 4(r), given by equation (2.16), may, on account of
relations (1.6) and (1.7), be represented in the form

.(.~). ^


V2 J


where kl, k2, and ql are determined from equations (1.25) and (1.7)
and cp(x) is the universal function:

p(x) (d-1 2)
x2 dx

S20 d~3 d 21
+ +
x dx dx2

(2 2
4 1

d 3 83
dx dx3

When equations (2.20) and (2.22) are substituted in equation (2.21)
and a change of variables is made (cf. equations (2.3) and (2.4))

1(r) = k24 2 ve2 (



I(x) J

. 2

+ X (px2 ()ds
wx 6



+ t ( )d(

o x 2I r)X I2 l x 4(()di
1- ()d + 2()d + L 4p()d +2
= 60



NACA TM 1350

Thus the universal function t(x) of equation (2.3) is connected
with the function 30 (x), which was computed in the preceding section,
by use of relations (2.25) and (2.23). Equation (1.26) expresses the
derivative dop/dx in terms of the function $p3(x). When this equa-
tion is applied several times, the second and third of
these functions can be expressed in terms of 37l(x) and therefore
also the function ((x). Thus, with knowledge of the function P11(x)
from section 1, p(x) can be determined and all the integrals in equa-
tion (2.25) can be numerically computed, that is, the function v(x),
which determines (due to a relation with equation (2.3)) the structure
of the pressure field can be computed. The graph of the function o(x)
thus obtained is given in figure 2.

The dotted curves in figure 2 show the asymptotic behavior of
it(x) for small and large values of x. Since the motion of the fluid
for scales much smaller than ql is laminar, for x<
x(x) = ax2 (2.26)

(See the analogous derivation for the structural functions of the vel-
ocity field.) The coefficient a in this formula can easily be
obtained from equation (2.25) as follows. From equation (1.28),
P 2(x) x2/2 for x< of x, may be considered as constant: p(x) 530. When this value is
substituted in equation (2.25), the term proportional to x2 gives
only the last of the integrals in equation (2.25) and

a 6= 0 / ()d 0.8 (2.27)

(This value has been obtained with the aid of numerical integration.)
The curve

t(x) = 0.83 x2

is the first of the asymptotic curves drawn in figure 2.

In the second limiting case, for x>>l1 equation (1.29) shows
P 3(x) 3x2/3/4, and therefore p(x) = 7x-873/18. Equation (2.25) is
now represented in the form

NACA TM 1350

-8/3 d + x 7 -8/3 d+

j+ ( +)( r8
^0 V 2 2 6x (t) 18-


- 7 -8/ d3

d +

+ + 2 (-8 -+

t (PC)

- 7 -8/3)a


It is not difficult to see that the values of the-integrals on the
right side of equation (2.28) for x-+ will not increase any faster
than the first degree of x, so that the principal term of the asymptotic
formula for ir(x) will be the term 9x4 //16. Thus, the numerical
coefficient in equation (2.5) is equal to 9/16 and the symbol of the
asymptotic equation (2.5) means only that

tx) (x) = 1

P2 Dz I (r)]2

The difference A(x) 9x4/3/16, however,
as x increases.

for x>>l

for r>>Tq

increases without limit

To obtain the succeeding terms of the asymptotic formula for
t(x), equation (2.28) is further transformed:

K (0)

NACA TM 1350

9(x)= x4/3 2 ( ) -3 do -
16 2 18 2 2

T31 Jo

(3 (7 -

L -8/3)d +

-x 2 + 1 3
2 2

1 x 2 (
6x 6

Here the integrals over the range from 0 to converge very
rapidly and may be numerically computed while the last integral over
the range from x to may be evaluated for x>>l with the aid of
the asymptotic formula given in a previous note. It should be noted
that this integral adds only an insignificant increment to the constant
term of the asymptotic formula for A(x). Finally, with an accuracy
up to terms approaching zero as x- ,

9 4/3
n(x) = -1 x

- 0.08x + 0.85

for x>>l (2.30)

This is the equation for the asymptotic curve for large values of
x plotted in figure 2.

