On stability and turbulence of fluid flows


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On stability and turbulence of fluid flows
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60 p. : ; 27 cm.
Heisenberg, Werner, 1901-1976
United States -- National Advisory Committee for Aeronautics
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Washington, D.C
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Aerodynamics   ( lcsh )
Turbulence   ( lcsh )
federal government publication   ( marcgt )
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This investigation is divided into two parts, the treatment of the stability problem of fluid flows on the one hand, and that of the turbulent motion on the other. The first part summarizes all previous investigations under a unified point of view, that is, sets up as generally as possible the conditions under which a profile possesses unstable or stable characteristics, and indicates the methods for solution of the stability equation for any arbitrary velocity profile and for calculation of the critical Reynolds number for unstable profiles. In the second part, under certain greatly idealizing assumptions, differential equations for the turbulent motions are derived and from them qualitative information about several properties of the turbulent velocity distribution is obtained.
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Werner Heisenberg.
General Note:
"Report No. NACA TM 1291."
General Note:
"Report date June 1951."
General Note:
"Translation of "Über Stabilität und Turbulenz von Flüssigkeitsströmen." Annalen der Physik, Band 74, No. 15, 1924."

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University of Florida
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Full Text
NAk -IpA'W \2q




By Werner Heisenberg


The turbulence problem, which will quite generally form the subject
of the following investigations, has been treated in the course of time
in so many reports from so many different viewpoints that it is not our
intention to give, as an introduction, a survey of the results obtained
so far. For that purpose, we refer the reader to a report by Noetherl
on the present state of the turbulence problem, where most biblio-
graphical data may be found as well.

For our purpose, a rough outline of the present state of the tur-
bulence problem will be sufficient. The investigations made so far are
divided into two parts; one part deals with the stability investiga-
tion of any laminar motion, the other with the turbulent motion itself.

The first-mentioned investigations led, at the beginning, to the
negative result that all laminar motions investigated are stable.
V. Mises2 and L. Ropf3 proved, on the basis of a formula by Sommerfeld ,
the stability of the linear velocity profile corresponding to Couette's
arrangement. Blumenthal5 reached the same result for a profile of the
third degree, upon which Noether invited discussion. On the other hand,
Noether6 later succeeded in specifying an unstable profile a profile
which is unstable even in the case of a frictionless fluid can never be
realized as a steady state of motion for actual conditions. More

*"Uber Stabilitat und Turbulenz von FlussigkeitsstrSmen." Annalen
der Physik, Band 74, No. 15, 1924, pp. 577-627.
1Noether, F.: Zeitschr. f. angew. Math. u. Mech. 1, 1921, p. 125.
2Mises, R. v.: Beitrag z. Oszillationsprobl.: Heinr. Weber-
Festchrift, 1912, p. 252.
3Hopf, L.: Ann. d. Phys. 44, 1914, p. 1.
4Sommerfeld, A.: Atti d. IV. congr. int. dei Mathem. Rom 1909.
Blumenthal, 0.: Sitzungsber. d. bayr. Akad. d. Wiss., 1913, p. 563.
Noether, F.: Nachr. d. Ges. d. Wiss., GOttingen 1917.

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recently, however, Prandtl7 has shown that indeed profiles exist which
possess unstable characteristics only if the friction is taken into

The other group of reports which achieved great success quite
recently by the calculations8 of Von Karman, Latzko, and others
investigates the turbulent motion itself proceeding by a semiempirical
method using the laws of similarity. Theoretically, the reports of
this group are based almost throughout on Prandtl's boundary-layer
theory. Their most important result for our purpose is the so-called
yl/7-law of turbulent velocity distribution which follows from Blasius's
law of resistance (examples can be found in Schiller's reportg)

The determination of the critical Reynolds number was always one
of the main aims of the first-mentioned reports, the stability inves-
tigations. So far, a satisfactory calculation of this number has not
been accomplished and it must be regarded as doubtful whether it could
be achieved by stability investigations. The tests of EkmanlO, Ruckesll
and Schiller9, together with the negative results of Hopf concerning
the linear velocity profile, rather suggest the notion that the critical
Reynolds number does not indicate the point where the laminar motion
starts to become unstable, but the point where, for the first time, the
turbulent motion is possible as steady state. From the viewpoint of
theory, we must thus be prepared to find eventually two critical
Reynolds numbers, one corresponding to the beginning of turbulence, the
other to the breaking down of the laminar motion.

The present investigation also will be divided into two different
parts, the treatment of the stability problem on the one hand, that of
the turbulent motion on the other.

7Prandtl, L.: Physik. Zeitschr. 23, 1922, p. 19, and Tietjens, 0.,
Dissert. G6ttingen, 1922.
See Zeitschr. f. angew. Math. u. Mech., 1, 1921, p. 233f.
9Schiller, L.: Rauhigkeit und kritische Zahl. Physik. Zeitschr. 3,
1920, p. 412.
loEkman, V.: Turbulent Motions of Liquids. Arch. f. Mat. och
fysik 6, 1919. p. 12.
1lRuckes, W.: Dissert. Wiirzburg 1907. See also Lecture by W. Wien,
Uber turbul. Bewegungen. Phys. Zeitschr. 8., 1904, and Verh. d. deutsch.
phys. Gesellsch. 9, 1907.

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The aim of the first part is to summarize all previous investiga-
tions under a unified point of view, that is, to set up as generally as
possible the conditions under which a profile possesses unstable or
stable characteristics, and to indicate the methods for solution of the
stability equation for any arbitrary velocity profile and for calcula-
tion of the critical Reynolds number for unstable profiles. This aim
can, of course, be attained only imperfectly by the use of approxima-
tion methods. Nevertheless, we hope to be able to clarify by such
calculations the qualitatively essential viewpoints. At first, the
investigation of any arbitrary profile seems physically meaningless
since only certain profiles actually occur; however, since we may
interpret any profile as finite disturbance of another, as for instance
Noether has done elsewhere, and since we must, on the other hand, later
extend the investigations to the (at first unknown) basic profile of the
turbulent motion, the investigation of an arbitrary profile seems, after
all, to be of great importance.

As application of the methods, the parabola profile will be cal-
culated completely.

In the second part, we shall attempt to derive, under certain
greatly idealizing assumptions, differential equations for the turbulent
motions and to obtain from them qualitative information about several
properties of the turbulent velocity distribution.


1. Statement of the Mathematical Problem

The most essential limitation we impose on our calculations con-
sists in the exclusive consideration of two-dimensional laminar motions
and only two-dimensional disturbances of these motions. Taking a rec-
tangular coordinate system X, Y, Z as basis, we therefore assume that
the velocity in the Z direction is zero and all remaining quantities
independent of Z. Furthermore, however, we shall only examine the sta-
bility of such laminar motions as occur between two straight parallel
walls. We assume the walls to be parallel to the X axis; therefore,
the laminar motion to be investigated also promises a velocity com-
ponent only in the X direction. This velocity w in the X direction
will, in some way, be dependent on y. Concerning the function
w = w(y) we reserve for later making a few assumptions about continuity,
symmetry, etc.; otherwise, however, this function is to be at first
quite arbitrary.

4 NACA TM 1291

If we put w = ay, our formulations become exactly identical with
those investigated by Hopf in the Couette case. The problem whether
the investigated profiles w = w(y) can be realized as steady motions
will not be dealt with for the present.

Before deriving once more the stability equation (already set up
elsewhere by Sommerfeld) briefly from Stokes's differential equations,
we introduce dimensionless variables in the known manner. Let h be
a characteristic length (for instance the distance between the two
walls), U a characteristic velocity of the profile, the viscosity,
p the density, and Uhp = R the Reynolds number; we introduce instead

of x, y, u, v, t, and p (u, v being the velocity in x or
y direction, respectively, t the time, and p the pressure) new
variables xo, yO, uO, vo, to, and po, according to the relations

x0 = ;

uo = U o = ; to = t u;

PO = pU

If the index 0 is subsequently omitted, Stokes's equations read

bu bu l1 p 2u
+ u-+v -= + --
bx by R bx x2

6v ov 1 6p 2v
+ u +v + -
bx by R by 6x2



Since we presuppose incompressibility, we write

u =

v = -

As is well known, we obtain by the elimination of p

LAt + A
6t 6y 6x 6x 6y R


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By A one understands here the differentiation symbol

62 62
2 a2
x2 + y2

Equation (4) does not yet contain anything about our special
problem, the stability investigation of a certain laminar flow. Accord-
ingly, equation (4) will form also the basis for the calculations of
part II. In order to pass over specifically to the stability investi-
gation, we divide the motion and therewith also the vector potential
into a basic flow and small oscillations superimposed over it. Thus we
set up the formula

r= (y) + c(y)ei(t-ax) (5)

= w(y) = w (6)

If we enter this formula into equation (4), omitting all terms not con-
taining q (since we regard equation (4) as satisfied for p = 0),
furthermore omitting all terms quadratic in p (since we assume c as
small), the corresponding differential equation for q reads

(V" a~q)(W ) Iw (qII"" 2c2q'p + M4p) (7)

The fact that we regard equation (4) as satisfied for q = 0 signifies
physically that we consider only such basic flows w which either, by
virtue of external forces, are really steady, or show a variation with
time which is slow compared to that of the small oscillations.

Equation (7) is in this generality already derived elsewhere by
Noether. It is an ordinary differential equation for cp of the fourth
order. It corresponds to the fact that the function p must fulfill
four boundary conditions; u and v, thus also c and c', must dis-
appear at the two walls. If we put P/a = c so that c essentially
signifies the wave velocity, the mathematical problem may be formulated
as follows: The solutions of the equation

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(p" 2)(w c) = 2cL2p" + c4(p) (7a)

are to be investigated with the secondary condition that at the bounding
walls (for instance, y = 1 and y = -1) p = 0 and cp' = 0. For each
value of a and R the corresponding value of c and p is to be
calculated; let a for reasons of simplicity always be positive.
According to whether the imaginary part of p is positive, zero, or
negative, we are dealing with a stable, undamped (undamped oscilla-
tions = neutrally stable oscillations), or unstable oscillation. The
conditions for profile w are to be found under which equation (7a)
admits only stable oscillations or, respectively, also unstable ones.

Before turning to the methods of solution we want to point out a
special property of the equation (7a). In the limit of the frictionless
fluid, R = o, equation (7a) is transformed into a differential equa-
tion of the second order for q

(c" a2q(w c) pw" = 0 (8)

Accordingly, only two boundary conditions must now be satisfied which
signify that the normal velocity component, thus v or p but no
longer p', is to disappear at the two walls.

