On the statistical theory of turbulence


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On the statistical theory of turbulence
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Physical Description:
36 p. : ill. ; 26 cm.
Heisenberg, Werner, 1901-1976
United States -- National Advisory Committee for Aeronautics
National Advisory Committee for Aeronautics
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Washington, D.C
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Turbulence   ( lcsh )
Aerodynamics   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
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A study is made of the spectrum of isotropic turbulence with the aid of the customary method of Fourier analysis. The spectrum of the turbulence motion is derived to the smallest wave lengths, that is, into the laminar region, and correlation functions and pressure fluctuations are calculated. A comparison with experimental results is included. Finally, an attempt is made to derive the numerical value of a constant characteristic of the energy dissipation in isotropic turbulence.
Bibliographical footnotes.
Statement of Responsibility:
by W. Heisenberg.
General Note:
"Report date January 1958."
General Note:
Cover title.
General Note:
"Translation of "Zur statistischen Theorie der Turbulenz." From Zeitschrift für Physik, vol. 124, 1948, pp. 628-657."

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N tkA Tif-143)




By W. Heisenberg

The interpretation of turbulence presented in the preceding paper
by v. Weizsaecker is treated mathematically with the aid of the custom-
ary method of Fourier analysis. The spectrum of the turbulent motion
is derived to the smallest wave lengths, that is, into the laminar
region; the mean pressure fluctuations and the correlation functions are
calculated. Finally, an attempt is made to derive the constant which is
characteristic for the energy dissipation in the statistical turbulent
motion from the hydrodynamic equations.

In the statistical theory of turbulence developed by G. I. Taylorl
and v. K&arman,2 the irregular turbulent motion of a fluid is described
by several characteristic functions between which simple mathematical
relations exist: the "spectral" distribution of energy to waves of dif-
ferent wave length, the correlations between the velocities at points
along a prescribed displacement in space or time, and the like. The
reports of G. I. Taylor contain detailed empirical and theoretical data
on these functions.

In his preceding report, v. Weizsaecker3 derived the most important
of these functions, the spectral distribution of energy for the limiting

*"Zur statistischen Theorie der Turbulenz." Zeitschrift fuir Physik,
vol. 124, 1948, pp. 628-657.
1Taylor, G. I.: Proc. Roy. Soc. A 151, 421 (1955); 156, 507 (1956);
164, 15 (1958); 164, 476 (1958).
2Karman, Th. v.: Journ. Aero. Sci. 4, 151 (1957).
SC. F. v. Weizsaecker: ZS f. Phys., being published. V. Weizsaecker's
paper and the present treatise have been written in close collaboration
during the time of our stay in England in 1945. Only after the articles
had been finished, Mr. G. I. Taylor kindly told us (spring 1946) that
essential ideas in these articles had been found and published already
by Kolmogoroff: Compt. Rend. Acad. Sc. USSR 50, 501 (1941); 52, 16 (1941);
and Onsager (Phys. Revue 68, 286, 1945). Compare a report by G. K.
Batchelor at the VI. Internat. Congress F. angew. Mechanik, Paris 1946.
Approximately at the same time we learned, furthermore, about a paper
by Prandtl and Wieghardt which contains similar concepts and has mean-
while been published in the Gottingen Academy reports (Nachrichten der
Akademie der Wissenschaften in G'dttingen, Math.-Physikal KLasse of the
year 1945, p. 6). The present paper may therefore be regarded only as
a supplement and completion of these earlier investigations.

2 NACA TM 1431

case of large Reynolds numbers on the basis of similitude considerations.
The following sections of this paper first will translate v. Weizsaecker's
considerations into the accustomed terms of Fourier analysis and, with
the aid of this translation, study the discontinuity of the spectrum for
high frequencies due to molecular viscosity. Then conclusions will be
drawn for the correlation functions and the pressure fluctuations, and
finally, a derivation of the fundamental constant of the energy dissipa-
tion will be attempted.

1. Representation of v. Weizsaecker's Considerations

in the Terms of Fourier Components

At sufficiently high Reynolds numbers the energy dissipation for
the turbulent motion takes place in such a manner that the large turbu-
lence elements lose energy due to the fact that, for them, the energy
and momentum transfer by small turbulence elements has the effect of an
additional viscosity (cf. for instance Prandtl4).

Under steady-state conditions, energy thus is continuously trans-
ferred from larger to smaller turbulence elements, with the spectral
region of a certain wave length always receiving from larger waves as
much energy as it gives off to smaller waves. For maintenance of this
equilibrium a certain energy distribution is necessary which, when the
molecular friction is neglected, is represented according to v. Weizsaecker
by the law

P = P = F(k)dk (1)
2 2 J

F(k) k-5/3

(k wave length signifies the wave number, vo = V- or measure of

the mean velocity.5) This spectrum F(k) k-5/5 is bounded on two sides.
For small wave numbers, that is, large wave lengths, for one reason or
another, it will no longer be possible to regard the flow as isotropically
turbulent, for the largest turbulence elements are governed by the geom-
etry of the devices which generate the turbulence. This end of the spec-
trum for small k therefore cannot in any case be the subject of a purely
statistical theory. For large k, in contrast, the spectrum is bounded

4L. Prandtl, Str6mungslehre (Flow theory). Vieweg, Braunschweig,
5rd edition 1942, p. 105 ff.
50ur vO differs from the vO in v. Weizsaecker's report by a
numerical factor of the order of magnitude of 1.

NACA TM 1451

by the molecular viscosity. For large k, finally, the molecular vis-
cosity will become larger than the apparent turbulent viscosity, and
the spectrum then will drop off very rapidly.