No knowledge of any experimental data on the structure of the pres-
sure field which could be compared with the results obtained herein is
known to the authors. It should be remarked that the computations pre-
sented previously show that the mean square values of the differences
in pressures are found to be so small, as a rule, that their measure-
ment would be associated with very great experimental difficulties. It
does not follow from this, however, that the computation of the struc-
tural function of the pressure field is practically useless. In the
following section it will be shown that the values of the local pressure
gradients thereby obtained are very large so that the accelerations pro-
duced by the fluctuations of pressure may play an essential role in
processes which arise in turbulent flow.

-8/3 dt +


18 ) d

t4 (p(t) -

NACA TM 1350

3. Computation of correlation functions of acceleration field. A
study of the acceleration field of the fluid particles in a turbulent
flow is now undertaken. This field differs from the fields considered
in the previous sections in that the very smallest and not the largest
vortices17 are essentially responsible for values of the acceleration at
a point, as is the case for the velocity and pressure fluctuations. For
this reason, in the case of the field of accelerations of the local
flow structure, not only the statistical characteristics of the differ-
ence in values of the field at two points (e.g., the structure function)
are determined, but also the statistical characteristics of the values
of the field. The most important of these characteristics is the corre-
lation function, the mean value of the product of the values of the
field at two points (i.e., in the case under consideration, the mean
value of the product of the acceleration components).18 The computation
of this correlation function is the main concern in this section.

The value of the correlation function at zero is determined first,
that is, the mean square of the acceleration of a fluid particle at a
single point. This magnitude is the numerical characteristic of most
interest of the acceleration field. From the equations of motion (1.14),
the acceleration components of the fluid particle
dv. ov. 3 3v.
W = d- = t + v (i = 1,2,3) (3.1)

17From considerations of the theory of dimensions it follows that
to vortices of the scale of 1, where I>>T, there corresponds the
characteristic period T2 = (2/e)1/3 such that the velocity character-

istic for these vortices is equal to v, = I/T2 = (el)1/ and the char-
acteristic acceleration is wI = Z/T,2 = (e2/2)1/3. Thus it is observed
that when the scale of lengths is decreased, characteristic velocity
decreases while the characteristic acceleration increases. From this
it follows that the very small vortices of scales V1 1 are mainly
responsible for the value of the acceleration at a point of the flow
(for such vortices, the dimensional considerations adduced herein do not
correspond, of course, to actual conditions, for the motion of these
vortices essentially depends on the viscosity)

18It is clear that the correlation function is a more significant
characteristic of the field than the structural function. Knowledge of
the correlation function always allows determination of the structural
function also. The converse does not hold true.

NACA TM 1350

are equal to

1 p


from which is obtained



2v p- Avi
p i= xi

The first and third term on the
expressed, without difficulty, in
the velocity and pressure fields,


i=lx i)


Z (Av.)2 =-
i=l 1

a2 11(0)

2 S

+ V2 (Avi)2


right side of this equation may
terms of the structural functions
equations (1.1) and (2.1):

S3 d211(0)
2 dr2




The middle term on the right side may be expressed through the
interrelated structural functions

D. (M,M') = [v.(M') v.(M)]Vp(M') p(M)](i = 1,2,3) (3.6)

of the velocity and pressure fields.
local isotropic flow these functions
tion (1.13)), the middle term on the
becomes zero, and therefore

i=l 1

32 d2 (0)
2p dr

Since in the case of incompressible
should be equal to zero (see equa-
right side of equation (3.3)


i= 1


But on account of equations (2.24), (1.7), (1.25), (2.26), and



NACA TM 1350

k2 2 -1/2 3/2 1 k2

0.15 2 -1/2 3/2 0.74 2 -1/2
SIs .96p v & = i v
-Is s,


d2 (0)