The conditions for the solvability of equation (8) have already
been investigated in detail by Rayleigh.12 Introducing a simple desig-
nation, one may distinguish basic flows "capable of oscillation" or
"not capable of oscillation" according to whether or not equation (8)
possesses a solution with real c which satisfies the boundary con-
ditions.13 If solutions with complex c exist, the stability problem
for these oscillations has, as will be shown later, already been
decided by equation (8), also in case of consideration of the friction;
the oscillations are then always unstable.

One is, however, beyond this led to the conjecture that the pro-
file w, under influence of friction, permits unstable or undamped

12Lord Rayleigh, Papers I., p. 361; III. pp. 575, 594; IV. p. 203.
13Here, however, it is by no means sufficient to approximate the
profile by tangents polygons; the result with respect to possible
oscillations would thereby be completely falsified.

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oscillations only in one case: when it belongs to the basic flows
capable of oscillation.

This supposition is the more obvious as it has been confirmed for
all profiles investigated so far.14 Nevertheless, it is by no means
motivated by the fact that equation (8) results from equation (7a) in
the limit R = since it has been proved, for instance, in the reports
of Oseenl5 that the limiting process R = o has led more than once to
false results in the differential equations, particularly with respect
to the boundary conditions of the frictionless fluid, and that one may
therefore apply the limiting process only to the integrals of equa-
tion (7a). Moreover, it can by no means be decided beforehand whether
the friction modifies the undamped oscillations of equation (8) in the
sense of a damping or an excitation.

Following, we shall attempt to prove our surmise mentioned above by
showing that the systems capable of oscillation are shown to possess
above a certain value of the Reynolds number and in general unstable
character, whereas all systems not capable of oscillation are shown to
possess a stable character.

By this principle the problem of the stability of a profile is
quite considerably simplified since, as is well known, the solutions of
equation (8) may be directly written down for very small values of a.

2. The Methods of Solution and the General Behavior of

the Integrals of Equation (7a)

The most important property of equation (7a) which permits an
approximate representation of its solutions consists in the fact that
R may be regarded as very large. It will become evident that if a
stability limit exists, this limit lies, in general, at very high values
Of R. Since it is, moreover, physically quite improbable that for
small values of R instability of the respective profile could occur,
it is sufficient for our next purpose to regard R as very large.

This assumption makes it possible to approximate the solutions of
equation (7a) by development in negative powers of R or, as will be
shown, of \fB. Furthermore, we shall assume a as small and shall
develop the solutions in a given case in positive powers of .2.

14Compare also Prandtl, Physikalische Zeitschrift.
15Compare, for instance, C. W. Oseen, Beitralge z. Hydr. Annalen
der Physik 46, pp. 231 and 623, 1915.

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Both methods of development in (aR)-l/2 and a2, respectively,
seem contradictory insofar as in the first case mR is assumed large,
in the second case a2 small; however, the contradiction is eliminated
by the fact that R rmy be regarded, in general, as extraordinarily
large so that for instance for R = 2000, a = 1/10, aR becomes equal
to 200, a2 = 1/100 which is fully sufficient for a satisfactory con-
vergence of the two developments. However, the convergence properties
of these approximation methods must be considered more exactly. The
investigation shows that the series in (aR)-1/2 are generally diver-
gent, yet show the well-known characteristics of the semiconvergent
series, that is, that the terms first decrease, then again increase,
and that one obtains the optimum approximation if one breaks off the
series with the smallest term. Our approximation method has, there-
fore, convergence properties similar to those of the series of the
perturbation theory used in astronomy, the behavior of which is
described in detail by Poincare, Meth. nouv. d. 1. mec. cel. II.

The use of the semiconvergent developments is rendered con-
siderably difficult by the fact that they lose their validity in the
neighborhood of a certain point so that it cannot be immediately
decided in what manner the approximate solutions on both sides of the
point must be joined in order to approximate a certain integral of the
equation (7a) on both sides. This question will be discussed in detail
in section 3.

The development in positive power series of a2 seems, in general,
to be actually convergent. For special profiles this development may
be strictly proved (for instance, for the linear profile); however, we
have not carried out an investigation of the problem under what con-
ditions for the profile w this convergence actually occurs.

We start with the derivation of the approximate solutions of
equation (7a)

(o2 asp)(w c) w"P = ( 2a.2" + ) (7a)

For this purpose we first put

fgdy 1
p= e g = _Rgo + gl + g + (9)

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We shall limit the development to the two highest terms in 2J. There

Rg02(w c) + R(go' + 2g0g1)(w c) = iaRgo4 + iR (g03g + 6g02g')

a2 and w" are presupposed to be of the order of magnitude 1 or, at
any rate, < \aR. By means of simple calculation there now results

go = -i(w c),

gl g
91= 2 .


1 dy = 2 log go

Thus we obtain two particular integrals of the equation (7a)

,1,2 = (w -

c) Y
C) -ifR(w-c)
c) e O


The point yo is to be determined by w being = c for y = yo.
Thus yO may, under certain conditions, be complex. The sign of the
root is to be chosen so that for

w c = -aei,

-iaxR(w c) = aRaei( 2)

the root becomes

(CRa)1/2e +2 V/

A remarkable fact about these two integrals is that ,2 in p
does not appear in this approximation (that is, only in the combina-
tion aR which, as follows from equation (7a), could in a certain
sense, be called the true Reynolds number).

As we shall see later, the two integrals (11) determine the
behavior of c in the boundary layer, and the nonoccurrence of a2
in equation (11) signifies physically that we consider only oscillations

10 NACA TM 1291

the wave length of which is large compared to the boundary-layer thick-
ness, which is certainly the case for the empirically observed unstable
For a complete system of solution of equation (7a), however, we
need two further integrals; naturally, we shall choose those which
result from the integrals of equation (8) by the development of
(aR)-1 in a power series.

For this purpose we first solve equation
of o2 in a power series. Thus we put

(8) by the development

(cp" a2c)(w c) cpw" = 0

cp = p(0) + a2c(1) + a4(2) + .



Hence follows

p(o)"( _- c) (p(0)w" = 0

(1l)"(w c) c(l)w"= cp(O)(w c)

(p(2)"(w c) c(2)w" = p(l)(w c)

By the method of variation of the constants the two integrals

(w c)(l + a2

I dy
S(w c)2

Sdy(w c)2 + .

dy(w c +

P3(R==) =

4(R=-) = (w c)

a2 fd,


y(w- c)2

f )2
(w c)2

NACA TM 1291 11

result. In addition, these integrals have now to be corrected by
quantities of the order (R)-', etc., if they are to satisfy
equation (7a).

Without writing the corresponding series development down in
detail, we give as result c with the quantities of the order (aR)-l

) = (w c) + 2 Y2 d dy(w c)2 + 4 +

(w c)2 y3 c)


= (w c) dy 21 + a2 dy(w c)2 dy +
S(w c)2 c)2

4 i d3 _dy_
a + ( c) + .
aR dy3 (w c)2

With equations (11) and (14), a complete solution system of the equa-
tion (7a) has been found approximately.

Before applying this system of solution for satisfying.the boun-
dary conditions, it will be useful to clarify the physical significance
of the four integrals Pl' T2' cP3, and p4, and to anticipate a few
results which we cannot establish until later.

The integrals cp1, q2 are very rapidly variable for the high
values of R which are of interest to us, as can be seen from the
exponent of the order V-T. If, therefore, for instance ci is at
one wall of the order of magnitude 1, it will vanish exponentially at
some distance from the wall. (In itself, it also could become extraor-
dinarily large; however, this is naturally prevented by the boundary
conditions.) Consequently, p is composed, except for the immediate
neighborhood of the wall, of p3 and cp4 alone, that is, is
very similar to the behavior of p in the frictionless fluid.

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The fact that a2 does not explicitly occur in cP1, q2, (com-
pare equation (ll)), but does appear in (3P, c4 (compare equation 14))
must evidently be interpreted physically to signify that, if a.2 is
assumed to be not very large (a2 < R, compare equations (7a) and (9)),
the wave length may be considered as infinitely large compared to the
boundary-layer thickness, but not compared to the width of the channel.
The characteristic difference between the "boundary-layer integrals"
p'l, 2, on one hand, and the integrals corresponding to the friction-
less fluid c3, 9c4, on the other, is therefore significantly expressed
in the occurrence or nonoccurrence of a2.

Concerning the convergence of the development in power series
of a2, we may hope that for values of a2 of the order of magnitude 1
it is still amply sufficient to enable a good approximation, since for
a linear profile the series for qp3, q4 become series of the type of
power development of cos a which in the neighborhood a = 1 still
converges very rapidly.

The flow pattern to be expected after all these conclusions corre-
sponds to the formulations made in Prandtl's boundary-layer theory.
Except for the immediate neighborhood of the walls, the motion obeys
very nearly the differential equation of the frictionless fluids. To
the walls themselves, however, adheres a boundary layer the thickness
of which is of the order of magnitude (aR)-1/2. In this boundary
layer the velocity u decreases toward the wall rapidly toward zero
whereas v is almost zero even outside of the boundary layer.

3. The Connecting Substitutions

If we want to study the course of the integrals of equation (7a)
from one bounding wall to the other, we must take into account that,
at a point y = yO in the channel w c = 0 (or that at least the
the real part of w c is zero), therefore the wave velocity there
agrees with the velocity of the basic flow. At this point, the
approximation formulas (11) and (14) for the integrals of equation (Ta)
cease to be valid.

Thus it is necessary to know the connecting substitutions for (p 1

p2' c3, and p4 which have to be applied for the transition from
Re(w c) > 0 to Re(w c) < 0. For this purpose, we develop w

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andp in the neighborhood of the critical point yO in the power
series of (aR)-1/3 and put therefore

y YO = (aR)-1/3

Furthermore we assume the imaginary part of yO to be of smaller
order of magnitude than (aR)-1/3. If it is of higher order of magni-
tude, the connecting substititions are self evident because then
nowhere in the entire range of real y does a "critical point" appear.
If the imaginary part is of the same order of magnitude, the behavior
of cp and w may be easily interpolated from the two limiting cases
just mentioned. At first we may even put

Im(YO) = 0

since p in our case

Im(yo) (R)-1/3

may be developed in power series of Im(YO) and at first only the
behavior of c for Im(YO) = 0 is needed.

Thus we now put

p = Q + EqP1 + E22 +

= (aR)-1/3

w c = Ean + b2 + .

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Then there results from equation (7a)

"" + + = + = -i t+ El" + )a+j + o' 'brjl -

2ETpob + .

