IAc4r M
NACA TM lo. 1218
TABLE OF CONTENTS
Chapter XIII. GENERAL REMARKS ON TURBULENT FLOWS .. 1
a. Turbulent Pipe Flow . . 1
b. Turbulent Boundary Layers ..... . 3
Chapter XIV. OLDER THEORTES ................... 4
Chapter XV. MORE RECENT THEORIES; MIXING LENGTHS . .. 11
Chapter XVI. PIPE FLOW .. . ... ..... 18
a. The Smooth Pipe . . 18
b. The Rough Pipe . . .
Chapter XVII. THE FRICTION DRAG OF THE FLAT PLAT? TN LONGITUDINAL
FLOW . . .. 30
a. The Smooth Pipe . . 31
b. The Rough Pipe. . .. .... 40
c. The Admissible Roughness . .... .41
Chapter XVIII. THE TURBULENT FRICTION LAYER IN ACCELERATED,
RETARDED FLOW .................. 4V,
Chapter XIX. FREE TURBULENCE . .... .... 49
a. General Remarks: Estimations . 49
b. The Plane Wake Flow . . .. ... 56
c. The Free Jet Boundary. .. . .. 61
d. The Plane Jet . ... .. 65
Chapter XX. DETERMINATION OF THE PROFILE DRAG FROM THE LOSS OF
MNDMENTUM .......................... ...... .68
a. The Method of Betz . . .. .69
b. The Method of Jones . .... .
Chapter XXI. ORIGTN OF TURBULENCE . . 74
a. General Remarks ....... . .74
b. The Method of Small Oscillations . ... .76
c. Results ............ .. . 91
Chapter XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION
LAYER ACCORDING TO THE MEETHOD OF GCRUSCHWITZ
(REFERENCE 78) . . 93
a. Integration of the Differential Equation of the Turbulent
Boundary Layer . . ...... 9
h. Connection Between the Form Parameters and I = /&* o
the Boundary Layer . .. 9
Digilized by hie Inlernel Archive
i1 2011 with ILnIdlng Irom
University ol Florida, George A. Smaihers Libraries willi support from LYRASIS and Ihe Sloan Foundation
hllp: www.archive.org details lectureserlesbouOOunii
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
T CHflJICPL E,24OCFPAITUd 4 NO. 1218E
LECTURE SERIES "BOUNDARY LAYER THEORY"
Part II Turbulent Flows*
By H. Schlichting
CHAPTER XIII. GENERAL REMARKS ON TURBULENT FLOWS
a. Turbulent Pipe Flow
The flow laws of the actual flows at high Reynolds numbers differ
considerably from those of the laminar flows treated in the preceding
part. These actual flows show a special characteristic, denoted as
"turbulence."
The character of a turbulent flow is most easily understood in
the case of the pipe flow. Consider the flow through a straight pipe of
circular cross section and with a smooth wall. For laminar flow each
fluid particle moves with uniform velocity along a rectilinear path.
Because of viscosity, the velocity of the particles near the wall is
smaller than that of the particles at the center. In order to maintain
the motion, a pressure decrease is required which, for laminar flow, is
proportional to the first power of the mean flow velocity (compare
chapter I, Part I). Actually, however, one observes that, for larger
Reynolds numbers, the pressre drop increases almost with the square of
the velocity and is very much larger than that given by the Hagen
Poiseuille law. One may conclude that the actual flow is very different
from that of the Poiseuille flow.
The following test, introduced by Reynolds, is very instructive:
If one inserts into the flowing fluid a colored filament one can observe,
for small Reynolds numbers, that the colored filament is maintained down
stream as a sharply defined thread. One may conclude that the fluid
actually flows as required by the theory of laminar flow: a gliding
along, side by side, of the adjoining layers without mutual mixing
laminarr = layer flow). For large Reynolds numbers, on the other hand,
one can observe the colored filament, even at a small distance downstream
from the inlet, distributed over the entire cross section, that is, mixed
*"Vortragsreihe 'Grenzschichttheorie. Teil B: Turbulente Str6mungen."
Zentrale fuir wissenschaftliches Berichtswesen der Lufitfahrtforechung des
Generalluftzeugmeisters (ZWB) BerlinAdlershof, pp. 154279. The
original language version of this report is divided into two main parts,
Tell A and Tell B, which have been translated as separate NACA Technical
Memorandums, Nos. 1217 and 1218, designated Part I and Part II, respectively.
This report is a continuation of the lecture series presented in part I, the
equations, figures, and tables being numbered in sequence from the first
part of the report. For general information on the series, reference
should be made to the preface and the introduction of Part I.