_ 1- 3 ( pY2 1.1 -1/2
P2 i= Tx

2 -1/2 3/2
P v 'e





Further use is made of the fact that for any choice of



Dii(r) = DIZ(r) + 2Dnn(r)



and of equation (1.12), the following is obtained:

A2 D d4
=l Dii(r) d

+ 3 )
r dr3)(_ I r

rdD (r)
dr 7

d5D, (r)
= r

d4D (r)
+ 11 dr

SdD (r)
r dr3

With the aid of formulas (1.6) and (1.7), the change from
to the nondimensional function 3,,(x) gives

19The computation of the magnitude Igrad p12 for locally isotropic
turbulence is also contained in the work of Heisenberg (reference 21).
The method of Heisenberg is based on the employment of the spectral
function E(p) and requires considerably more complicated computations.
Moreover, in the final formula of Heisenberg, magnitudes enter which
cannot be separately measured in tests.





NACA TM 1350


(r) = -5/2

1I VI r +

3/2r_ r V r ) +
71 Ill


It is now noted from equation (1.25) that


s12 Vr


and that 0 (x) is an even function of x which may be expanded in
the neighborhood of zero in a power series in x2:

oiZ(x) = blx2 + b2x4 + .


From equations (3.12), (3.13), and (3.14) the following is obtained:

2 D( s -5/2 3/2 7 t S -5/2 5/2
A I D 0 J= D o 2840 b2 = b V5/2 3/2
i= ii 120^54 2

By use of this method, only the determination of the coefficient
b2 in equation (3.14) remains. From the first of equations (1.28) it
follows that bl = 1/2. When the expansion (3.14) is substituted in
equation (1.26) and the coefficients of r3 are equated (or, what is
equivalent, differentiating equation (1.26) with respect to r three
times and then setting r = 0), the following equation is readily

b2 = (3.16)

The substitution of this value of b2 in equation (3.15) gives

- A Dii(0
v1=1 /

S2 (A)2 71SI -1/2 3/2 0.3 -1/2 5/2
i=l 6 AF15

24 1

NACA TM 1350

Since IS I 0.4, it follows from a comparison of equation (3.9)
with equation (3.17) that the acceleration of the fluid particles in a
turbulent flow is essentially determined by the fluctuating pressure
gradients and not by the friction forces. The term with II''(0) in
equation (3.7) is more than 20 times as large as the term depending on
the viscosity. It shall be seen that this greatly simplifies the compu-
tation of the correlation functions of the acceleration field.

When equations (3.9) and (3.17) are substituted in equation (3.7),
the following formula is obtained for the computation of the mean square
of the acceleration wO:

'O2= w2 1. + O.31S -1/2 3/2 (3.18)
i=l JS(

Since jIS = 0.4, equation (3.18) may be replaced by the simple
-1/2 3/2
w02 = 3v 8 (3.19)

This general relation permits the estimation of the order of mag-
nitude of w0 in specific cases of turbulent flow without difficulty.

As an example, formula (3.19) is applied to the computation of the
mean square acceleration in certain turbulent flows behind a screen (or
grid) in wind tunnels and in turbulent atmosphere. In the case where
isotropic turbulence was produced by screens in wind tunnels, the dissi-
pation a may be defined either as
3 dv'2
-fV dx

where v'2 is the mean square of the velocity fluctuation, V the
mean velocity, x the distance from the screen, or as


where X is the length introduced by Taylor, experimentally determin-
able by inscribing a parabola in the graph of the correlation function
B11(r). When the dissipation e is known, w0 can be computed from
the formula

WO = 2.77 e3/4 cm/sec2 (3.20)

NACA TM 1350

obtained by substituting the air viscosity v = 0.15 sq cm/sec in equa-
tion (3.19).