Thus in first approximation

0P"" = -i(P" "a


in second approximation

pI"" = -i[l"ua + 'o"bq2 2o0b]


For the integrals qpl, c2, equation (11), we infer from equa-
tion (15) that they behave in the critical range (ri order of magni-
tude 1) like the integrals found by Hopf for the linear profile, that
is, like certain cylindrical functions. Thus, we may conclude at
this point that the connecting substitutions for (pl, p2 from equa-
tion (11) except for quantities of the order (aR)-1/3 must be the
same as for the linear profile

Pi -+ 'Pi + ip


P1 icp2 cp1

corresponding to a transition of

Re(w c) < 0 Re(w c) > 0

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However, for the study of the integrals p3, cp4 in the neighbor-
hood of y yo = 0, the simple calculations made so far are not suf-
ficient since for the latter the approximate solution (15) would
read p" = 0; however, we know from equation (14) that, in the limit
R ---h, c"', in general, becomes logarithmically infinite at the
point y yO = 0. Equations (15) and (15a) are therefore in this-
form unsuitable for expressing this singularity.

Instead, we now set a = 0 and w'" = 0 (that is, we break off
the development of w with the second term); otherwise, however,
integrate equation (7a) exactly at first. In doing so, we notice that
T = w c must be a particular integral of this simplified equa-
tion (7a) and we make, therefore, for cp the statement familiar from
the theory of linear differential equations

S= (w c)f' dy

Then there follows from

i"" = -iaCR(("(w c) w"p)


c = (w c) f dy

1t"'(w c) + 4"w' + 6t'w" = -iaR(2w'(w c)4 + (w c)2f')

which after repeated integration becomes

T'(w c) + 3*'w' + 3wrw" = -iaR((w c)2* C)


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C is an integration constant. If one now again introduces

n= (y- y0)(aR)1/3 E = (R)-1/3,

w c = Ea + e2b2, = 0 + 1 +

there results

*o'"a + 34o'a = -i(a220 C)

1 anq + 3*1'a + OQ"bT2 + 6i'0bn + 60b = (17a)

-i(2ab*on + a2i2 *)

Of course, these differential equations still contain all solu-
tions of equation (7a). We intend to study particularly 94 ( 9p
shows for a = 0 at the point y = yo regular behavior); therefore,
we select the one solution of equation (17a) which behaves at some
distance from y0, thus for large (w c)aR, like since
(w c)2
we know from equation (14) that 94 at some distance from yO is

given by dy
(w c) C2
J(w c)

Thus we obtain according to equation (17a)

0 a2


1 n + 31' = -i= CT + aiq2) (18)

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'1i is again fully determined by the fact that it is to behave "at
infinity" like -

We now ask for the transformation substitutions for cp4 (and cp3),
meaning thereby the following: In the asymptotically valid repre-
sentations for cp3, cp4 (equation (14)), we always find the integral
dy which loses its sense if it is to be extended beyond the
J (w c)2
point y = yO (w c = 0). Actually, is, near the critical
(w c)2
point, replaced by the function *. Thus the behavior of (partic-
ularly il) in the critical neighborhood is the solely decisive factor.
If this behavior and therein the magnitude of the integral / dy
(extended beyond the critical point) be known, this knowledge is
equivalent to knowing the transformation substitutions for T3;, (p.

The solution l characterized by equation (18) and the boundary
condition at infinity reads:

() -- 12/3(1) -i / H2/3 )2 dT -
3a2j P3 f+

H2/3(2) 2/3 (1) T2d (19)

Therein Bankel's cylinder functions of the index 2/3 and the
argument !(-ia0T )3/2 appear (a0 = al/3). The sign of (-iaoT)3/2

3/2 (ni)/2
is to be taken so that (-io0 )3/2 becomes positive for n = re(i)/2
A closer investigation of equation (19) shows that behaves in the
entire upper semiplane, and partially, even in the lower one, namely,
for r = re i within the limits -i < < 7 at infinity like
6 6

HACA TM 1291

if a0 or a is positive. If a is negative, the upper and
lower semiplane are interchanged

S 2bC 5it 13Bi
lim il(rei ) is valid, if 6 < <
r 40 a3T 6 6
a < 0
Sir 7iti
or < <-6

a > 0
o I(20)
Hence we infer the important result:

b-- ir a.> 0
p0 a3
I ti dr = *
0- 2bC 0.
a3 ir a < (21)

Thus the transformation substitutions for cp3, cp4, accurate up to the

magnitudes of the order (aR)-1/3, are now found for finite values
of a also; we now know and that is sufficient what, according

to equation (21), the integral dy extended from w c < 0
(w c)2
to w c > 0, signifies.

The formulas (16) also may be derived once more from equation (17);
to the asymptotic solutions (11) of equation (7a) correspond the

1 2/3( (2 io)37 (19a)

of the homogeneous equation (18) (C = 0). The problem of finding the
transformation substitutions of the "asymptotic" integrals (11) and (14)
is therewith completed with the required accuracy (except for quantities
of the order (oR)-1/3).

NACA TM 1291 19

4. Fulfillment of the Boundary Conditions and the

Stability of the Oscillations Corresponding

to the Solution System I

Our considerations so far have been quite independent of the type
of profile except for a few limitations concerning the singular points
which had to be imposed on the profile. In order not to lose our-
selves in an excessive number of different possibilities, we shall
further specialize the character of the basic flow. The considera-
tions, however, have much more general validity. We thus assume that
the bounding walls are represented by the equations y = +1 and
y = -1, that, furthermore, the wall y = 1 possesses, with respect to
the other, a relative velocity in the positive X direction (of the
magnitude w(+l) w(-l)) and that the laminar flow adheres to the
walls (which corresponds to Couette's test arrangement); finally, we
assume that in the range -1 < y < +1, that is, in the fluid, Re(w c)
once and only once is zero. Moreover, we shall presuppose in the
entire region continuity for w and the derivatives of w and, beyond
this, make the additional assumption that the functions w, w', w",
etc., always are of normal magnitude, that is, that they do not, for
instance, at certain points, assume a magnitude of the order (aR)1/2

Furthermore, for the following calculations, we at first regard
a as so small and aR as so large that we may put with sufficient

T3 = w c


p = (w c) ( dy
=f-1 (w- c)2

The fixing of the lower limit of the integral in p4 obviously does
not signify a limitation of the generality of our solutions. Rather
we determine thereby P4 as that linear combination of q3 and p4

which disappears at the point y = -1. In case w c should dis-
sppear there also, p4 obviously is replaced by the function
rp = w c which now for y = -1 becomes zero.

NACA TM 1291

In order to satisfy the boundary conditions first at the wall
y = -1, we form two aggregates fl, f2 from pl', p2' q3, T4 for
which really = cp' = 0 for y = -1


go(-l)[w(-l) c] -9 w'(-l)
gO(-i)[w(-) c] + vw'(-1)

T3 _1
w(-l) c p(-l)

S 3 _P I
w(-1) c (-1)

Therein we understand from now on by go the root V-iaR(w c), not

as in equation (10),

4-i(w c),

in order to save writing down the

In order to satisfy the boundary conditions at the other wall as
well, one must attempt to determine two constants A and B so that

Afl(+l) + Bf2(+l) = 0

Afl'(+l) + Bf2'(+l) = 0

The condition for the possibility of such a determination is



f 2(+1)
= 0



fl = 9 +

f2 = 94 -

* (22)

NACA TM 1291

By this condition c or 0, respectively, is determined if R and a
are giv.en. Thus it is now a question of solving equation (23) for c
and of determining the sign of the imaginary part of p. Equation (23)
forms the perfect analogue to Sommerfeld's turbulence equation for the
linear profile.

From equation (16) we infer

fl(+l) = T4(+1) +

S_____ P3(+1)
g0(-l)[w(-l) c] w'(-l) (1) c

p(+l) + iqp2(+l)
91( )

fl (+l) = q4'(+l) +

go(-l)[w(-l) c]

- w'(-1)

w(-l) c

pl' (+l) + ip (+l)

> (24)


f'(+1) = cr'(+1)
f2'(i) = i4 (+)

go(-1)[w(-) -] + w'(-) W


g (-l) [(-l) C] +

9 (-i)

w(-l) c

p3' (+1)
w(-l) c

q^n ir

NACA TM 1291

We insert these values of fl, f2, f'1

f2 into equation (23)

after having made an estimate of the magnitude of the individual terms
in order to eliminate unnecessary complications of the calculation by
writing down unessential terms. For this purpose we note that there
will be, in general, either 92(+1) << 92(-l) or 92(+1) >> 2(-l).

This is caused by the factor aR- in the exponent of pl, q2 in
equation (11) if there does not exist the equality

Re -i(w c)dy = Re

\j-i(w c)dy
f o

which we exclude.

Which one of the two cases will occur cannot be decided before-
hand; generally, both are possible and yield both solutions. In the
case of an obliquely symmetric profile, one case gives the solutions
symmetrical to that of the other. At any rate, the two possibilities
behave principally quite analogously and it is therefore sufficient
to investigate one of the two. Thus we assume

92(+1) << q2(-1)

that is (compare page 9), the point w = c
w = w(+l) than to w = w(-l).

is to lie nearer to

Hence it follows that

cP2( -l)

is extraordinarily small, thus is very large.

remain in fl and fl'
denominator; in f2 and

Thus there

only the terms which have p1(-l) in the
f2' the terms containing 92 are eliminated.

liACA TM 1291 23

From equation (23) we thus obtain

+ + i2(+l c'(+1) -

O(-I)[W(-1) c + w'(-A w-1) c]

S(+) + i2(+ p2 (+ l+) -

3(+i) ci] (25)

O(-1) w(-l) -c + w'(- [w(-1) c

Even in this form the equation for c is still rather complicated.
We therefore further simplify equation (25) by cancelling now not only
quantities of the order of magnitude e N, but also quantities of the
order (k)-1/2

For this purpose we determine that go(+l) is of the order
( aR) I thus at first excluding the possibility of w(+l) c being
ver,, sail, and that furthermore

i '(+1) + i2'(+1) = w (+l) c (+l) + ic2(+l] +

go(+l) [q1(+1) ic2(+l]

Thius i.'- retain only those terms of equation (25) which are multiplied
b-,- the factor go(+l).