For the calculations, we shall use the following notation: the
velocity v in a normalized volume V is to be expanded into a Fourier

v = e c = nx, nx, ny, nz integers) (2)


Therein vk = v*-k and the number of the "natural vibrations" between
6 __2 Z'S
k and k + Ak is given by 4itk Then one obtains

IvT,.21 =. h,2d V Iv,..21 = F(k)dk (3)
22 k (2(r)3


F(k) = (2,)-2k2VIk2l (4)

From div v = 0 there follows

(.kt) = 0()

Let us call the coefficient of the molecular- viscosity, p. The
mean energy loss then is, because of div v = 0 with the assumption
that the bounding surfaces are at rest:

6This method is somewhat less apparent, but mathematically more
convenient than the customary expansion with respect to sin and cos.
It amounts formally to the limiting condition that vjv/6x, at a
bounding surface of the volume V are to have the same values as at the
opposite bounding surface.

NACA TM 1451

S = P(rotv)2 (6)

= '- i- 2 = ^ F(k)2k2dk (7)

If the spectrum obeys in a large region the law F(k) k-5/5, the
total energy is determined by the largest turbulence elements. We may

assume for instance that the law k-5/3 is valid down to a minimum wave
number ko; for smaller k we shall assume F(k) = 0. Then

V02 = 2 0 F(k)dk = 2 CO C dk = 3Cko2/3 (8)

CVO 2/3
3 kO
2 k 32/5
F(k) = k02/ for k > kO

vk2 6lVk-l/ (10)

V. Weizsaecker considers the energy loss Sk of that portion of
the total spectrum the wave numbers of which lie below k. For these
turbulence elements, the turbulence elements of smaller wave length
(<2n/k) have the effect of an additional viscosity. One may therefore
generally write

Sk + 7k)f F(k') 2k'2dk' (11)

where rk is to designate the additional turbulent viscosity; it iq
produced by the cumulative action of all turbulence elements with wave
lengths <21r/k. With respect to dimensions, iTk is according to
Prandtl the product of density, mixing length, and velocity where the

NACA TM 1431

mixing length will be comparable to the diameter of the turbulence ele-
ments in question whereas the velocity of the turbulence elements is
given for instance by vO(kO/k)1/5. With reference to v. Weizsaecker's
report one will therefore put

Ik = oP / dk' (12)

(K is a numerical factor)
The expression below the integration sign is essentially determined by
dimensional considerations; but one could of course imagine that, for
instance, the waves k' in the proximity of k enter into the integral
with somewhat different weights than the waves with large k values;
that is, the integrand could depend, in addition, on the dimensionless
number k'/k. However, because of the homogeneous form of the spectrum

F(k) k-5/3, one may include all these uncertainties in the numerical
factor K and give to the integral arbitrarily the exact form (12).

This method is unobjectionable in the region of th6 k-5/3-law but
becomes inaccurate at the ends of the region where the geometry or the
molecular friction modifies the spectrum. But even at the latter limit
(12) will still be a good approximation which at least qualitatively
correctly represents the effect of friction.

The constant K in equation (12) must be exactly determined by
the hydrodynamic equations; it has the same numerical value in all cases
where one may speak of statistical isotropic turbulence, and does not
depend in any way on the geometry of the flow. The theoretical deter-
mination of this important number will be attempted in section 5. There,
it will also be shown that the turbulent energy dissipation actually may
be written as a double integral of the type
k po
Sdk' dk with the integrand signifying the energy
0 k
transferred from k' into k'' per unit time (equation (89)). This
integral is more complicated than the simplifying expression (13) which
results from (11) and (12); however, for the following considerations (11)
and (12) may be regarded as sufficient approximations. For Sk one thus

Sk = + pK dk" ) F(k') 2k'2dk' (15)

The decisive step of v. Weizsaecker's consideration is the state-
ment that this expression for k > ko must be independent of k:

6 NACA TM 14351

Sk = S = const (for k > ko) (14)

because the total energy lies, almost entirely, in the long-wave region
of the spectrum, and the energy "transport" thus must become independent
of k.

Equation (15) may therefore be interpreted as the determining equa-
tion of the turbulent spectrum F(k) which must yield for the region of
large Reynolds numbers the k-5/5-law, and for still larger k values
the fading of the spectrum due to the molecular viscosity.

2. Shape of the Spectrum in the Region of the

Smallest Turbulence Elements

We put first [i/P = v and differentiate (15) with respect to k.
Then there results

/ r m

dk. F(k'))F(k) k2 =


(k F(k')k'2dk
k J5' 0


Then we define new variables x and w by the equations

x -= g

F(k) = F(ko)eW-; w = w(x)


Therewith (15) is transformed into

72 2

w x)
e 2 2

ex -wdx


The constant v = (cf. equation (9)) is essentially the
reciprocal Reynolds number of the total flow and therefore always very
small; if the Reynolds number itself were small, the flow could not be
turbulent at all. By repeated differentiation there originates from (17)

- 1 dw +

w x
e 2 x

w x
= 2e 2 2 (18)

NACA TM 14351 7

In this equation, one may approximately evaluate the integral
m .w+x


w(x) + (x-x) + and breaking off after the second term. Since

the exponential function rapidly decays, one thus obtains a good approx-
po w+x -2
dxe 2 g 2 e (19)
x 1 + (9

By substitution into (18) there finally results

(i fi w+x

One recognizes from (20) immediately the variation of the spectrum. For
not too large x and w the first term in the sum may be neglected and
one obtains

7( x 2 (1 + dw, i.e. F(k) ^= FO (21)
dx) dx 3 kO

as must be the case according to v. Weizsaecker's theory. For large x
and w, in contrast, the first term predominates; therefore one must
then have

7 F const k-7 (22)

In the region of the smallest turbulence elements the spectrum therefore
decays very rapidly, namely with the seventh power of the wave number.

Only in the transitional region from (21) to (22) are numerical
calculations necessary in order to determine the solution of (20). Since
one may put for smaller x, that is, for the region

1 << x << 12

A (25)
w = x

NACA TM 1431

(therewith not only (18), but also (17) then is satisfied with sufficient
accuracy), one may calculate, progressing from point to point, dw/dx
according to (20) from w, and therewith derive w for higher x. It
is sufficient to perform the numerical calculation for a particular

large value of the constant, for instance 2K FO = a. For another
value b one may then obtain w by a simple similitude transformation:

wb(x) =wa + lg+ ) Ig (24)

as one recognizes by substitution into (20) and (25).