NACA TM No. 1218
to a great extent with the rest of the fluid. Thus the flow character
has changed completely for large Reynolds numbers: A pronounced.
transverse mixing of adjacent layers takes place. Irregular additional
velocities in the longitudinal and transverse directions are superposed
on the :ainr velocity. This state of flow is called turbulent. As a
consequence of the mixing the velocity is distributed over the cross
section more uniformly for turbulent than for laminar flow (compare fig. 17,
part I). For turbulent flow there exists a very steep velocity increase
in the immediate neighborhood of the wall and almost constant velocity in
the central regions. Consequently the wall shearing stress is considerably
larger for turbulent than for laminar flow; the same applies to the drag.
This follows also from the fact that in turbulent flow a considerable
part of the energy is used up in maintaining the turbulent mixing motion.
The exact analysis of a turbulent flow shows that at a point fixed
in space the velocity is subjected to strong irregular fluctuations with
time (fig. 72). If one measures the variation with time of a velocity
component at a fixed point in space, one obtains, qualitatively, a
variation as shown in figure 72. The flow is steady only on the average
and may be interpreted as composed of a temporal mean value on which the
irregular fluctuation velocities are superimposed.
The first extensive experimental investigations were carried out
by Darcy (reference 60) in connection with the preliminary work for a
large waterdistributing system for the city of Paris. The first quanti
tative experiments concerning laminar pipe flow were made by Hagen
.(reference 95). The first systematic tests regarding the transition from
laminar to turbulent flow were made by Osborn Reynolds (reference 61).
He determined by experiment the connection between flow volume and pressure
drop for turbulent flow and investigated very thoroughly the transition
of the laminar to the turbulent form of flow. He found, in tests of
various velocities and in pipes of various diameters,.that transition
always occurred at the same value of the Reynolds number: . This
v
Reynolds number is called the critical Reynolds number. The measurements
gave for the pipe flow:
Recrit = crit 00 ( 1)
For Re Recrit, turbulent. Later'
on it was ascertained that the numerical value of Recrit is, moreover,
very dependent on the particular test conditions. If the entering flow
was very free of disturbances, laminar flow could be maintained up to
Re = 24000. However, of main interest for the technical applications
is the lowest critical Reynolds number existing for an arbitrary disturbance
of the entering flow, due cit]evr to irregularities in the approaching flow
or to vortices forming at the ripe inlet. Concerning the drag law of the
pipe Reynolds found that the pressure drop is proportional to the 1.73
power of the mean flow velocity' :
1.73
Ap ",, u
NACA TM No. 1218
b. Turbulent Boundary Layers
Recently it was determined that the flow along the surface of a body
(boundary layer flow) also can be turbulent. We had found,for instance,
for the flat plate in longitudinal flow that the drag for laminar flow is
proportional to \Uo3 (compare equation (9.18),Part I.) However, towing
tests on plates for large Reynolds numbers carried out by Froude
1.85
(reference 96) resulted in a drag law according to which W Uo More
over, the drag coefficients in these measurements remained considerably
higher than the drag coefficient of the laminar plate flow according to
equation (9.19), Part I. Presumably this deviation is caused by the
turbulence of the boundary layer.
A clear decision about the turbulent flow in the boundary layer was
obtained by the classical experiments of Eiffel and Prandtl concerning
the drag of spheres in 1914 (reference 62). These tests gave the following
results regarding the drag of spheres (compare fig. 73). The curve of
drag against velocity shows a sudden drop at a definite velocity Vcrit,
although it rises again with further increasing velocity. If one plots
the drag coefficient cw = W/F 2 U (F = frontal area) against the
Reynolds number Uod/V, cw shows a decrease to 2/5 of its original value
at a definite Reynolds number (Recrit). Prandtl explained this phenomenon
in 1914. He was able to show that this drag decrease stems from the laminar
boundary layer changing to turbulent ahead of the separation point. The
resulting considerable rearward shift of the separation point causes a
reduction of the vortex region (dead water) behind the sphere (fig. 74).
This hypothesis could be confirmed by experiment: by putting a wire ring
on the sphere (sphere diameter 28 centimeters, wire diameter 1 millimeter)
one could attain the smaller drag at smaller Vcrit and Recrit. The wire
ring is put on slightly ahead of the laminar separation point; it causes a
vortex formation in the boundary layer, which is thus made turbulent ahead
of the separation point and separates only farther toward the rear. By
means of the wire ring the boundary layer is, sotosay, "infected" with
turbulence. Due to the mixing motions which continually lead high velocity
air masses from the outside to the wall, the turbulent boundary layer Is
able to overcome, without separation, a larger pressure increase than the
laminar boundary layer.