I1ACA TM 1291

Thus the simple result is found

[l(+l) ic2(+1lIp (+l) = O


2J = -iR(w-c)dy ( ) (26)
yO dy
e 0 -i y 0 (26)
J-1 (w c)2

This equation possesses two completely different solution systems
2 \-iaR(w-c)dy
e 0 = i (I)

1 dy
TOd (II)
S(w c)2

System I represents the perfect analogue to the solutions Hopf
obtained for the linear profile and has discussed in detail elsewhere,
section 4. Actually it is shown that the oscillations corresponding
to system I always are of stable character. From

2fV -iaR(w-c)dy
e = 1


2 -iaR(w c)dy = ii + 2n) (27)
yo (2

NACA TM 1291

where n is a positive (compare page 9) integer. It is easily seen
that this equation can only be satisfied when ac = P possesses a
positive imaginary part. Thus the oscillations characterized by equa-
tion (27) are actually damped, the amount of the damping being of the
order of magnitude w(+l) c, and therefore by no means need be small.

5. The Solution System II and the Conditions for

Instability of a Profile

The solutions in the system II are identical with the solutions
of the Rayleigh equation (8) and satisfy the condition

dy = (28)
i ( c)2

or (compare the remark to equation (14a), page 19) quite generally

y dy
cp4 = (w c) (-cy =
J (v c)2

y = +1

y = -1

The latter form differs from the first in certain exceptional cases
which will be discussed later; moreover, equation (28) represents, of
course, only a first approximation (a = 0). For the solutions of
equation (28) one must now distinguish four different possibilities:
Either, (1) equation (28) has solutions with complex c, then the
profile is always unstable since the conjugate complex value of c
also always represents a solution; (2) there exist solutions of equa-
tion (28) with real c. Then we designate, as Prandtl did elsewhere,
the profile as "capable of oscillations." This can, according to
equation (21), occur only if, at the point (w = c), w" = O, if,
therefore, the profile either possesses a point of inflection or is

NACA TM 1291

composed of linear pieces; (3) real values of c exist which make
at least the real part of

f1 dy

[1 (w- c)2

zero; or, finally, (4) if none of these three cases occur, equation (28)
has no solution. In cases (3) and (4), we call the profile "not
capable of oscillations." We contend that case 1 always results in
instability, cases (3) and (4) always in stability, case (2) generally
in instability of the profile taken as a basis. For cases (1) and (4),
this has already been proved above. In case (3), we put c = cr + ici
with cr signifying that real value of c for which the real part of

P1 dy
1 (w c)2

disappears. Then we know from section 3 that for ci < 0, the imaginary

part of the integral becomes 2b i, for ci >>(aR) 3, however,
i. Thus, for reasons of continuity (compare section 3), a
point ci > 0 must exist where the imaginary part of the integral (28)
also disappears. The four solutions of equation (28) thus character-
ized yield therefore a quantity c with a positive imaginary part,
thus stable oscillations.

Case (2) finally requires somewhat more detailed calculations.
Before performing them we note that to case (2) pertain two types of
solution for equation (28) which cannot be represented in the form

i y = 0
S(w c)2

If w(+l) = w(-l) a solution of equation (28) is p = w w(+l); in
fact, here q = 0 for y = +1 and y = -1. Furthermore, it

NACA TM 1291

may happen that, for instance, w' becomes infinite for w = w(+l).


1 [w w(+l12

also is a solution of equation (3) which satisfies the boundary con-
ditions. We shall not treat this case here in more detail since it
will be discussed more thoroughly in part II; however, it must be
noted as essential that the difference between cases (2) and (3) is
very large and that it is, for instance, by no means sufficient to
approximate, according to Rayleigh, a curved profile by a polygon.

For an illustration of this difference

ReJ(c] = Re dy
S -1 (w c)

is represented qualitatively as a function of c in figure 1 where
the solid curve corresponds to the curved profile, the dashed curves
to the one consisting of linear pieces. One sees that every break
causes a new root Re(J) = 0 because J at the point c = Wbreak for

the broken profile varies like 1 This corresponds to
w break
Rayleigh's well-known theorem that there are as many oscillation roots
as breaks. Nevertheless, the curved profile does not possess an oscil-
lation root. After this comment, we revert to our contention that the
profiles capable of oscillation generally become unstable if the fric-
tion is taken into consideration.

For a proof of this instability, we return to equation (25) and
to the more exact solutions in system II. Since we know that c is
real, except for quantities of the order of magnitude (cR)-1/2, we
may assume

cp2(+l) >> (+l)

NACA TM 1291

If we furthermore neglect the terms of the order (aR)-1/2
tion (25), we obtain after slight transformations

in equa-

S dy 1 1
(w c)2 go(-1)[w(-l) c 2 Og(+l)[w(+1) C2

We put further c = cO + 5 where cO is the zero of J, 8 a small
quantity of the order (aR)- /. We assume for reasons of simplicity
a to be positive; then we may on the right side replace c by cO
and may develop the left side into a Taylor series in 8. Thus there
results, if we break off the Taylor series with the second term which
we presuppose as sufficient approximation

J(c) = J(co) +


and from equation (25) because of

J(co) = 0;

go = -iaR(w c)

(concerning the sign, compare page 9)

de (c=co)

R-- 1 i + i
S-2aR ( +
C w(-15/2 [w(+1) clS/2

Hence follows, because of

cO w(-1) > w(+l) cO

NACA TM 1291

(cO is to lie nearer to w(+l)) that the imaginary part of 8 and
thus also that of c and of P has the same sign as

dc (c-.c )

and that oscillations corresponding to a negative value of have
an unstable character. If therefore our partly linear profile still
has the property that d < 0 at the point w = c, it is unstable.
This condition < 0, however, is satisfied very frequently, for
instance, always when the point w = c lies near one wall (for
instance, y = +1) and the profile is linear from the point w = c to
the boundary.

Summarizing, we conclude: The instability or the stability of a
profile can be decided for all profiles considered so far by their
behavior in the case of frictionless fluid. Profiles which are capable
of undamped oscillations in the latter case and where the friction is
taken into account become, under certain presuppositions, unstable.
The latter profiles must have very special properties as shown above;
they must, for instance, be partly composed of linear pieces or they
must have a point of inflection w" = 0. (Compare above.)

At the same time, however, these profiles of type 2 are the only
ones still to claim physical interest since they are the only ones
whose behavior with respect to their stability corresponds approximately
to Reynolds' conjectures. Following, we shall show that these profiles,
in general, really have a critical Reynolds number (with the exception
of the broken profiles).

6. The Reynolds Number of the Stability Limit; Numerical

Calculation on the Parabola Profile

If, therefore, a profile is prescribed which, for frictionless
fluid, permits undamped oscillations and with friction is unstable,
the question arises, for what minimum value of the Reynolds number does
instability occur? The simplified equations (25), (26), etc., do not
suffice for answering this question. We must revert to equation (23)
and to the forms (11) and (14) for the integrals pl' P2' p3, and
74; however, it is, of course, quite impossible generally, for an

NACA TM 1291

arbitrary profile w, to represent the critical Reynolds number as a
function of w and of integrals over w; it will only be our task to
indicate the way by which one arrives at the critical velocity and
then to perfc'rm the calculation on a special example.

Since in our last calculations a and R had appeared only in
the combination aR (because we had assumed a2 as small), these
calculations can yield at best a critical value for aR only, not
for R alone. Thus we must first investigate the behavior of the
roots of equation (23) for increasing a2. Of the roots Qf equa-
tion (23), only those in the solution system II which satisfy the
equation qP4(+l) = 0 are of interest.

Instead of equation (23) we must therefore discuss the equation

P = (w ( c) 2 dy + a2 f dy(w c)2. y
S-c) Jc) ) (28a)

S= 0 for y = -1, y = +1

If the profile consists, as in Rayleigh's example, of linear pieces,
there exists (compare page 27) always a root of equation (28a) for
every break and these roots remain in existence for every value of
a2. Thus, the broken profile yields no maximum value of a2 and
therefore cannot ever lead to a critical Reynolds number.16

This is different if (cf. pp. 26 and 27) a solution of equations (28)
or (28a), respectively, with real c is possible for the reason that
either somewhere in the profile w" = 0 or that w(+l) = w(-l),
(p = w w(+l) represents a solution of equation (28). These latter
types of solution always yield a solution of equation (28a) only for
a very definite value of a2. For w" = 0, c is determined by the
very fact that for w" = 0, w is to be w = c; thus the equation (28a)
defines a quite definite value of a2; however, for the case
w(+l) = w(-l) a solution of equation (28a) obviously exists only
for a2 = 0.

16It is still presupposed that R and aR are large and a < Thus critical Reynolds numbers will possibly appear if these presup-
positions are no longer valid; however, the respective Reynolds num-
bers R would then probably assume values so small that they certainly
would be of no physical significance.

NACA TM 1291 31

For this type of solution of equations (28) or (28a), which are
characterized in the limit R = o by a very definite value of a2, we
shall expect that, with the friction taken into consideration, a also
may vary from its definite value only by small amounts. For these
profiles the appearance of a maximum value (and in the case w" = 0
also of a minimum value) for a is very understandable. Thus all
oscillations, the wave length of which is smaller than a certain
critical wave length, are in such cases damped for all values of aR.

After having found an upper limit for a2, one will attempt to
determine the approximate magnitude for the lower limit of aR. A
simple investigation of equation (25) shows that essential variations
in the imaginary part of c occur only after the exponent of e in
the approximate representation (11) in (p(+l), p2(+l) has decreased
to values of the order of magnitude 1; however, if this is the case,
we very soon reach the critical value (for which the imaginary part
of c is changed from negative to positive values) as will be shown
in the numerical example. If we assume that w is essentially linear
between w = cO (co = real part of c) and w(+l) the condition for
the approximate magnitude of aR reads

(aR)1/2(+) co] 3/2
~ 1 (29)


1/3 w'(+1)2/3
w(+l) CO

Since in the cases of interest to us w(+l) co will probably be
small, we may by assumption form a conclusion as to high critical
Reynolds numbers. At the same time we note that for a certain value
of R there will always exist not only a maximum value but also a
minimum value for a of the unstable oscillations. This follows from
the fact that we did find a minimum value of aR (not R).

As numerical example for our general calculations made so far, we
select the parabola profile because it is physically the most
interesting one. It is to be classified as "profile capable of
oscillation" of the type w(+l) = w(-l).