Figure 1 shows the result of the numerical calculation for

L -F = about 1000. The numerical integration shows that for this

value in the region

x > 5
w(x) 7x 21.85

More generally, one obtains therefore in this region of the k7' law:

w(x) = 7x + 5.0 4lg( F/0) (26)

that is,

F(k) = 0.0496 F 0 (27)

A serviceable interpolation formula which is correct in the two
limiting cases and does not result in any large errors in the transitional
region, either, reads:

F(k) = FO- 1 (28)

If one defines IL = it/kg as the "diameter of the largest turbulence
elements" and introduces as the Reynolds number of the total flow

RO pv (29)

NACA TM 1431

one obtains according to (9), (27), and (29)

kg = 0.16ko(RO)5/4 (30)

One may denote Ls = it/ks as the "diameter of the smallest turbulence
elements" and obtains

Ls = 6.25IO(ROK)/4 (51)

By (9), (28), and (30) the form of the spectrum is determined for
the entire k-region. For the actual flows, of course, the shape of
the spectrum will be different for small k-values (k~-ko) since there
the geometry of the tests plays a role, for instance, the shape of the
grids by means of which the turbulence is produced. In order to be able
to sensibly carry out the comparison with experiments, one will then
introduce a quantity ko in such a manner that, for instance in the

domain of the k-5/-law (thus for ks >> k >> ko), the formula

F(k) = kO2( ) (32)

becomes correct. The quantity ko thus defined then does not give any
direct statement regarding the variation of the spectrum for the smallest
k-values. Generally, though, the spectrum will greatly deviate from the
k 5/3-law in the region k -~kO

For k > ks, too, the spectrum will not unlimitedly retain the form
k-7. The well-known investigations of BurgersY make it very probable that
at sufficiently small Reynolds numbers finally no turbulent motions what-
ever exist. On the other hand, the k-7-law shows such rapid decay that
the region k > k. is practically insignificant. A somewhat larger
error will arise, particularly in the transitional region, due to the
inaccuracy of equation (15) itself; but it is probably not worth while
to apply already at this point the much more complicated equation of
section 5 to the problem stated here. The correct equations would at
any rate lead to somewhat different numerical factors in (27), (30), and

7Burgers, J. M.: Verh. d. Kgl. Nied. Akad. d Wiss. 17, Nr. 2, 1
(1959); 18, Nr. 1, 1 (1940).

NACA TM 14531

For the comparison with experiment one needs the energy distribution
with wave number for a certain direction, for instance with kx, since
the spectra have been measured experimentally by Simmons8 and Dryden9
by means of the fluctuations of the velocity with time in an airstream
which is guided past the measuring point at a constant velocity U which
is large relative to vO. Also, this spectrum has a different form
according to whether the Fourier expansion of vx or Vy is concerned.
Experimentally, first the spectrum for yx is required; however, we
shall also derive the spectrum for vy since it will be necessary
later on, in the calculation of the correlation functions. Since
according to equation (5) (vkI) = 0, one obtains
B v 2( kx2)
vx 2 1 2 33)

The spectrum of x in kx which we shall designate by Fx(kx)
becomes therefore

xx moo y z 1 F(k)
_0- 41tk2 2 k2 /

= dk (k2 _- k2)F(k) (34)

In a similar manner, the spectrum for Vy becomes

F+ + +k dkydkz 1 (k)

= kx (k2 + kx2)F(k) (35)
8o kx 0 /

From (34) and (35) there follows for

(a) ko << k << ks:

(ko 5/3 9Fotko 5/3 6Fnfko 5/3
F(k) = FO )/ thus Fx(kx) = and Fy(kx) = 6


Simmons, U. Salter: Proc. Roy. Soc. A 165, 73, (1958).

9Dryden, Schubauer, Mock u. Skramstad: Nation. Adv. Comm. Aero.,
Nr. 581, 1938; Dryden, H. L.: Proc. V. Intern. Congr. f. Applied. Mech.
Cambridge, Mass., p. 362, 1958.

NACA TM 1431

(b) k >> ks:
k05/5k 16/5
F(k) = F k s

Fx) EE kO5/ksl6/5k-7

PM kx) = 1 k/ l6/3k (^

As a serviceable interpolation formula (which, however, is somewhat
less accurate than (28) in the transitional region), one may again put

Fxx) 9FO/ko5/ x \8/5 (38)
x = xg j + ()) (ks8)


kgxx = 0.645 ks (59)


Fy(kx) 6F (O/[1 +( (40)


ksYX = 0.793 ks (41)

Before the comparison with experiment is carried out in detail, we want
to raise the question at what critical Reynolds numbers the transition
from the k-5/-law to the k-7-law takes place, that is, if one wants
to express it in this manner the transition from the really turbulent
motion proper to the laminar motion. One may regard as the critical
Reynolds number for this, perhaps, the expression

RS = (42)

wherein, according to v. Weizsaecker, vs = v( ) From (29), (50),
and (51) one then obtains

R 10.2
Rs K


12 NACA TM 1431

The numerical value of K will be discussed later on. At any rate, the
transition therefore takes place at a certain numerical value of the
Reynolds number, as was to be expected from general similitude considera-