The turbulence of the friction layer is of great importance for all
flows along solid walls with pressure increase (diffuser, wing suction
side). It is, however, also present in the flow along a flat plate where
the pressure gradient is zero. There the flow in the boundary layer is
laminar toward the front, experiencing transition to the turbulent state
further downstream. Whereas the laminar boundary layer thickness increases
downstream with xl/2, the turbulent boundary layer thickness increases
NACA TM No. 1218
approximately as x4/ ; that is, for the turbulent boundary layer the
increase of the boundary layer thickness is considerably larger, (fig. 75).
The position of the transition point x rit is given by (fig. 75):
( )cr = 3 to 5 x 105 (13.2)
v /crit
In comparing the critical Reynolds numbers for the pipe and the plate
one must select r and 5, respectively, as reference lengths. The
equation for the flat plate is, according to Blasius (reference 8)
(compare equation (9.21a))
vx
or (13.3)
v v
Thus, with ) = 25000:
Scrit
(5) = 5.500 = 2500 (flat plate) (13.4)
v crit
U r
This critical Reynolds number must be compared with mx at the
transition point for the pipe. Due to the parabolic laminar velocity
distribution in the pipe umax = 2 u, and because r = d, then, for
the pipe, umaxr/v = ud/v. According to equation (13.1), () = 2300
crit
for the pipe. Thus the comparable critical Reynolds numbers for pipe and
plate show rather good agreement.
CHAPTER XIV. OLDER THEORIES
The first efforts toward theoretical calculation of the turbulent
flows go back to Reynolds. One distinguishes in the theory of turbulence
two main problems:
1. The flow laws of the developed turbulent flow:
The space and time velocity variations affect the time average of
the velocity; they act like an additional internal friction. The problem
is to calculate the local distribution of the time average of the velocity
components, and thus to gain further information concerning, for instance,
the friction drag.
NACA TM No. 1218
2. Origin'of turbulence:
One investigates under what conditions a small disturbance,
superposed on a laminar flow, increases with time. According to whether
or not the disturbance ir reases with time, the laminar flow is called
unstable or stable. The vestigation in question is therefore a stability
investigation, made to cla ify theoretically the laminarturbulent transi
tion. These investigations aim particularly at the theoretical calculation
of the critical Reynolds number. They are, in general, mathematically
rather complicated.
The first problem, since it is the more important one for general
flow problems will be our main concern. The second will be discussed
briefly at the end of the lecture series.
As to the first problem, that of calculation of the developed
turbulent flow, one may remark quite generally that a comprehensive
theoretical treatment, as exists for laminar flow, is not yet possible.
The present theory of the developed turbulent flow must be denoted as
semiempirical. It obtains its foundations to a great extent from
experiment, works largely with the laws of mechanical similarity, and
always contains several or at least one empirical constant. Nevertheless
the theory has contributed much toward correlating the voluminous experi
mental data and also has yielded more than one new concept.
For the numerical treatment one divides the turbulent flow,unsteady
in space and time, into mean values and fluctuation quantities. The mean
value may a prior be formed with respect to either space or time. We
prefer, however, the time average at a fixed point in space, and form
such mean values of the velocity, pressure, shearing stress, etc. In
forming the mean values one must not neglect to take them over a suffi
ciently long time interval T so that the mean value will be independent
of T. Let the velocity vector with its three mutually perpendicular
components be
*w = iy + jv + kw (14.1)
For a turbulent flow the velocity components are therefore functions of
the three space coordinates and the time:
u = u (x, y, z, t)
v = v (x, y, z, t) (14.2)
w = w (x, y, z, t1
The time average for the component u, for instance, is formed as follows:
u(x, y, z) = u it (14.3)
*Throughout the text, underscored letters are used in place of
corresponding German script letters used in the original text.
NACA TM No. 1218
If u, v, w are independent of to and T, the motion is called steady
on the average, or quasisteady. A steady turbulent flow, in the sense
that the velocity at a point fixed in space is perfectly constant, does
not exist. The velocity fluctuations are then defined by the equations
u = u + ut
v = v + vt (14.4)
w = W + W
and in the same way for the pressure:
p = P + p' (14.5)
The time average of the fluctuation quantities equals zero, according to
definition, as the following consideration will show immediately:
to+T to+T to+T
S u'dt = u dt dt = u = (14.6)
T T T
Ito to to
Thus:
u' = v' = w' = 0 (14.7)
The Additional "Apparent" Turbulent Stresses
As a result of the velocity fluctuations additional stresses
( = apparent friction) originate in the turbulent flow. This is readily
illustrated for instance by the case of the simple shearing flow U = U(y)
(fig. 76). Here V = 0; however, a fluctuation velocity v' in the
transverse direction is present. The latter causes a momentum transfer
between the adjoining layers across the main flow. This momentum transfer
acts like an additional shearing stress T. Whereas in laminar flow the
friction is brought about by the molecular momentum exchange, the turbulent
exchange of momentum is a macroscopic motion of, mostly, much stronger effect.