In particular, when use is made of some of the data given by
Townsend (reference 4) (these data refer to the flow in a wind tunnel
behind a square screen with size of mesh M = 6 inches at a distance
x = 30.5 M from the screen for various values of the velocity V), the
following values for e and w0 are obtained:

V m sec-1 a cm2 sec-3 wO cm sec-2

12.2 60.5 60.4
24.4 312.4 206.8
30.5 559.8 320.3

From this table it is observed that the instantaneous values of
the acceleration in turbulent flow behind the screen will be of the
order of several meters per second per second.

The application of formula (3.19) or (3.20) to the computation of
the accelerations in a turbulent atmosphere is rendered difficult by
the fact that at the present time there are no available measurements
of energy dissipation for this case. However, for the degree of accur-
acy of the computations, much justification exists for employing an
estimate of the magnitude of & for a turbulent atmosphere by the
formulas of the theory of the logarithmic boundary layer. It is known
(reference 15) that for the logarithmic boundary layer
e (3.21)
x y

where y is the distance from the wall, x is a nondimensional con-
stant (Karman constant) equal approximately to 0.4, and v. =O /P
(To is the friction stress, p the density) is the so-called dynamic
velocity determined by the difference of the mean velocities at two
points or by the mean velocity at one point and the magnitude of the
roughness. Substitution in formula (3.19) of expression (3.21) for
the dissipation and v = 0.15 sq cm/sec gives a computational formula
which determines the mean square acceleration in a logarithmic boundary
air layer:

v /4cm
= 5.5 cm (3.22)
y 3 sec

NACA TM 1350

Since v*. is proportional to the mean velocity V,

WO- V9/4 (3.23)

that is, w0 increases rapidly with V. For the example, the magnitude
of the roughness is assumed to be ho = 5 cm (it is noted incidentally
that the computations following depend relatively little on the magni-
tude of the roughness) and the mean velocity of the wind at the height
150 cm is denoted by V. Then

v n(/h 0.1 V (3.24)

and for the mean square acceleration wo at various velocities V the
following values are obtained:

V, m sec-1 1 3 5 6 8

wO, cm sec-2 22 260 830 1200 2400

The mean square acceleration under the conditions considered for
a mean velocity of the wind V = 5.5 m/sec thus attains the magnitude
of the acceleration of gravity g, and for a greater wind velocity may
considerably exceed this acceleration. It is natural to assume that
such large accelerations may play a significant part in many physical
processes in the atmosphere (e.g., in the phenomenon of the condensation
of fogs).

The computation of the correlation function of the acceleration
field is now considered:

Aij(M,M') = wi(M)wj(M') (3.25)

Again, substitution of equation (3.2) gives

A..(M,M') = 1 p Ep' V P A'v, +Av. P'E + v Av.A'v,
10 2 o x p 3 1 o 1 3


The magnitudes without the primes refer to point M and those with
primes to the point M'. The middle term on the right side may be<
neglected for the same reasons for which the middle term on the right
side of equation (3.3) was previously rejected, and the first and third

32 NACA TM 1350

terms may easily be expressed in terms of the structural functions (1.1)
and (2.1). Therefore,

)p op'

= a. (M,M')
2- ti


where (i and (

are the components of the vector MM' and

Av.A'v = -AD (M,M')
I j j

S 2

'3 2)

The transformation of equations (3.27) and (3.28) follows.
2 2 2
fl(M,M') depends only on the distance r = +i +2 + 3

= dr


1 dll(r) i j
r dr J2+



1 d1(r)
r dr


Replacement of Dij(M,M') by means of equations (1.3) and (1.12)

[Di(r) + D(r)ij

- 1 D (r)
2r I 22 ti ij

which gives the following:

A2Dij(M,M') = Dl(r) -


r3 dr

4 dD2
r3 dr






+ D2(r)6ij

- r4 4

+ dr3D
r dr3

+ 5 d1z


Aij(M,M') = Al(r) + A2(r)

2 n(M,M')




+ r d5DZ
2 dr5


NACA TM 1350


Sl- id2r(r) dlfl(r) ( )
Al(r) 2p2 dr2 r dr -- Dl(r) (3.34)
2p r rdr r

A2(r) 2 dl(r) D- D2(r) (3.35)
2p r

and Dl(r) and D2(r) are determined by formulas (3.31) and (3.32).