In figure 2, the measurements of the spectrum Fx(kx) by SimmonslO
are compared with the theory. The measurements in question are intensity
measurements on an airstream which flows past the measuring point at the
velocities U = 456 cm/sec (0), 608 cm/sec (X), 1060 cm/sec (0), and
which has been made turbulent by a grid of 7.6 cm mesh width; the meas-
urement was made 2.1 m behind the grid. The measured points of Simmons
are plotted individually only in the right part of the figure, in the
left half the approximate scatter of the measured points is indicated
by a vertical line. The abscissa is k in cm-1 (in logarithmical
scale), the ordinate Fx(kx), likewise logarithmically, in arbitrary
units. If one assumes U/vO to have the same value in all three meas-
uring series, which is confirmed by other measurements by Taylor, one
obtains, in the case of a suitable selection of this ratio, the three
curves plotted in the figure. Qualitatively, the Simmons data are well
represented by the curves, particularly also the divergence of the three
test series in the short-wave part of the spectrum. In details, however,
there exist considerable discrepancies; one recognizes from the figure
that the range of validity of the k-5/3-law is here so small that a
reliable check is not possible. The reason for this is the smallness
of the Reynolds number RO. For k = 1 cm-1 the diameter of the turbu-
lence elements is 5 cm, thus about half as large as the mesh width of
the grid; in this region, the turbulence is not yet fully isotropic,
therefore the k-5/3 -law cannot yet be valid. However, already for
k = 4 cm-1 the influence of the molecular viscosity becomes noticeable,
and the intensity drops off markedly. The related measurements by Dryden
quoted before which extend over a large spectral region have been made
at Reynolds numbers so small that the validity of the k~5/5-law can
hardly be checked. Therefore it would be desirable that similar meas-
urements be carried out at very much larger Reynolds numbers. For the
ratio U/v0, one obtains from the adjustment of the theoretical curves
to the measuring points of U/vO = 553 if one identifies TO with the
mesh width of the grid. This value agrees well with measurementsll of
this ratio in similar tests if one assumes K to be about 0.5.

10Simmons, U. Salter: Proc. Roy. Soc. A 165, 753 (1938).
11Cf. G. I. Taylor, Proc. Royal Soc. A, 164, 486, (1958).

NACA TM 1451

Another and probably more accurate determination of K is obtained
from the damping with time of the turbulence, already theoretically treated
completely by Taylor (cited before). For the total energy loss per cm5
and second: S, there results from (9), (15), and (14)

S = pK v3k (44)

For the damping with time of VO there must thus apply

d.(4KF)= ,O3k (45)

with the solutionl2

vo(t) = V (46)
1 + K kOVO(0)t

Taylor who essentially derived this equation reports on measurements
by Simmons in which U/vO(t) was ascertained as a function of t = x/U
(x equals distance of the measuring point from the grid). From (46)

U U x U 68 x (47)
--- + -- + 0.68r (47)
vo(t) vo(O) 8 JL ()M LO

If one puts vo = u' 5 (u' = 2v according to Taylor) and identifies
LO with the mesh width, there follows from Taylor's measurements K = 0.85,
from the corresponding measurements by Dryden a somewhat smaller value.
However, because of the uncertainty regarding the value to be inserted
for L(), this determination is probably still uncertain by about 50 per-

35. The Correlation Functions

Taylor and von Karman (cited before) studied the correlations which
exist between the velocities at two points at a given distance. The two

12Footnote at the time of proof correction: For this solution,
ko = const. is presupposed which certainly is not the case for larger
times. The problem of damping is investigated more closely in a paper
of the author about to be published (Proc. Roy. Soc. A.).

NACA TM 1431

correlation functions R1(x) and
role, are defined as

R1(x) = VL VL2
RL~xJ= *X(P)~

R2(x) which therein play the main

R2(x) vy(Pi)vy(P2)
v 2

with the point P2 displaced with respect to the point Pi
distance x in the x-direction.

These functions are, according to Taylor, in
with the spectra:

a simple relationship

S 00

kxFx(kx)cos kxx




dkxFy(kx)cos kxx

0 O



With the aid of equations (34) and (55), (49) is transformed into

Rl(x) =

R2(x) =

3 00

dkF(k)(sin kx kx cos kx)k~5x-3



S 00

dkF(k)(k2x2sin k x + kx cos kx sin kx)k~3x-3




From these expressions one recognizes immediately the correctness of
von Karma'n's relationship

R2 = R1+ + (51)


by the


R2 (x)

NACA TM 14351

The formulas (50) may be approximately evaluated in the two limiting
cases x < 1/ks and 1/ks < x < 1/kO. When x < -, it is advisable
to expand the integrands with respect to powers of x. The first terms
of the expansion then lead to the quantity

j l dkF(k)k2
k2 0=-- (52)

which may easily be calculated from (13), (44), and (29):

T2=_ ,Roko2 (55)

Thus one obtains:

R1(x) = 1 + .

For x < 1/ks (54)

R2(x) = 1 x 2k2 +

Taylor defined a length A by the equation

2 5 (55)

and designated it as a measure for the magnitude of the smallest turbu-
lence elements. According to (55), A becomes

A = 2.71 LO= 0 .4.4Ls(ROK) (56)

It must be emphasized that A is not identical with the quantity
L, (equation (51)) which we have denoted as "diameter of the smallest
turbulence elements" and that A also depends on 1L and VO in a

NACA TM 1431

manner different from that of Ls. A comparison of (51) and (56) shows
that for sufficiently large Reynolds numbers Ls becomes A.

In the opposite limiting case 1/k5s < x < l/kO one obtains
from (50)

Rl(x) = 1 0.645(kox)2/5 +
R2(x) = 1 0.858(kOx)2/5 +

Here these first terms of the expansion do not depend on the special form
of the spectrum in the proximity of kO; only for x ~ 1/kO does the form
of the spectrum in the proximity of kO becomes important; there, however,
the problem may no longer be treated with purely statistical methods.
The formulas (54) and (57) thus give essentially a complete description
of the correlations in so far as they may be regarded as a consequence
of statistical isotropic turbulence.

The formulas (54) and (57) also show clearly that the correlation
function does not have the same form in all flows but that rather, in the
case of variations of the parameters, the inner and outer parts of the
function undergo different similitude transformations. This point has
been stressed particularly by Taylorl5 in contrast to a different con-
jecture of von Karman (cited before).

For comparison with experiment, the measurements of R1(x) and R2(x)
made by Simmons have been plotted (circles and dots, respectively) in
figure 5; furthermore, the theoretical curves calculated according to
the exact formula (50) are shown. Here again LO has been identified
with the mesh width of the grid and A has been calculated from the
spectrum for U = 1060 cm/sec. The experimental values agree, in fact,
with the theoretical ones very exactly at the smaller values of x,
actually more exactly than could have been expected in view of the uncer-
tainty of IO. Beginning from xk ,~ 1 the deviation of the experimental
points from the theoretical curves becomes noticeable which was to be
expected from the derivation. The variation for larger x-values depends
on the behavior of the spectrum in the proximity of kO which cannot,
in principle, be represented by our formulas. But even for larger x-values
the deviations from the theoretical curves remain small.