I1ACA TM 1291

Here too we shall consider only the two-dimensional motion, that
is, not Poiseuille's flow in tubes but the flow prevailing between two
parallel walls at rest (y = +1, y = -1) under the influence of a
constant pressure gradient. Thus we put

w = 1- y2 (30)

The symmetry of w and w c permits the deduction that q must
be an even function of y.17 Thus we single out, from among the solu-
tions of equation (7a), two symmetrical particular integrals and attempt
to satisfy the boundary conditions at one of the walls, for instance,
y = -1. Those at the other wall then are fulfilled automatically.
Obviously we may take simply Cp3 as one of those symmetrical integrals.
For the other we choose

(Pl(Y) c 2(y)
1p(0o) p2(O)

It follows from equation (29a) that for our profile near critical
velocity c will be small of the order (aR)-1/3; in the following
calculations we shall thus cancel terms of higher than first order
in c. Furthermore, we state that cp2() will be >> pl(0) so that
in the neighborhood of w = 0 and w = c the second symmetrical
function T simply is reduced to p1(y). From equation (16) then
follows that we have the two integrals C3 and p1 iq2 at disposal
for fulfillment of the boundary conditions for y = -1. Equation (23)
is therewith transformed into

17If one divides cp into a part even in y and a part odd in y,
each part of cp by itself must satisfy the differential equation (7a)
and the boundary conditions because of the symmetry of w c and w.
For the general stability investigation of the profile 1 2 it is
therefore sufficient to treat the two cases "cp even" and "odd"
separately and only these two cases; however, it may easily be seen
that the assumption of symmet-rical oscillations, that is, "cp odd" does
not lead to a solution of equation (23) and thus not to unstable
oscillations. The assumption "cp even" therefore suffices for the.
stability investigation. This is noteworthy insofar as, accordingly,
all symmetrical oscillations are stable and only unsymmetrical dis-
turbances unstable.

NACA TM 1291

9p3(-l-l) icp2(-l)
3 (-1) 91'(-l) icp2'(-l)


In (p3 one must always take y = 0 as lower limit for the occurring
integrals in order to guarantee the symmetry of (3. Furthermore, we
shall develop in c3 only up to magnitudes of the order a4 and in
the development in (aR)-1 break-off with the terms of the order
(aR)-1. We now write equation (31) in the form

(pl'(-l) iq'P2'(-)
cp1(-l) i(p2(-l)

If one inserts equations (11) and (14),

2 I -iaR(w-c)dy

9 3'(-1)

there results


+i Jia 9+

2 -14
2f -iaR(w-c)dy

[ r-1
2 Jo
c2 0f

dy(w c)2 + a2 +


(dy c)
(w c)2 d

2 4 i
dy(w c) + a + -.

Since c becomes very small, we assume in first approximation w
from 0 to c as linear; w 2(y + 1). Then we obtain

1 + am2

NACA TM 1291

1 S f-5 dy =- 3 3/2 (34)

If we put

z = 1 c3/2(2R)1/2

there arises from equation (33)

/-(l~i~z a2 JOI dy(w c)2 + .
e-(+i)z i 3z(1 + 9 + a2 dy( wc)
-(l+i)z 2 2 c 1+
e i 1 + a2 .

This equation is perfectly analogous to equation (26). We are
interested, above all, in the limiting value R or z, respectively,
for which the unstable oscillations are transformed into stable ones;
thus the imaginary part of c is exactly zero. This limiting value z
will, of course, also be a function of a. Thus we now assume c as
real and thereby obtain the limiting value of z or R, respectively,
as a function of a. The minimum value of R on this R(a) curve
will be denoted as the characteristic Reynolds number for the parabola
profile. Detailed calculation shows that one obtains by means of the
form (35) of the stability equation only the upper part of the curve
R = R(a) (solid line in fig. 2) which was to be expected according to
the deliberations of section 6; however, one can calculate the lower
part of the curve R = R(a) only by using for q 1, C2, approximations
other (compare equation (19a)) than the asymptotic formulas (ll). The
critical Reynolds number denotes just the range where the asymptotic
formulas cease to be valid. Since this circumstance would lead to very
complicated numerical calculations and since we cannot attach, in
general, (compare section 7) any essential physical significance to the
type of instability here characterized, we used rough estimates for
calculation of the lower part (dashed line in fig. 1) of the curve
R = R(a) which, of course, cannot yield quantitative results; however,
the qualitative behavior of the curve is surely reproduced correctly.
Thus, we conclude from figure 2:

NACA TM 1291

1. There exist both a maximum value of a and a minimum value
of R for instability.

2. For a definite value of R there exists a maximum as well as
a minimum value of a; instability prevails within these values, sta-
bility outside.

3. The maximum value of a lies approximately at a = 0.7
(a2 = );t the magnitude of the minimum value of R is of the order

of 103. A calculation of this minimum value with some degree of
exactness is not possible from the figure.

7. Physical Discussion of the Results of Part I

Let us summarize once more in detail all results found concerning
the stability problem. Above all, it became clear in the course of
the calculation that the problem of the stability of a profile for a
viscous fluid can generally be decided by treating like Lord Rayleigh
the limiting case of frictionless fluid (equation (8)). Profiles which
are unstable then (that is, for R = o) remain so for sufficiently
large finite values of R (section 4) as was to be expected beforehand.
Likewise profiles which, in the frictionless case, are not capable of
oscillations are found to be stable (section 4) and profiles, which
according to the investigation by equation (8) permit undamped oscil-
lations, generally to be unstable (section 5). This latter case
obviously is the only one which, physically, signifies something new
compared to frictionless hydrodynamics; however, it should be emphasized
that this case, contrary to what one might conclude at first from
Rayleigh's reports, represents an exceptional case. If one disregards
the possibilities w(+l) = w(-l), w'boundary = (section 5), it is
a necessary condition for the occurrence of this exceptional case that
somewhere in the fluid w" = 0. The broken profiles consisting of
linear pieces introduced by Rayleigh belong to those exceptional pro-
files; however, the only permissible conclusion is that curved profiles
for the purpose of stability investigation may not be approximated by
polygons according to Rayleigh (page 27). It is true that one may
find also for profiles curved everywhere (w" 1 O) (section 5) when
using the differential equations with friction terms oscillations
which are damped for every value of aR for which, however, in the
limit R = m the amount of the damping like (aR)-1/3 tends toward
zero; thus one has here also undamped oscillations for R = o. However,
these oscillation roots are lost if one takes the simplified differ-
ential equation (8) without friction terms as a basis. Insofar, there-
fore, this also is not a case of exception to the rule according to

NACA TM 1291

which consideration of friction, in cases where the frictionless equa-
tion (8) permits undamped oscillations, results in an excitation;
however, as said above, the possibility of undamped oscillations for
equation (8) must be regarded as an exceptional case. The parabola
profile belongs to these exceptional profiles (section 6). If we
investigate further for the unstable profiles concerning the range of
values for R within which instability occurs, it is found that
generally, too, only profiles of the last class may lead to critical
Reynolds numbers, that is, only profiles which permit without friction
undamped oscillations; however, among the latter, Rayleigh's broken
profiles nevertheless did not yield (compare also page 30, footnote 16)
a critical Reynolds number. For Rayleigh's profiles a minimum value
of aR does exist but no maximum value of a; therefore, no minimum
value of R for the neutral stability either (section 6). Only those
profiles of the last class for which in the frictionless case only a
definite value of a leads to undamped oscillations (for instance,
the types w" = 0 for w = c, w(+l) = w(-l)) result in a maximum
value of a and a minimum value of aR, thus also in a minimum value
for R. For a definite value of R there exists therefore for those
profiles a maximum as well as a minimum value of a for the unstable
oscillations. All these results are in agreement with the stability
investigations of hydrodynamic profiles made so far.18

The question is now how these mathematical results will manifest
themselves experimentally. It seems surprising that the stable pro-
files (for instance, Couette's19) and the unstable ones (for instance,
Poiseuille's) empirically show exactly the same behavior. Above a
certain Reynolds number turbulence occurs in case of sufficient dis-
turbances; if the disturbances are made as small as possible, the
laminar profile may be obtained for arbitrarily high Reynolds numbers.
Especially the last fact, which has been tested by Ekman (footnote 10,
p. 2), on the parabola profile seems to contradict the theory for
unstable profiles; however, it can easily be seen that this contradic-
tion is only illusory:20 The smaller the external disturbances, the

18Compare the reports quoted in the introduction.

19However, compare the interesting investigations of Couette's
motion concerning its stability against three-dimensional disturbances.
G. J. Taylor, Stability of a Viscous Liquid Contained between Two
Rotating Cylinders. Phil. Transact, of the Royal Society London 223.,
pages 289-343, 1922.
20This possibility of interpreting Ekman's tests as a sort of
starting effect has been pointed out to me by Professor Prandtl. I
should like here to express my deepest gratitude to him for this and
many other valuable suggestions.

NACA TM 1291

longer it takes (particularly for high Reynolds numbers since the
excitation there is of the order (aR)-1/2, compare section 5) until
they noticeably influence the motion. Thus it will always be possible
to postpone, for flow in tubes, this moment so long that the quantity
of fluid, the stability of which is dealt with, has already left the
tube when its instability becomes apparent. The Reynolds number we
calculated could therefore be tested only on a closed system of tubes
where the same quantity of fluid always flows. On the other hand,
the tests by Schiller (footnote 9, p. 2) which show that below a
certain Reynolds number only laminar motion exists cannot be included
at all in stability investigations. The original motion here is not
laminar; one rather deals'with existence or nonexistence of a turbulent
form of motion. At any rate, one may conclude from all these con-
siderations only that a solution of the turbulence problem by stability
considerations alone is absolutely not possible.

Still, the previous investigations may yield important qualitative
results for our real purpose, the calculation of the turbulent motion.
If we interpret the turbulent motion as a certain basic flow with
superimposed undamped oscillations, we may conclude from our calcula-
tions that the minimum value R for which this type of motion is
possible probably also lies at values of the order of magnitude 103; that
the wave length of the undamped oscillations lies at 2nh/2, namely a
at values of the order 1, and that a for a prescribed R is confined
to certain occasionally very narrow limits; that furthermore these
oscillations have the character of a wall disturbance as may be con-
cluded from the smallness of w(+l) c. These qualitative results
are quite independent of the special form of the basic flow; however,
beyond such qualitative criteria the calculations made so far do not
contribute anything toward the actual solution of the turbulence


1. Statement of the Mathematical Problem

The Reynolds number usually denoted as critical (which is, for
instance, measured in Schiller's tests and indicates the appearance

21However, the disturbances in stability observed by Ruckes
(footnote 10, p. 2) for rather small Reynolds numbers are perhaps
caused by instability according to section 7. This would be quite
conceivable when the critical Reynolds number according to section 7
lies below the one for which turbulence (compare part II) is possible
for the first time.