The functions Al(r) and A2(r) are expressed in terms of the
longitudinal and transverse correlation functions of the acceleration
field determined by the equations
A, (r) = w (M)w,(M') (3.36)

Ann(r) = wn(M)wn(M')

where wl(M) and wl(M') are the projections of the accelerations at
the points M and M' on the direction of the vector MM' and wn(M)
and wn(M') are the projections of the accelerations at these points
in a direction perpendicular to the vector MM'. In fact, the acceler-
ation field of a locally isotropic turbulent flow is isotropic in the
usual sense, and therefore
A, (r) Ann(r)
Aij(M,M') = r2 n j + Ann(r)ij (3.37)

(see reference 19 and equation (1.3) herein). Comparing equations
(3.33) and (3.37) and taking into account equations (3.34) and (3.35)

A,(r) = Al(r) + A2(r) = dr2I (Dl(r) + D2(r))
2p dr
Ann(r) = A2(r) 1 dII(r) D2(r) (3.39)
202r dr 2(r)
2p r

In formulas (3.38) and (3.39) it is possible, in the usual manner,
to pass to nondimensional functions. These may be further computed
with the aid cf the results of sections 1 and 2.

34 NACA TM 1350

It may be noted that in these computations the terms with Dl(r)
and D2(r) may be neglected without introducing any appreciable error.
In 'act, it was shown previously that for r = 0 the terms depending
on the viscosity, that is, the terms containing Dl(r) and D2(r), are
negligibly small compared with the terms determining the pressure gra-
dients. With increasing r b'cth terms decrease asymptotically, the
terms depending on the viscosity decreasing much more rapidly than
those determined by the pressure gradient. From formulas (2.6) and
(1.9) it follows that for r>>l

d2I(r) r-2/3

1 dIl(r) r-2/3
r dr

D1(r) ~ r10/3
D2(r) ~ r

Thus, for both small and large r, the terms of equations (3.38)
and (3.39) containing v are considerably smaller than the terms
depending on Il(r). In this connection, the investigation of the struc-
ture of the acceleration field in a turbulent flow permits the rejection
of terms with viscosity in the equations of motion, and the assumption

w. =-1 (i 1,2,3) (3.42)

Aij(M,M,) 1 I2I(M,M') (5.45)
1J 2p2 2p0

For the longitudinal and transverse correlation functions (3.36),
there is then obtained

A (r) 1- d211(r)
2p dr (3.44)

Ann(r) 1 d1l(r)
nn 2p2r dr

NACA TM 1350

With the aid of formulas (2.24), (1.7),and (1.25), the change to
nondimensional magnitudes is made,and using equation (2.25)

A,(r) = -1/2 3/2 ( 0 -1/2 63/2 (345)

k2 -1/2 3/2
Ann(r) = 2- v 6

where al (x)
the formulas

"- r

0.45 -1/2 3/2 (-)
VT 6/ nn (l)


and ann(x) are universal functions which are given by

1 r6
6x Jo

a nn(x) = 1
nnv 4x

4 1

x x
2(()d 1 1-- 3
12x 0 x


S4e(P)d +


As in the case of the velocity and pressure fields, for x< x>>l, it is possible to obtain for the functions introduced in the
theory described herein simple asymptotic formulas. It is clear first
of all that

(0) = Cnn() = 1 p()d = 0.83


If in formulas (3.47) and (3.48) x is assumed much less than 1
(x< shown in the first formula of equation (1.28), P12 (x) = x2/2, and there-
fore P(x) = 30; whence

j (x) _= 1(0) x2

for x<

1 2
a (x) = a (0) 2 x
nn nn 2

NACA TM 1350

In the second extreme case, for x>>l, the asymptotic behavior
of a~LI(x) and ann(x) is determined with the aid of formulas (2.30)
and (3.44).