15G. I. Taylor, Jour. Aero. Sci. 4, No. 8, 311, 1957.

NACA TM 1451

4. The Pressure Fluctuations

While studying the diffusion in a turbulent airstream, Taylor (cited
before) has derived a relationship between the correlation function for
the diffusion and the root mean square value of the pressure gradient.
We shall investigate therefore also the root mean square values of the
pressure fluctuations from the viewpoint of the theory here described.

With reference to equation (2) one may expand the pressure into a
Fourier series

p = pkeir,p = pk


and the fundamental hydrodynamic equation

i = (Vv) VP + [I Av
p p -

is transformed into

-k t
ixt~v k =P k2vk
k? -


Because of

k'k =

0 it follows

i [ k -
i(yk'k) YI k -k'


Pk = -(Yk')(Ivk-k:,1)
- k -

For the root mean square values of pressure and pressure gradient there

grad2p =2 k2j p2


We are interested, first, in this latter mean value:

- I k2v
p -k

p2 lk21

NACA TM 1451

grad2 p = X k' 1 (Ykpk)(2k-kk)( ktsk)(Skrt") (62)
k k' k 1kk' ~

The superscript bar indicating approximation to the mean signifies
here simply approximation to the mean in time. If one wants to calculate
mean values of the type (62), one must take as a basis some kind of
"assumption of disorder" regarding the turbulent motion. One may start
from the fact that the amplitudes yk in the course of time fluctuate
by a value given by (10) or (28), respectively, so that the time average
-v -v is given simply by the spectrum (28). The phases of the vk,
however, will in the course of time pass through all possible values; all
values of the phase will occur, on the average, with the same frequency.
If one could regard the phases pertaining to different wave numbers as
completely independent statistically, there would, in taking the mean of
such products of four factors v 1 v2 v, be left only the terms in

which every two wave numbers are equal and opposite; thus, terms of the
type v k k k and these mean values could be replaced by the pro-
ducts of the mean values of the squares of the amplitudes:

vvkl k vk = k ik v2 v2 (65)
vk'tlr :17 -117--1 -z!!2-7-42

Actually, however, statistical correlations will exist between the phases
pertaining to different wave numbers since the waves mutually influence
one another. In section 5 we shall attempt to estimate such correlations
in a simple case. In spite of the presence of the correlations, however,
we are inclined to assume that in a sum of the kind (62) the terms of the
type (63) make the largest contribution since their mean value is differ-
ent from zero even in the first approximation, without any assumption
regarding the behavior of the waves with respect to time whereas the
other mean values attain a value different from zero only because of the
finer fluctuations of various waves. We believe therefore that one obtains
a serviceable approximation if one takes only the terms of the type (65)
in (62) into consideration. Then there results

grad2p = 2p2 v )("- ) -- (64)
k'k" (k)' k")2

NACA TM 1451

In taking the mean with respect to the directions of the vk one assumes
again all directions perpendicular to k for vk to be equally probable -
it is expedient to use the relation

(a2)(-2kb) 1 (65)

Thus one obtains

g 2T p2 [k|2k2 (k 2]' 2 (6)
graAdp jv 2 vk ,2 (66)
2 k ,k'1 I (kv _)2

If one sets (k'k?") = k'k"'t, one can perform the integration with respect
to t and finds

gradp >1,2 Y\ k,,2| I k'k" k ) (67)


(s) + 1 i[.s2 + .- s4 s6 + (g2)lg
s 16s3 3 3 2s 1 __sj


For 0 s < 1 there applies approximately

Vy(s) P:s s2 + (69)
S15( 7 21

If one transforms the sums into integrals and substitutes the spec-
trum (28) into (67), there follows finally

grad2p f2= 21 21 81- F k 2.
k r k ', k F r r,,


NACA TM 1451

One recognizes from (70) that the integrals converge at small k-values,
and that one may therefore perform the integration from k = 0 without a
considerable error. This shows that grad2p is actually determined by the
behavior of the spectrum at large k, that is, by the smallest turbulence
elements. Had we calculated p2, we would have found, on the contrary, that
the integral diverges at small values of k, thus that its value is deter-
mined entirely by the largest turbulence elements. Therefore, the value of
p2 cannot at all be calculated according to the method used here; for,
first, in the case of small k, the spectrum has a form dependent on the
geometry; and second, it would surely by quite unjustified to consider, for
the largest turbulence elements, only the mean values of the type (65) since
the geometry certainly impresses definite phase relations upon the system of
the largest eddies.

Equation (70) now becomes

grad2 = 2 2 4, 4/5,3 2/3 f 0 d / dq T (71
grap Ok k 0 OT3 0 12T3 + 8/12l + 8/2 (71)

The double integral at the right was estimated, according to a gra-
phical method, to be 0.765; thus there follows finally (cf. (50))

grad2p = 0-17p2v4kO S4ks2/5 (72)

0= 05.Op2V k02 (73)

Taylor (cited before) had expressed the conjecture that grad2p
should have the same order of magnitude as p2 v0 2(v/ox)2 -thus the order
of magnitude p O 4kO2/5ks/5. One recognizes now from (72) that grad2p
must be considerably smaller, the more so, the larger the ratio ks/kO.
The length AT defined by Taylor:

ap 2 ( )2
)= 2p2 (74)

NACA TM 1431

that is,

7 4
grad p = 2 p2 (75)

must, therefore, for large Reynolds numbers, become considerably larger
than the length ?h of equation (55). From (75), (75), and (56) follows

AT = X0.42ROS (76)