The equations of motion of the turbulent flow, with this turbulent
apparent friction taken into consideration, can be obtained from the
NavierStokes differential equations by substituting equation (14.4) into
the latter and then forming the time averages in the NavierStokes
differential equations. To that purpose the NavierStokes differential
equations (3.16) are written! in the form:
NACA TM No. 1218
a uv)
+
8x
P + Ux
6t 6x
o(uv)
av2
+C+
y C
5Y
oy
CY
ox
Cx
6(uw)
r
6p
Op
oy
02
[o
iCt
P t
+82u
/2,
\2x
6z P 
\x
6v aw
+ 0
oy dz
By introducing equations (14.4) and (14.5) and forming the time averages
one first obtains from the continuity equation:
and thus also:
6u dv 6w 0
ax 6y 6z
0 '.1 a aw '
 ++ = 0
6x By 6z
By introduction of equations (14.4) into the left side of equation (14.8)
one obtains expressions as, for instance,
2 2 2 2
u =( + u')2 = u + 2 u u' + u' etc.
In the subsequent formation of the time average the squared terms in
the barred quantities remain unchanged since they are already constant
with respect to time. The mixed terms, as for instance u u', .
and also the terms that are linear in the fluctuation quantities are
eliminated in forming the average because of equation (lb.7). However,
the terms that are quairatic in the fluctuation quantities as u'2
uv', remain. Thus one obtains from the equation system
(equation (14.8)), after forming the time average the following system
of equations:
6(ww')
6z
+ 6
oz
O?
C2u
S
oy
'2
S2v
4I
2w
+ +
2
Oy
+ _
2
62V
z
.2
on
(14.8)
(1l.0)
(lI.10)
NACA TM No. 1218
P +
1 dx:
 +
6y
az
w
6zu
I =
'N7
'I =
ax
 + GVl p
ay
6T 
+z PA p
+ +
___ +
I x
2twt
6z
6utWI
+y
Iu'w'
ax
(14.11)
The left side now formally agrees with the NavierStokes differential
equations for steady flow if one writes instead of u, v, w the time
averages of these quantities. On the right side additional terms which
arise from the fluctuations have been added to the pressure and friction
terms.
Remembering that in deriving the NavierStokes differential equations
one could write the resultant surface force per unit volume by means of
the components of a stress tensor according to equation (3.7) in the form
R = i +
R+ +
xy
\ z
+k 8xz
6x
one recognizes by comparison with (equation (14.11))
for the quantities added by the fluctuation motion a
tensor in the following manner:
that one may introduce
symmetrical stress
a = put,
XY
= pu'vw
T =pu'xz
xz
T = pu'v'
xy
y = pv''2
T = yz
yz
yz = PV
Crz = pwt2
One has therefore, for the mean values of the quasisteady flow, the
following equations of motion:
+ 6z/
ay
yz
+ yz

(14.12)
(14.13)
NACA TM No. 1218
/ou ou 6U 6p x 6xy
pu + v + w = + u + +
\x dy dz ox dx Oy
ov O'
p u + v
\ ax dy
N N
O + 6
o dx c.y
+ w +
+ = +
OI +6
+ xy
6x
v+ + xz +
ox
a dOT
6y ~z
Y z + z
oy 6z
oz
as y
The continuity equation (equation (14.9)) also enters. The boundary
conditions are the same as for laminar flow: adhering of the fluid to the
wall, that is, on the solid walls all velocity components equal zero.
According to equation (14.14) the mean values of the turbulent flow obey
the same equations of motion as the velocity components of a laminar flow,
with the friction forces, however, increased by the apparent stresses of
the turbulent fluctuation motion. But since the fluctuation velocities
u', v', and particularly their space distribution are unknown,
equations (14.14) and (14.13) are, at first, rather useless for the
calculation of a turbulent flow.
Only when one will have succeeded in expressing the fluctuation
quantities u'2, u'v', in a suitable manner by the time averages
u, v, ., will it be possible to use equations (14.14) to calculate,
in particular, the mean values u, v, w.