cl,(x) x-2

for x>>l (3.51)

M W 5 -2/3
n(x) = x-
nn 8

The computation of the functions cax(x) and %nn(x) for x-1
may be carried out numerically by using the data contained in sections
1 and 2. It is convenient in place of ajj(x) and Lnn(x) to intro-
duce the normalized functions
R(x) = t(x)

x (3.52)

Rnn(x) = ann(o)

These functions are equal respectively to the correlation coeffi-
cient of the longitudinal and transverse components of the acceleration
at two points a distance r = xi-l from each other. The graphs of the
functions R3 (x) and Rnn(x), which were determined by numerical inte-
gration of the integrals appearing in the right sides of equations (3.47)
and (3.48), are shown in figure 3. It is seen that the longitudinal
correlation function R2n(x) rapidly decreases, and for x>1.1 it
may practically be considered equal to zero. The function Rnn(x), on
the contrary, decreases at a relatively slow rate, and for x = 3 is
approximately equal to 0.17. When the magnitudes of these functions are
estimated for relatively large values of x (of the order of 10 and
above), formulas (3.51) may be used. From these formulas, when x = 10,
for example, RT.(10) = 0.03. (In fig. 3 the range of applicability of
formulas (3.51) is not represented, since to do so it would be necessary
to choose a much smaller scale.)

It may be noted further that the form of the correlation functions
of the acceleration field shown in figure 3 differs sharply from the
form of the correlation functions of the velocity field for isotropic
turbulence. In the case of the velocity field, the graph of the longi-
tudinal correlation function is generally located above the graph of
the transverse function and the axis of the abscissas intersects the
second and not the first of these curves. This difference in behavior
of the correlation functions for the velocities and accelerations is

NACA TM 1350

explained by the fact that the velocity field in an incompressible fluid
is a. solenoidal vector field, whereas the acceleration field is con-
sidered as a potential vector field (see equation (3.42)). From this
it follows that the functions R,,(x) and R nn(x) are interconnected
by the relation
dRnn(x) (
(x) = Rnn(x) + x (3.53)

[This relation, which is a necessary and sufficient condition for the
isotropic potential vector field having the correlation functions
R,,(x) and Rnn(x), was obtained by A. M. Obukhoff,while the correla-
tion functions B,1(r) and Bnn(r) of the velocity field satisfy the
Karman condition (cf. reference 19 and equation (1.13)]:

Bnn(r) = B,(r) + dB(r) (3.54)
2 d2 ar

Conditions (3.53) and (3.54), in addition to the factor 1/2 in
the second term on the right, differ in the interchange of the roles
o-f the longitudinal and transverse functions. It is not surprising,
therefore,that the functions Rl,(x) and Rnn(x) behave in a manner
opposite to the behavior of the functions B1(r) and Bnn(r).

In conclusion, the authors wish to express thanks to
A. V. Perepelkina and Y. V. Prokhorovaywho carried out the numerical
computations for sections 2 and 3.

Translated by S. Reiss
National Advisory Committee
for Aeronautics


1. Kolmogoroff, A. N.: The Local Structure of Turbulence in Incom-
pressible Viscous Fluid for Very Large Reynolds Numbers. DAN
(SSSR), vol. XXX, no. 4, 1941.

2. Kolmogoroff, A. N.: Dissipation of Energy in the Locally Isotropic
Turbulence. DAN(SSSR), vol. XXXII, no. 1, 1941.

3. Obukhoff, A.: On the Energy Distribution in the Spectrum of a. Tur-
bulent Flow. Izv. AN SSSR, ser. geogr. i geofiz, vol. XXXII,
no. 1, 1941.

NACA TM 1350

4. Townsend, A. A.: Experimental Evidence for the Thecry:, of Local Iso-
tropy. Proc. Cambridge Phil. Soc., vol. 44, pt. 4, Oct. 1948,
pp. 560-565.