It is true that this result does not agree with the experimental
findings. Taylor indicates, for a test which Simmons had performed
adjoining similar experiments by Schubauerl4, /I\ = 0.5 where one must
assume approximately 5ROK -5.9 according to the test conditions.
Thus one must raise the question whether the result (72) has perhaps been
falsified by the fact that only the terms of the type (65) were taken into
consideration in taking the mean. However, one can easily see that this
may perhaps affect the numerical factor in (72) but that the dependence of
kO and ks, that is, the dependence of A\ on (ROK) is in no way con-
nected with this approximation. For already equation (62) shows that on
the right side the normalization factor v0 4k04' must appear, because
of equation (10). After the mean has been taken, this factor is supple-
mented by a factor of the dimension k2/5 which obviously can be at most
of the order ks2/5 ; it must be, because the pertaining integral with
respect to k would diverge like k2/3 if the decay of the spectrum
would not set in for k ~ kg with k-7. There would remain the possi-
bility that only the numerical factor in (72) has been estimated as too
low due -to the consideration solely of the terms of the type (65). But
it is hard to imagine that the correct expression would increase by more
than a tenfold which would be necessary for interpretation of the

Perhaps the contradiction may be cleared up in the following manner:
The main contribution to grad2p stems from wave numbers of the order ks,
thus, from turbulence elements whose diameter measures a few millimeters.
In Simmons' test, the airstream is heated by a hot wire of 20 cm length
stretched across the wind tunnel, and then the distribution of the heated
air is measured at a certain distance behind the hot wire. Precisely the
smaller distances (5 to 15 cm) are decisive for the determination of ? .*

14Schubauer: Rep. Nat. Adv. Comm. Aero. Nachr. Nr. 524, 1955.

22 NACA TM 1431

It suggests itself to assume that the hot wire itself produces in
the airstream a small vortex street and additional turbulence, with the
turbulence elements probably having a length of a few millimeters that
is, the wire increases precisely the intensity of the turbulence in the
spectral region which exerts the strongest influence on grad2p. The
additional turbulence in the immediate proximity of the wire is probably
much more intense than the original turbulence of the same wave range.
However, this additional turbulence is rapidly damped, of course, and it
is surely difficult to estimate whether this turbulence alone can explain
the discrepancy between (76) and the empirical -,.-value.

5. Energy Dissipation for Normal Isotropic Turbulence

The investigations of the preceding section are already closely con-
nected with a basic problem of the statistical turbulence theory: namely,
with the determination of the energy dissipation in the case of normal
energy distribution, that is, the determination of the constant K in
equation (12). In this problem the molecular friction may be neglected
entirely. The fundamental hydrodynamical equations could therefore be
presupposed in the form

Y = -(x)Z 1vp v = 0

Furthermore, a suitably singled-out partial volume of the fluid is to be
selected as normalization volume which under certain circumstances is
moved simultaneously with the fluid, corresponding, for instance, to the
mean value of the velocity with respect to the volume. We assume therefore
that the volume moves with the velocity u. Then (60) is transformed into

= --iZ (') kk i-s(ki ) + i (1) (77)

For the calculation of the energy dissipation, one has to ascertain
how the intensity x21 of a certain natural vibration (or perhaps

better: the sum of such squares of amplitudes with respect to a small

spectral region Ak:/ .2 varies in the course of time. One

NACA TM 1451

recognizes from (77) that one needs for this purpose time averages with
respect to products of the type

Yk v1, v5 (78)

wherein k = k. Because of the statistically uniform distribution
of the phases, these mean values would disappear if there would not exist
statistical correlations between the phases pertaining to different k
which stem from the mutual influence of the various waves, as has been
explained already in section 4. In order to ascertain these correlations,
one must somehow express in the equations the influence exerted upon a
wave with given k by waves with a different k; one may do this for
instance by representing one of the three amplitudes in (78) as a time
integral over vk and expressing vk in turn by a sum over two other vk
according to (77). Then one obtains products of four amplitudes yk each
of which, however, must partly be taken at different times. For such pro-
ducts the considerations of section 4 are valid according to which one
obtains a first approximation by taking only products of the type (65)
into consideration. Of course, one could continue the procedure in prin-
ciple and attempt to calculate the other mean values of quadruple products
by tracing them back to six-fold ones, etc. However, such calculations
would probably become much too complicated; the higher terms probably also
would make a lesser contribution, and we shall thus be content with the
first step.

For these calculations, one will obviously need mean values of the

Yk(t)vk(t + T)

and we define therefore

v t + T.)vk(t -
Rk(t,T = -k kA(79)

k --

NACA TM 1451

The summation over a small spectral region Ak has been included
into the definition of Rk(t,T) so that the magnitude of the normaliza-
tion volume does not directly enter into Rk(t,T) and that the mean is
taken equally over all directions of k. Evidently the spectral region Ak
must be selected wide enough that many natural vibrations of the normaliza-
tion volume still may be accommodated in it (that is, k2AkV > 1), yet
very small comparedi to k itself. These requirements are for the turbu-
lence elements of the order of magnitude V themselves no longer compatible
but for those, the statistical methods cannot be applied anyway. The whole
procedure thus can be carried out only if it is found that the large turbu-
lence elements practically do not any more contribute to the mean values to
be investigated.