NACA TM 1291

of turbulence in case of sufficiently large disturbances) has no con-
nection with stability problems and with the laminar flow; it is
absolutely a characteristic constant of the turbulent motion. Like-
wise, the Blasius drag law and the well-known conclusion derived from
it (that the turbulent velocity in the proximity of a wall increases
with the 1/7 power of the distance from the wall) show clearly that
the so-called turbulent motion has its own well-defined regularities
and that it represents a second possible form of motion of the viscous
fluids. Thus the only way to a solution of the turbulence problem is
to attempt to eliminate the indefiniteness of the turbulent motion
and to idealize it until it permits mathematical analysis by Stokes's

The turbulence problem of hydrodynamics is a problem of energetic,
not of dynamic stability. There exist two different forms of motion
of the viscous fluid, each of which has a certain range of values of
Reynolds numbers within which it is possible. Laminar flow is possible
from R = 0 to R = o but becomes, however, under certain conditions,
above a certain value of R dynamically neutrally stable. The turbulent
motion on the other hand exists only above a certain critical value
of R and is always22 energetically more stable than the laminar
motion. Thus one may in the range of R in which both forms of motion
are possible always let the fluid drop from the laminar to the tur-
bulent state by means of sufficiently large disturbances.

In order to make approximate mathematical treatment of the tur-
bulent motion possible, we again consider the flow between two parallel
walls and make, first, the following assumptions:23 The flow is to
be (a) symmetrical about the X axis with the bounding walls at rest,
(b) periodic in the X direction with the period24 2n/m, and
(c) periodic with time with the period 2A/P, and (d) all disturbances
are to propagate with a speed p/a relative to the X axis, that is
if the motion is developed into a Fourier series, only products of
i(pt ax) are to appear in the exponents of25 e.

22Compare F. Noether, elsewhere.
23This statement, too, which represents a simple generalization
of Sommerfeld's stability theorem was indicated, for investigation of
the turbulent motion itself, by F. Noether without further conclusions
in his paper entitled "Zur Theorie der Turbulenz" (Concerning the
theory of turbulence), Jahresberichte des deutschen Math. Vereins 23,
page 138, 1914.
24The assumption of a definite a is justified by the result of
part I that a is confined between certain limits which are, partic-
ularly in the proximity of the minimum value of R, very, narrow.
25We need not emphasize the part that the actual motions are doubt-
lessly much more complicated; nevertheless, one may well expect these
statements to permit qualitative statements concerning the turbulence.


The Fourier development of the stream function should therefore

'I = cp(y) + P(y)ei(t-~x) + 1p(y)e-i(t-.x) + p2(y)e2i(pt-ax) +

The mathematical problem then consists in the determination of the
odd (according to a) functions cpO cp1, and qc2 (0l 2 con-
jugate to (qp, c2). At first, the degree of convergence of the
series (36) is completely unknown; the question of convergence can be
decided only after calculation of pO, q1 If we want to carry
accuracy so far as to the nth'approximation, that is, if we want to
calculate O pn, we obviously obtain (n + 1) simultaneous
differential equations for the (n + 1) unknown functions p cp.
Following, we shall need partly the first, partly the second approxi-
mation. Thus we enter equation (4) with the statement equation (36),
compare the coefficients of the periodic functions on both sides, and
break off with the term e2i(Pt-ax) and thus with q2,, T2. For (0p
we write w (as in equation (6) for 0 ). Thus three simultaneous
differential equations are produced (the simultaneously obtained con-
jugate equations need not be written down)

d2 '. i w''
-dy2 ) 2((P2 P2 = -w2

2' ( ," a2) + ((C1P" 4a2 ) =


T2" 422 _(w ) 2 + I"- ) =l

S,,("- -2 Y -" + 16ma4 )

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The first of these equations may be integrated twice and yields

1 1 p1 + 222' 222) = R(w' c Cly) (37a)

C and C1 signify arbitrary integration constants.

Since the left side of equation (37a) and w are, according to
requirement (a), odd in y, C must disappear in our case.

If we go back from the second to the first approximation, our
system of equations is reduced to two simultaneous differential equa-
tions for w and cpl

1 1 -fL = -- w- Cy)
(cp ak2(w w1cp, (l 1"1 2m21" i +CL )
_91 a 1 1P I

By way of an interpolation we shall now reflect what replaces equa-
tion (38) if we do not consider a flow symmetrical about the X axis
(requirement a), (that is, the flow of a fluid under a pressure
gradient between two walls at rest), but instead a flow antisymmetrical
about the X axis (that is, a flow between two walls moved relative to
each other without pressure gradient as in the Couette case). Require-
ments (b) and (c) are to be maintained. The statement (36) will then
ng longer be satisfactory since m1p, (p2, etc., for arbitrary 3/a
are no longer even functions; in order to obtain the entire flow
pattern in terms of odd functions, we must also include the symmetrical
oscillations of the form ei(-Pt-ax) in the formulation for I, that
is, must start with the terms

+ cp(y)ei(t-ax)+ c -ye-i(pt-ax) + 1(y)e-i(t-ax) +
cP + cpi( y)e '+cl-ev p(e h /+...

NACA TM 1291 41

As a consequence finally all the terms of the form ei(mpt-nax)
appear in i (elimination of requirement (d)).

In place of equation (33) there results

:1(yJP(y)' cpl(y)'%l(y) + Pl(-Y)CP(-y)' -

Sl(-y'1(-y) -1 (w' C) 1 (39)

( 2w)(w w"1 -

The two equations of the system (38) and (39), respectively, are of
simple illustrative significance.

The second equation is none other than our former stability equa-
tion (7) which determines the amplitude of the oscillation superimposed
on a basic flow w and which formed the basis for our investigations
in part I. The first equation, however, represents the theorem of
momentum. The left side of this equation essentially indicates the
momentum transferred on the average by the turbulent vorticity26, the
term with w' on the right represents the laminar momentum transfer,
and the constant C or Cly, respectively, is the constant of the
momentum theorem.

Due to the boundary conditions at the walls q~ = 91' = 0. There-
fore there w' = C or ClYwall, respectively; thus at the walls the lami-
nar momentum transport surpasses the turbulent one, W'wall will gen-
erally be very large. (Compare the next section.) At the channel
center, however, that is, in the entire tunnel outside of the immediate

26We are referring here to the mean momentum in the X direction
which, for our problem, is transferred in the Y direction. The
momentum in the X direction equals, on the whole u, the velocity of
the particle transporting the momentum in the Y direction is v; thus
the momentum transferred during the unit time uv, on the average uv
which for the case (36) results in
UV = -ia(cpj ncp1l

NACA TM 1291

wall proximity, w is of the order of magnitude 1, thus very small
compared to C or Cly, respectively. The turbulent momentum trans-
port will therefore here completely overbalance the laminar one.

It corresponds to the structure of the systems (38) and (39)
that we are able to give immediately a trivial solution of them,
namely qp = 0, w' = C or, respectively, w' = C1y, that is, we thus
revert to the laminar motion.

Our problem now is to obtain definite results concerning the non-
trivial solutions of equations (38) and (39).

2. The Turbulent Motion in Wall Proximity and

the Law of Resistance

The most important result concerning the behavior of w in the
immediate proximity of the walls is the law derived by V. Karman
(elsewhere) from the Blasius drag law by means of considerations of
similitude that w in the proximity of a wall increases with 11/7
(j representing the distance from the wall). We repeat briefly
Von Karman's train of thought since we are thereby enabled to a general
visualization of what to expect, even without knowing the Blasius law,
concerning the behavior of w of the wall.

As can easily be seen from considerations of similitude, it must
be possible to represent the shearing stress T acting at a wall
(that is, the drag) in the form

T = K1 U f(R)

where K signifies a certain dimensionless constant.

If we specially assume a power law there is

U R= U Tipt
E hF F

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From equation (40) follows inversely

U = 1 h- 3 1+(

The velocity distribution in wall proximity must then be represented
by an equation of the form


w =

(Here again (compare equations (1) and (6)) w has been selected
dimensionless and therefore contains U in the denominator; q denotes
the distance from the wall )

We again assume in first approximation a power law (let a be a
dimensionless constant)

w 1

If one now requires the velocity distribution in immediate wall proxi-
mity to be a function only of T, p, p, but not of h which is
physically very plausible, there follows

1+ s

For as corresponds to Blasius' law there results e .
4 7

In order to understand the physical significance of this result
we note: w 1/7 signifies that w' = w is infinite at the

boundary, that therefore w infinitely clings to the wall; however,
it is clear that actually w' at the wall cannot be infinite since
w', on the contrary, essentially denotes the shearing stress at the

NACA TM 1291

wall and is therefore, according to equation (40), to be equated
to Rt except for a numerical factor independent of R.

w edge~ R (42)

w' at the wall is therefore very large for the large values of R
which are of interest to us. Corresponding to its derivation the
velocity distribution w ~ r1/7 will be strictly valid only in the
limiting case of infinitely large distance from the wall or of friction-
less fluid (R = 0). These facts can be still more easily comprehended
if the law w r11/7 is written in the form ~ w7. From the fact
that the shearing stress is finite we know that the first term of the
power development T(w) must be of the form 71w. This term, however,
is very small, essentially equalling the reciprocal value of w' and
thus being of the order R-t (compare equation (42)).

The meaning of the derived law w T11/7 is thus obviously that
the series development of r(w) is to start with the terms

S= 71w + 7w7 + (43)

where 71 is extraordinarily small and that therefore the first
term 71w for somewhat large values of w may be cancelled compared
with the second 77w7.

According to the explanations above we expect independent of the
validity of the 1/7-law for the basic flow of the turbulent motion
small curvatures at the centers, and in the wall proximity, clinging
of the basic flow to the walls.

For such a profile the investigations of part I do not directly
apply since there w', w", etc., had been presupposed as finite;
however, these investigations can easily be generalized to include
profiles like the one considered here. (Compare section 5, page 26.)
Particularly, the solution of the reduced equation (3) (thus lim R = m)
with satisfaction of the boundary conditions becomes especially simple
here; the profile characterized just now belongs, according to sec-
tion 5, page 26, to those capable of oscillation; a solution of equa-
tion (28) with real c exists. This is extremely important because

NACA TM 1291

it shows that the turbulent profiles are always unstable according to
section 5, or, in other words, that it is just the deviation (42) from
the laminar resistance law which makes an unstable profile and thus
makes turbulence possible.