5. Von Karman, Theodore: Progress in the Statistical Theory of Turbu-
lence. Proc. Nat. Acad, Sci., vol. 34, no. 11, Nov. 15, 1948,
pp. 530-539.

6. Richardson, L. F.: Atmospheric Diffusion Shown on a Distance-
Neighbour Graph. Proc. Roy. Soc. (London), ser. A, vol. 110,
no. 756, 1926.

7. Obukhoff, A.: Local Structure of Atmospheric Turbulence. DAN SSSR.,
T. 67, No. 4, 1949.

8. Stommel, H.: Horizontal Diffusion Due to Oceanic Turbulence. Jour.
Marine Res., vol. 8, no. 3, 1949.

9. Krasilnokov, V. A.: On the Propagation of Sound in Turbulent Atmos-
phere. DAN(SSSR), T. 47, No. 7, 1945.

10. Yudin, M. I.: Problems of the Theory of Turbulence and Wind Struc-
ture with Application to the Problem of the Vibrations of an
Airplane. Gidrometizdat, 1946.

11. Obukhoff, A.: Structure of the Temperature Field in a Turbulent
Flow. Izv. AN SSSR, ser. geogr. i geofiz, vol. XIII, no. 1,

12. Yaglom, A. M.: On the Local Structure of the Temperature Field in
Turbulent Flow. DAN(SSSR), T. 69, No. 6, 1949.

13. Krasilnokov, V. A.: On the Effect of Fluctuations in the Coeffi-
cient of Refraction in the Atmosphere on the Propagation of Ultra-
short Radio Waves. Izv. AN SSSR, ser. geogr. i geofiz, T. 13,
No. 1, 1949.

14. Krasilnokov, V. A.: On the Fluctuations of the Angle of Incidence
in the Phenomenon of Star Twinkling. DAN(SSSR), T. 65, No. 3,

15. Landau, L. D., and Livshitz, E. M.: Mechanics of Dense Media.
Gostekhizdat, 1944.

16. Obukhoff, A.: Pressure Fluctuations in Turbulent Flow. DAN(SSSR),
T. 66, No. 1, 1949.

NACA TM 1350

17. Yagloni, A. M.: On the Field of Accelerations in Turbulent Flow.
DAN(SSSR), T. 67, No. 5, 1949.

18. de Karman Theodore, and Howarth Leslie: On the Statistical Theory
of Isotropic Turbulence. Proc. Roy. Soc. (London), ser. A,
vol. 164, no. 917, Jan. 21, 1938, pp. 192-215.

19. Loitsianskii, L. G.: Some Basic Laws of Isotropic Turbulent Flow.
Trudy TSAGI (CAHI) Rep. No. 440, 1939. (Central Aero-Hydrodynamical
Inst. (Moscow), 1939.) (Available as NACA TM 1079.)

20. Yaglom, A. M.: Homcgeneous and Isotropic Turbulence in a Viscous
Compressible Fluid. Izv. AN SSSR, ser. geogr. i geofiz. T. 12,
No. 6, 1948.

21. Heisenberg, von W.: Zur statistischen Theorie der Turbulenz.
Zschr. f. Phys., Bd. 124, H. 7-12, 1948.

22. Kovasznay, Leslie S. G.: Spectrum of Locally Isotropic Turbulence.
Jour. Aero. Sci., vol. 15, no. 12, Dec. 1948.

23. G6decke, K.: Messungen der Atmospharische Turbulenz. Ann. d.
Hydrographie. Heft 10, 1936.

24. Millionshtshikov, M. D.: On the Theory of Homjogeneous Isotropic
Turbulence. Izv. AN SSSR, seria geogr. i geofiz, vol. XXXII,
no. 9, 1941.

25. Obukhoff, A.: Theory of the Correlation of Vectors. Uchenye
zapiski MGU., no. 45, 1940.

NACA TM 1350

Figure 1

Figure 2

NACA TM 1350 41

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