In order to obtain from the hydrodynamical equations information
about the quantities Rk(t,T), it suggests itself to examine the following

t+ -k 2 -1 -i v2 vk(k ) + i(uk) k 7 (80)
22) k'

In this expression one can replace v, by a time integral with respect

to k'

r)+ T T
t4 j t +2 t+--T'
k' = d' + k(-T) = d'k'2 + kr(-T) (81)
-T 0

If T is selected sufficiently large, the correlation between
vkt(t) and vk'(T) will disappear; it is therefore expedient to per-
form, after substitution into (80), the limiting process T--)m. If, in
addition, one takes the mean with respect to the directions thus elimi-
nating the term with u one obtains from (80) and (81):

NACA TM 1451

k+Ak k+Ak t+-Tr

k k k' YO k" -
I g t+- k(t ) Z 4 dT'(kt k) L1" k -

/ t+1--T '
(k k') v2k k' t+ I t-i\
2 1S ) k- k (82)
k'2 K V- k

If one furthermore, as in section 4, takes into consideration the terms
of the type (65) only, also replaces k k' by k', and integrates with
respect to the cosine of the angle between k and this vector, there

k+6k k+Ak V Z I
Z (t + 1) 1 = d'2 ( dT 11 Rk1 t ,T x

Rk,(t + -,T)k-'k-5(k2 k'2)2kk(kI + k k2k'2 -

(k2 + k'2)(k2 k,2)1g k + k' (85)

This equation presents the possibility of expressing the differential
dRk(t,T) dRTk(t,)
quotients with respect to time k(tT) and k(,) by the Rk them-
dt dT
selves; when the Rk are known, one can, moreover, calculate the energy
dissipation from (853), putting T = 0. For this purpose we shall assume
that the entire turbulence phenomenon is either stationary or is damped
very slowly so that the times during which the intensity vk2 noticeably
varies are very long compared with the fluctuation period of vk. The
notation vk22 represents, therefore, the mean value over a time which is
certainly much longer than the fluctuation period but is very much shorter
than the damping time.

d~k ((t, T)
The equation for gives a measure for the fluctuations of
the quantity Rk(t,T) as a function of t about its time mean:

Rk(T) = Rk(t,)


NACA TM 1431

One may surmise that these fluctuations are small in the region of small T
which is determined by the small turbulence elements, and that they increase
with growing T; this question will be further investigated later on.

Before carrying out the further calculations one has to determine how
the partial volume V and its velocity u are to be chosen. One could
first try to put u = 0 and to identify V with the total volume. How-
ever, one would obtain an erroneous picture of the actual conditions: The
decrease with time of the correlation function Rk(t,T) = Rk(T) as a
function of T is determined in this coordinate system by the largest
turbulence elements, and is therefore very rapid. One can show that the
correlation function in this coordinate system is given with sufficient
approximation by

k2v2T2 kV
kVOT i0
kvT -1 2
Rk(T) = 1 ----e dxex

The calculations which have led to this expression need not be discussed
in more detail since the expression is not further used later. The
physical interpretation of the expression is given by the following con-
sideration: The function Rk(T) in it decreases after a time of the

order 21(; that is the time during which, for instance, precisely an
eddy of the wave length 23, due to the high velocity in the largest tur-
bulence elements, passes by the point of observation. The fact that the
correlation function decreases after that time signifies therefore simply
that the velocity in the largest turbulence elements is of the order of
magnitude vO, but, statistically, fluctuates about values of this order.
This phenomenon is not connected with the disintegration of the eddy of
the wave length 2-. If the F(k) ~- k-5/3-law is valid, it is rather to

be expected, according to the similitude considerations of v. Weizsaecker,
that the disintegration of the eddy takes place only after time intervals
of the order 21cvO-k-2/3kO-1/. On the other hand, however, the energy
dissipation is connected with the disintegration of the eddies, not with
the motion on a large scale. If one wants to describe the disintegration
of the eddies in equations, one must move the coordinate system at the
same time. One then must make the linear dimensions of the partial vol-
ume V somewhat, but not very much larger than 2J and move it simul-
taneously in proportion to the mean velocity within it. We shall assume

NACA TM 1451

experimentally that one can give for every k a volume V corresponding
to it in such a manner that Vk5 becomes independent of k and that for
the thus selected, simultaneously moved volume elements, the correlation
function R(t,T) is a universal function of the variables vOk2/ k1/3t
and vOk2/5k1/5T as is to be expected according to v. Weizsaecker's
similitude consideration. We shall show that the relations for Rk(t,T)
arising from (85) actually can be satisfied by this assumption if the Fk
are distributed according to the k~5/3-law.

If one puts Rk(t,T) = g(,Tj) wherein

v= 2/ 1/3t

v= 2/5k 1/

k '
ii = y


there follows from (85)

dg( ,n) = 3- yf(y) d P g( Lrr ^k+ n f- )y2/3, -

S- 2, +

S ( ) ,y2/3 5,

dyf(y) df,' [ -T' i)g +


g L +

T- + ')2/3I ,2/3


f(y) = (l y2 + y) ( y2)(l y4)lg + Y-
y 14/3 \ (l 3 y ]



8 Jo

I Iy2/5,,,y2/3)

NACA TM 1451

These equations actually do no longer contain the constants ko, vO.
The reason is that the integral with respect to y converges for small as
well as for large y; f(y) disappears sufficiently for y = 0 as well as
for y--y. Therefore one may take the integral over k', instead of from
kO, simply from 0, without considerable error; moreover, the convergence
of the integral for large values of k shows that the molecular friction
actually is of no importance in this problem; the behavior of the spectrum
in the region of the smallest turbulence elements is unimportant for the
correlation functions R(t,T) and the energy dissipation for medium

Before attempting the numerical solution of (86), we shall use the
equation (85) for calculating the energy dissipation in the approximation
here aspired to. For this purpose we put T = 0 and integrate the equa-
tion (85) with respect to k between two arbitrary limits K1 and K2:

d f2 4,k2dk = 2 j I J dT'R1(t ,T)Rk,(t ,T)
Tt K 2 'k2n)5jK: k k' JQ L( 2 2

(k2 k'2) kk( 4 + 2k2 (k2 + k2 )(k2 k Ik k1] (88)

The integrand on the right side is an antisymmetrical function in k
and k'. If one calculates the variation with time of the total energy,
that is, if one puts K1 = 0, K2 = m, there results therefore zero, as
far as the integral on the right converges at all. That is, the total
energy is constant in time; this is a necessary requirement since the
molecular friction has not been taken into consideration. However, if
one considers the variation with time of the energy which is contained in
the part of the spectrum lying between K1 and K2, the integral on the
right side may be transformed in the following manner (we shall call the
antisymmetrical integrand simply J):

fK2 2 f K2 rK2 k K1 K2 (
k2d k = dk dk'J = dk dk'J dk dk'(-J) (89)
Kt K 2 0 j o j J
1 K 1 1 S2