The solution of equation (28) is for a = 0


-1 Cw W(+1) 2
CD [w w(+l| / w -------

By selection of the lower
zero for y = -1; that it
selection c = w(+l) can

limit of the integral we made p become
becomes zero also for y = +1 due to
be seen easily from the following

CP = Lw w(+l) [ dw
fw(-1) w w(+1 l2w

The integral of the right side becomes at the point w = w(+l)
infinite of lower order than since w there (in the
w w(+l)
limit R = m) becomes infinite. Thus cp = 0 for y = +1. By equa-
tion (44) we have represented in the limit of frictionless fluid the
amplitude of the turbulent oscillations and derived from the boundary
conditions the value for P/a, namely B/a = w(+l). It is, however,
self-evident that the solution symmetrical to equation (44)


1 cp w -1] 2

also satisfies
here. In case
equations (44)
systems exist,
the velocities

the boundary conditions; thus c = w(-l) is valid
of the Couette arrangement we therefore conclude from
and (44a) that two mutually symmetrical oscillation
the wave velocities of which agree respectively with
of the two walls (w(+l) and w(-l)).

NACA TM 1291

For the symmetrical flow between two walls at rest, on the other
hand, w(+l) = w(-l) = 0. From equations (44) and (44a) we then con-
clude that every integral of the form

Pa =W fdy

satisfies the boundary conditions; however, from the requirement (a)
that p is to be odd there results that we must select as lower limit

of the integral y = 0. Thus

P = d (44b)
fo w

In the case of symmetrical flow there is therefore, particularly
for p/a

w(+l) = w(-l) = = 0 (44c)

In the turbulent basic flow, the type of its singularity at the
walls is of foremost interest to us; thus for the assumption w ~ 16,
the exponent e. We shall attempt to show that from the differential
equations (38) and (39) respectively in the limit R = o at least in
immediate proximity of the wall such a power law with the exponent
E = 1/7 actually follows. It is true that the domain of convergence
of the power series used is not established so that the conclusions,
as far as they apply to the shape of the profile at some distance from
the wall, are uncertain. We develop cp and w in the neighborhood
of T = 0 in integral and positive powers of T this is possible for
any finite value and then inversely T in integral powers .of w.
Thus we are led directly to the formula (43) for q(w).

We contend, and this is the most important result we shall need
later, that il in first approximation may be represented by a series
of the form

p1 = ~212 + c515 + agTl3 +

NACA TM 1291

where a2, a .. are real, all purely imaginary con-
stants; furthermore, w is of the form

w = 0 + P7T7 + (45)

This contention may be proved for the differential equations (38)
directly by expressing cp and w in undetermined coefficients if
the terms aO., am, a2 a3, and 0, 01 are prescribed. Thus we
will, above all, attempt to determine these terms. First, p1 and
p' for n = 0 must be zero because of the boundary conditions; thus
the series for pl starts with m2n2 (ao = a,. = 0). We can verify
afterward that, furthermore, the following term a3m3 is eliminated,
that is, becomes very small compared tc the other terms. By way of an
interpolation we shall prove here for this purpose by a single approxi-
mate integration of the second equation (38) that a3 assumes the
order of magnitude aR. For a2 = 0 equation (38) reads

il(w T- i i""

whence follows

Pl I )-1 w pRl' + A (46)

The constant A is here of the same order of magnitude as the left
side of equation (46) at the center of the tunnel, thus almost of the
order of magnitude 1. (Compare part I, section 2.) The term cp '"
at the edge is therefore of the order aR due to the boundary condi-
tions. The same is valid for a3.

Thus we shall meanwhile assume a3 as small and later attempt to
justify that assumption. Of the constants 0, 01, the first, 00,
equals zero because of the requirement (a), section 1.

NACA TM 1291

The constants a2 and pl are, at first, arbitrary27 and there
is no possibility of deriving them from the solution of the differential
equations (38) and (39) in the proximity of the wall. This possibility
would.arise only if we should succeed in continuing the solution (45)
analytically up to the other wall; however, this is an extremely com-
plicated mathematical problem if only for the reason that, as will be
seen, the simplified equations (38) and (39) are not sufficient for
determining ql and w at the center of the tunnel. Although we
must therefore forego the solution of this problem, we may still expect
to obtain, by merely developing qp and w in the proximity of one
wall with undetermined coefficients a2, l1, those qualitative char-
acteristics of w and ql in wall proximity which are, according to
experience, quite independent of the behavior of the fluid at the
tunnel center as, for instance, the law w -~ '1/7.

We enter equation (38) with the statement

qp n2 + a44 + +C5 +

w = 01 + 02T)2 + 3 + .

replacing the second equation by (46). We therefore again assume a
as very small which here only signifies (compare part I, section 2)
that the wave length of the oscillations is to be large compared to
the boundary-layer thickness; moreover, we put, according to equa-
tion (44c)


For the first equation (38) we write furthermore

-iaR R( 1' l -C = 2TR (li'(lr PliOlr) = w C1y

Therein Plr denotes the real, gli the imaginary part of "1'

27We shall assume a2 as real. This does not imply a limitation
of generality since cpl is determined only up to a factor of the form
eiX as the initial point of the time coordinate in equation (36) may
be chosen arbitrarily.

NACA TM 1291

From equations (46) and (47) now follow the recursion formulas

n(n l)(n- 2)a = -ia.R s(n 2s) an-i-sps-l (48)

npn = 2aR s(n 2s)an_ s (r)

in addition

1 = ClYedge

282 = C1

Therein asr denotes the real, asi the imaginary part of ms.

From equation (48) there follows first

a4 = 0

From equation (49) there then results

03 = 04 = 8505 P6 = 0

The term 02 may also be approximately equated to zero.

From equation (49) there follows


NACA TM 1291

For very small T the term 0212 is therefore to be neglected com-
pared to the first term Pjlq for larger q, however, the higher terms
p7T,7 etc., are completely predominant.

Let us thus assume also 82 = O and
terms of the series for p1 and w.

thus calculate the higher

There follows

5 = -iaR 3x 4x5; a6 = 7 = 0; a8 =
^ ~ 3x 4 x5 6

_()2 3x la2x1 2
3 x 5 x 6 x 7 x 8

a9 = m10 = 0;

al = -(aR)3( 7 32 -
2 x 7 x 9 x 1o x 11

3 x 5 x 6 x 8 x 9x x 11

el]2 '13 = 0

2 "2 B1
07 = -(R)2 70 8 =9 0 = 11 = 12 = 0;

Sa22P13 "2 1
p13 = (acR) -a22p13 a.10
5 x 5 x 6 x 8 x 11 x 13 x 14 7 x 10 x 11 x

The representation (45) for w we contended is therefore
it may easily be shown too that of the further terms only
sixth term has 0 a finite value.

proved and
in every

Hence follows for the representation of I as power series in w

S= 1w + 77w7 + 71w13 + .

2 22
7 = (aR)2
7nB 7


Y13 (acR)f4 4 x 3 x 3 x 17 M2 2
13 xlOxOXll xl3x 0113 5x5x6x8x11x13x14xP111


71 = -1

NACA TM 1291

The terms y2 to 76, 8 to 712' 714 etc., all equal

The development (50) now actually completely agrees with equa-
tion (43) and we seem thus to arrive, even without knowledge of the
constants a2 and 81, to the law T w7 semiempirically derived
by von Karman. The coefficients l7 and 77, however, cannot be
calculated. Inversely, we may perhaps conclude from the empirical
findings for the coefficients 01 and c2 that 77 is of the order

of magnitude 1, y7 of the order (aR)3/ thus a2 ~ (aR)3/8
Subsequently, we thus also confirm our former assertion a 3q << a2.
Raising the question of what order of magnitude are the values of w
for which the third term in equation (50) is small compared to the
second-for which therefore the w7 profile actually is valid, one
finds w v p1-1/6, thus -R-1/8. Accordingly, the profile w ~ l1/7
follows from the differential equations (38) only qualitatively at
first. No information about the fact that the 1/7 profile has been
observed almost up to the tunnel center is given in our calculations;
however, this was not to be expected since the other constants entering
the law also depend on the behavior of the fluid at the opposite wall.

As an interpolation, we shall once more briefly summarize what
factors we have neglected in deriving equation (50) from equations (48)
and (49) and attempt thereby to determine within what accuracy the
conclusions drawn from (50) are correct. First, we used system (38)
instead of (37), thus cancelled magnitudes of the order cp2/w. Further-
more, we equated a3 = 0, pB/ = 0, P2 = 0 and therewith neglected

magnitudes of the order -, -, --, and -- respectively.
aM aw pi 0771 6
The accuracy of our calculations will be determined by the largest of
the terms here neglected. Simple considerations of the order of magni-
tude, not executed here, make it probable that of these terms (2/w
is the largest but that this term, too, goes toward zero with R-~ o.

2This power series n(w) may, of course, also be derived directly
from equations (46) and (47) without the detour over the series of w(q)
if w is introduced as independent variable; however, the calculations
required for this purpose are somewhat more complicated.

NACA TM 1291

Selection of a sufficiently large value for R will therefore make
it possible to carry the accuracy of the results derived from equa-
tions (48), (49), and (50) arbitrarily far.

As to Blasius' law of resistance, it can, of course, be derived
inversely according to the method described above from the law T w7
by means of consideration of similitude if one assumes, as we did, that
the behavior of w in the proximity of the wall is independent of the
tunnel width; however, for the reasons stated above (impossibility of
analytical continuation) we must leave the question unanswered whether
this latter physically very plausible assumption also follows from
the differential equations (38) and (39), respectively.

We are, however, able to draw a noteworthy direct conclusion con-
cerning the law of resistance from equations (38) and (39) by means of
consideration of similitude. In the tunnel, except for immediate wall
proximity and the point y = 0 (compare below equation (66)) one may
write instead of the first equation (38) because of the magnitude of
C1 (compare pages 41 and 42)

iaLR( 9q)11 p1 = Cly (51)

Since the amplitude cp cannot go toward infinity with R-4-. this
would render all our calculations devoid of physical sense there
follows that C1 is at most of the order of magnitude aR, that there-
fore the exponent of equation (40) must be <1 (which in a certain
manner also can be seen from equation (41)). Hence follows (compare
equations (42) and (40)) that the law of resistance T = const.U2
usually assumed in hydraulics represents an upper limit for all imagi-
nable laws of resistance of turbulence which is independent of the wall
characteristics. One may conclude as an assumption that the law
T ~ U 7 is valid only for smooth walls it was for those only that
we obtained n w7 that the law of resistance for rough walls,
however, more and more approaches the quadratic law.29 For rough walls
the amplitude qp will be independent of R and of the magnitude of
the wall disturbances; moreover, for rough walls the boundary conditions
will no longer cause pl to be real in first approximation as corre-
sponds to equation (44).