NACA TM 1451

In the first of the two integrals on the right J is always posi-
tive; in the second (-J) is always positive. From this notation there
follows that the first integral may be interpreted as the energy which,
per unit time, flows from smaller wave numbers (k' < Kl) into the region
between K1 and K2; the second integral as the energy which flows toward
larger wave numbers (k' > K2). If one puts, in particular, K1 = 0 and
K2 > kO, the second integral represents the entire energy dissipation;
for the normal spectrum [F(k) ~ kb'5 it must prove to be independent of
K2. Thus one obtains from (85), (86), and (89) for the energy dissipa-
tion the expression


S-Lv3k oJ' ikf dyf(y) fr ( d -) g I2/5,2/3)

= vo b dy(-lgy)f(y) J ad s( g y2/5,IY2/3) (90)
T4 o o 2 2)

This expression is actually independent of K2 as it must be. Since
the entire energy dissipation according to equation (44) is also given by

pK vgkOx, there follows

K = 1J dy(-lgy)f(y) J drg( g y2/5y2/5) (91)

From this equation K can be calculated numerically if the function
g(,f) is known.

We now turn to the treatment of the equation system (86). This sys-
tem represents a considerable simplification compared to the initial
equation (77) in so far as it does not any more contain any dimensional
quantities, and is already derived from the equilibrium spectrum k 5/.
On the other hand, (86) also still contains statements regarding the
fluctuations of the g(t,r) as a function of the and is, for that
reason, doubtlessly too complicated to permit rigorous solutions. One
could attempt to completely neglect the fluctuations in a first approxi-
mation, and to calculate with the mean values. Unfortunately, however,

NACA TM 1431

it turns out that the contribution of the fluctuations in certain regions
is large. One recognizes this from the second equation (86).
For, if one puts

g(,r) = g() + ng(Q,T)


where g(r) signifies the mean value over of

g(r) = g(,T])


there follows from the time mean of the second equation (86):

Sdyf(y) d.T1j[g(r' rTO + gk' + Tr)gri y2/ )1

dyf (y) fdr f rAg(^

g(S n' +

- 2 -

)A ( y2I3, TI/5 I y2/5)]


This relationship shows that the
always be small. It is true that the
- = 0; this follows from the relation

fluctuations Ag((,Tj) cannot
left side of (94) vanishes for

f(l) = -y 4/f(y)


3 f
T8 J

+ 1- TV.y21,,y13

NACA TM 1431

which will have to be discussed later, and signifies that the spectrum
k is actually in equilibrium; however, for larger qld the left side
assumes appreciable values. Therefore, it is doubtful whether one will
obtain a sufficient approximation if, in the transition from (82) to (83),
one takes only the mean values of the type (65) into consideration. How-
ever, I did not succeed in improving here the approximation or in
obtaining more'than a very crude estimation of (86).

One may perhaps assume for such an estimation that, for large values
of 7T, the first term of the summation on the right side of (94) is much
larger than the second. For in the first one, the integral is taken over
the region 7T' T] which probably contributes a great deal whereas in the
second, for large T1, the factors Ag have already strongly decayed in
the entire integration range. One may therefore attempt the assumption,
at least for large -q, that the second term of the summation on the right
side of (94) may be neglected. In this approximation the time average of
the first equation (86) then becomes, with use of (94):

dg I)= _- (fy dy) dTI'g( + T1)g iy2/5) (for > > 0) (96)

One may utilize this equation, for instance, in such a manner that
one assumes a plausible form for g(Q), leaving the scale in the
r-direction undetermined at first, and determining it subsequently so
that the equation (96) is valid as exactly as possible for large T1. In
this way one will describe the steepness of the decrease of g(Tr) for
large T with some correctness, and precisely this steepness is decisive
for the value of K.

In the practical execution of the calculation it is expedient to
introduce, in the place of y and f(y), new variables

s = y2/; cp(s)ds = f(y)dy (97)

Then there applies as one recognizes from (87) (compare also (95)):

cp) = -sp(s) (98)

NACA TM 1431

This equation is based on the fact that the energy dissipation from
the wave number lk to the wave number k coincides with the one from k
to ak, except for a factor qualified by the similitude transformation.
Furthermore, there then applies in good approximation

(s) = (1 (for 0 s l) (99)

and for larger s one can reduce cp(s) by means of (98) to the range
o < s < 1.

Figure 4 gives a plausible curve for g(Tr), and in addition the right

side of (96) as J d and g'(rj). The scale is selected in such

a manner that the two last curves coincide for large il. Considerable
differences then exist for small r1 but there the equation (96) also can
no longer be correct. If one substitutes the function g(r) thus
obtained into (91) and neglects the fluctuations, there results for K:

K J ds(-lg s)cp(s) 1drg(T)g(Ts) = 0.98 (100)

This crude estimation therefore gives the correct order of magnitude
for K, but the exact value may well differ from 0.98 by as much as a fac-
tor 2. The calculations of this section thus have not led to an exact
calculation of the constant K but they did provide a qualitative mathe-
matical representation of the processes on which the energy dissipation
is based. Perhaps it will be possible to arrive at a rather exact experi-
mental determination of K by means of a comprehensive discussion of the
various experiments of Simmons, Dryden (cited before), Prandtll5, and
others regarding the spectrum and the damping of turbulence.

Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics

15Prandtl, L.: Proc. VI. Intern. Congr. f. Appl. Mech. Cambridge,
Mass., 1938, p. 340.

NACA TM 1451 55

0 2 4 6



Figure 1.- Representation of the function w(x).

NACA TM 1451

S" 0


0 0

x 0

0 1 2 3 4 5 10 20
kx- cm-1

Figure 2.- The turbulent energy distribution as a function of the wave

NACA TM 1431 35




^ 0.4-
S\R|( x)

0 2 3 4 5 6
x -

Figure 3.- The correlation functions.

NACA TM 1431



-0. I


0 5 10 15 20

Figure 4.- The correlation function g(r).

NACA Langley Field, Va.

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