29Compare the more exact investigations by Von Karmxn, elsewhere,
and the experimental investigations by Schiller, same periodical 3,
page 2, 1923.

NACA TM 1291

Nothing is changed in the conclusions of this paragraph if the
equations (39) are taken as a basis instead of the differential equa-
tions (38).

3. The Turbulent Motion Outside of the Immediate

Proximity of the Wall

It is essential for the motion at the tunnel center that pl here
is composed of those two integrals of (7a) which appear in case of
frictionless fluid, thus for equation (8). (Compare part I, section 2.)
The most important characteristic of cpl following from this fact is
that it satisfies except for magnitudes of the order p2 and
(aR)-1 the condition

pi 1- = Const. (52)

This results, according to Abel's theorem, from the fact that,
except for magnitudes of the order q2 and (aR)-l, Plr' and (Pli
(the real and imaginary part of (pi) are solutions of the differential
equation (8).

Hence it can be concluded that the equations (38) and (39) are not
sufficient for establishment of the motion over the entire tunnel width
but that we have to go back to equation (37) and to the system of equa-
tions which corresponds to it for Couette's case.

This, in general, involves a complication of the mathematical
problem. Only in Couette's case may the problem be solved comparatively
easily because the first equation (39), except for magnitudes of the
order w'/C, thus (aR)3/4 and (22, compare equation (37), agrees
with equation (52). Whereas, therefore, equation (52) in consequence
of its derivation from Abel's theorem is correct only up to magnitudes
of order P2, in Couette's case equation (39) should still be valid up to
magnitudes of the order (p22. This requirement is satisfied if we put


P2 = 0

NACA TM 1291

This equation is therefore to be regarded as solution for Couette's
case, of the differential equation we took as a basis.

According to equation (53) it would follow for ql from (37)

91 '91 "1?Pl = (04)

The system applying to Couette's case is not equation (37) but a more
complicated one which we are not going to write down. We do, however,
state about it that it leads, like (37), for 92 = 0 to the solu-
tion (54) and thus to the result

91 = aeTY + be-)' (55)

Here, a, b, and y are any complex constants. For w then follows
from the second equation (39) or, respectively, from its reduced
form (8)

w c = ale72 + be71 (56)

Since at one of the walls there should be w c = 0, and since, on
the other hand, w should be odd in the neighborhood of y = 0, it
follows that w, simply by the vanishing of 71 and a suitable
increase of al and bl, must degenerate to a linear profile.

Thus, we obtain the important result that for Couette's case the
basic profile w of the turbulent motion takes an essentially linear
course over the entire tunnel width however, strongly deviating from
the laminar profile, it will be much flatter than the laminar one -
that, however, (compare II, section 2) at the edge it clings again
like 11/7 to the walls].

We shall now turn to the more complicated case of a flow between
two walls at rest, thus exactly to the system (37). For a solution
we must naturally be content with rough approximations. First, we

NACA TM 1291 55

can cancel in equation (37) the right sides of all three equations,
namely the friction terms; this is fully justified by the considera-
tions of Part I, section 2. Then we equate O/a = 0 (compare
equation (44c)'.

We thus obtain for pi in the place of the second equation (37)

'(1w w 'fWl a2 2 1)- 2p2 (l"' a2i) + 2' ( 2" -

^ T,) + (T2 "' I'

- aa2 ') = 0


If we develop cp as solution of the equation (57) in powers of a2
on one hand and powers of p2 on the other, and if we further note
that I is to be odd (compare (44b)) and write p1 = q10 + 11 there
results with only the linear terms taken into consideration


10 = aw Y d
10 Jo w2

11i = aw 2 O

dy (2wplo I 2 ~ 10" + 22Tp10

'10 2" 102"')


Naturally cp is herein not fully determined the constant factor a
assumed as real which does not signify a limitation remains undetermined.
If we substitute this value of p1 into the simplified first
equation (37.), namely

cpcp1 1 = Const.


56 NACA TM 1291

we obtain, with q92i denoting the imaginary part of (2, the result

(2i' 10o + 22i10"' 2o10 '2i 102i") f Const. (61)

Now, however, as follows from the third equation (37) and from the
fact that q~ is real in first approximation, q2i satisfies the

p2i" C2iw" = 0 (62)


2i bw = (63)

If we substitute this value of qP2i into equation (63) and if we
further consider that for y = 0 the left side of equation (61) and
therewith the constant on the right side is zero (this signifies for
the constant of the right side of equation (60) only that it is in
first approximation zero, that is, small of the order qpl22 or
a2~1cp2, respectively, or a. 1 we obtain

910 10 '10 = 0 (64)

which fully agrees with (54).

This equation, it is true, becomes, like equation (54), trivial
in the neighborhood of the point y = 0; it is there fulfilled
identically since p is an odd function of y. Thus it cannot permit
there a determination of w. This leads for the symmetrical profile
(64) to a remarkable discontinuity at the point y = 0. (For the odd
profile such a discontinuity cannot be seen from the differential
equations.) If one integrates (37a) one obtains, as shown above,
after a single integration the equation

2aR cplq 1 I + 2(2" + CCP2 + w" (65)

NACA TM 1291

where C (compare pp. 41 and 42), Blasius' law of resistance being valid,
is of the order of magnitude (aR)3/4, thus at any rate very large
The left side of equation (65) disappears, however, with cl and c2
(which, as we know, are odd functions of y) at the point y = 0. Thus

v = C (66)

must be valid there. This signifies that wy0" is very large

(~(aR)3/4 and that therefore w at the point y = 0 shows a sharp
break30 (radius of curvature -(aR)-3/4). At a small distance from
this point the course of w must, according to equation (64), again
be essentially linear.

We obtain the result: For the flow between two walls at rest as
well and surely this may be applied also to the flow in the tube -
the profile is linear approximately over the entire tunnel width; at
the center, however, it shows a sharp break (it clings to the walls
with the yl/7 law). (Compare figure 3.)

The physical cause of the sharp break is the fact that the gradient
of the turbulent momentum transfer for y = 0 disappears for reasons
of symmetry and that therefore, because the gradient of the entire
momentum transfer over the tunnel width is constant, the gradient of
the laminar momentum transfer, that is w", must be very large there.

4. Final Remarks and Summary of the Physical Results

Our investigations still show two important gaps. First, they do
not yield the transition from the ~1/7 profile to the linear profile
valid in the center part. Second, they are limited to large values
of R and thus do not yield the minimum value of R, either, if such
a minimum value exists for which the turbulent motion is still possible.
The first of these two gaps is most difficult to fill in (compare
page 48); we cannot even indicate a method which would satisfactorily

30Professor Prandtl was so kind as to point out this break to me
on the basis of empirical material. The break seems less sharp empiri-
cally than according to calculation results, which is easily explained by
the fact that the assumption (a), page 38, concerning the symmetry of the
vortices and disturbances also does not exactly correspond to actual

NACA TM 1291

solve this particular problem. One may attempt to piece the two
approximations together that come from the wall and from the tunnel
center. This would have to be done by means of the condition that
at the respective junction (pl, [1' p", I' w, nd w' are to
be continuous; however, the convergence of the developments (45) and
(50) is hardly sufficient thus to guarantee a somewhat defined approxi-
mation. At any rate the ultimate result, the profile w, is still to
a great deal dependent on the type of joining the two approximations.
Finally, it must be regarded as dubious whether such an exact carrying
out of the formulation (page 39) would yield essentially new physical
results in agreement with experience since these statements certainly
represent a very strong idealization of actual conditions.

In contrast, filling in of the second gap does not offer any
basic difficulties whatsoever; all necessary expedients are contained
in Part I and once the profile w is completely known, the methods
described in Part I are, on principle, sufficient to calculate according
to Part I, section 6, the minimum value of R for which the turbulent
motion is possible. One could, for instance, calculate the critical
Reynolds number for a profile obtained, according to the method
mentioned above, by piecing together the two approximations, or one
could base this investigation on the empirically observed profile and
thus calculate the Reynolds number in a semiempirical manner. In any
case one will the investigations in Part I made this probable and
direct calculations, here not reproduced, confirmed it arrive at
the same order of magnitude of the critical Reynolds number, namely
R ~ 103. The exact value of R will, it is true, still be too
dependent on the manner in which the profile was obtained to permit
comparisons with experience. For that reason we did not perform here
such a calculation of R.

Let us finally summarize what may be concluded as physical result
from our investigations concerning the turbulence problem. In Part I
we recognize that the laminar motion and its stability condition are
not of essential significance for the turbulence problem and the
critical Reynolds number. In Part II, however, we investigated the
turbulent motion itself and may hence give a few data on the turbulent
state of motion. In general, the velocity distribution over the
entire tunnel is of the simplest type; it is according to the test
conditions linear or constant (section 3). At the center there is,
for symmetrical flow between two walls at rest, a sharp break; at
the walls the flow clings, for the T1/7 profile, to the walls
(section 2). The calculations do not disclose anything about the fact
that the 1/7 profile is valid until far into the tunnel interior. The
turbulent oscillations are for Couette's case almost harmonic in the
interior of the tunnel (section 3, equation (53)); in the proximity

NACA TM 1291

of the walls all oscillations will occur. The velocity of the waves
agrees with the wall velocity (section 2, equations (44) (44c)); for
Couette's case there exist two groups of turbulent oscillations, one
of which agrees, with respect to its velocity of propagation, with one
of the walls, whereas the other group possesses the velocity of the
other wall. Thus the turbulent disturbances show, superficially, the
character of a wall disturbance. It must, however, be emphasized that
these disturbances are capable of existence as free oscillations, inde-
pendently of wall roughness and similar influences. The amplitude of
the turbulent waves considerably increases toward the walls (this
follows from equation (44), section 2) and goes toward zero only
directly at the wall.

The wave length of the occurring oscillations (Part I, section 8)
is, with respect to order of magnitude, equal to (rather somewhat
larger than) the tunnel width. The minimum value of the Reynolds num-
ber (Part I, section 8) for which turbulence is still possible, lies -
with respect to order of magnitude near 103. From the profile 1/7
Blasius' T u7/4 seems to result, under certain presuppositions, as
the law of resistance for smooth walls. For rough walls it probably
approaches (section 2) the hydraulic law T ~ u2. The purpose of the
present report was not so much to establish these regularities, to a
great part known before, as it was to prove that all results obtained
so far (seemingly partly contradicting each other) can be uniformly
described mathematically with the aid of simple basic assumptions.

I wish to express here my deepest gratitude to my revered teacher,
Professor Sommerfeld, for suggesting this report and for frequent

Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics

60 NACA TM 1291

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