A first expression of this kind which brought however little
was originated by Boussinesq (reference 64). He introduced, aside
the ordinary viscosity coefficient, a new viscosity coefficient of
apparent turbulent friction. In analogy to the stress tensor for
laminar flow which is, according to equation (3.13):
ax xy Txz
1xy oy Tyz
''xz Tyz a
pO0
0 p 0 + 1
0 0
success
from
the
(14.15)
Boussinesq puts for the apparent turbulent friction:
x ox
T P E( +
0z ..
TXY ay
Then there corresponds to the laminar viscosity coefficient j
factor pe:
the mixing
xz
az
(14.14)
(14.16)
10 NACA TM No. 1218
C l, p or V e
The kinematic viscosity of the turbulent flow (apparent friction) is
usually very much larger than that for the ordinary laminar friction.
(Hundred or thounsandfold or more). In general, one may therefore
altogether neglect the ordinary viscosity terms iU, in
equation (14.14). Only at the solid walls where due to the noslip
condition
u = v = = as well as u' = v' = w' = 0
the apparent turbulent friction disappears, does the laminar friction
again become dominant. Thus there exists in every turbulent friction
layer in the immediate neighborhood of the wall a very narrow zone where
the flow is laminar. The thickness of this laminar sublayer is only a
small fraction of the turbulent boundary layer thickness.
One can easily understand from the example of the simple shear flow
according to figure 76 that in a turbulent flow the mean value u'v' is
different from zero. For this case, a correlation exists between the
fluctuation velocities u' and v' in the following manner: The
particles with negative v' have "mostly" a positive u', since they
come from a region of larger mean velocity u.* The parts with positive
v', on the other hand, have "mostly" a negative u', because they come
from a region with smaller U, and retain in the transverse motion
approximately the xmomentum of the layer from which they come. Thus,
"mostly" u'v' < 0 and, therefore, the time average u'v' < 0. Therefore,
the shearing stress is this flow is:
T = p u'v' > 0
xy
In measuring turbulent flows one usually measures only the mean
values U, v, since only they are of practical interest. However,
in order to obtain deeper insight into the mechanism of the turbulent
flow, the fluctuation quantities have recently been measured and also
their mean squares and products:
u' v'
According to measurements by Reichardt (reference 65) in a rectangular
tunnel (width 1 millimeter, height 24 centimeters) the maximum value
*"Mostly" is to indicate that particles with different signs,
thoui' h not excluded, are in the minority.
NACA TM No. 1218
of u for instance, equals 0.13 Umax, the maximum value of v'
equals 0.05 iax. Both inaxima lie in the neighborhood of the wall.
One may say, therefore, that in this case the turbulence is strongest
near the wall.
In a flow that is homogeneous (wind tunnel), turbulent fluctuations
are also always present to a varying degree. They determine the socalled
degree of turbulence of a wind tunnel. Since the measurement of the
fluctuation quantities is rather difficult (hotwire method), a more
convenient measuring method has been chosen, for the present, for determin
ation of the degree of turbulence of a wind tunnel: namely, the determin
ation of the critical Reynolds number for the sphere from force measure
ments or pressure distribution measurements. One defines as critical
Reynolds number the one where the drag coefficient c, = 0.3. It
becomes clear that a unique connection exists between the critical Reynolds
number and the turbulent fluctuation velocity in the sense that the
critical Reynolds number of the sphere is the lower, the higher the
turbulent fluctuation velocity. According to American measurements
(reference 97) the connection between the longitudinal fluctuation and
the measured critical Reynolds number for the sphere is as shown in the
following table:
V' /u 0.004 0.0075 0.012 0.017 0.026
Recrt 10 2.8 2.4 2.0 1.6 1.2
crit
In addition to the apparent increase of viscosity, the turbulent
fluctuation motion has other effects: It tends to even out any tempera
ture differences or variations in concentration existing in a flow.
The diffusion of heat, for instance, is much larger than for laminar flow,
because of the exchange motions which are much stronger in turbulent
flow. A close connection therefore exists, for instance, between the
laws of flow and of heat transfer from a heated body to the fluid flowing
by.
Ninth Lecture (February 2, 1942)
CHAPTER XV. MORE RECENT THEORIES; MIXING LENGTH
In order to make possible a quantitative calculation of turbulent
flows, it is necessary to transform the expressions for the apparent
turbulent stresses (equation (14.13)) in such a manner that they no longer
contain the unknown fluctuation velocities but contain the components of
the mean velocities. Consider, for that purpose, a particularly simple
NACA TM No. 1218
flow, namely a plane flow which has the same direction everywhere and a
velocity varying only on the different stream lines. The mainflow
direction coincides with the xdirection; then:
u = u(y) = 0 = (15.1)
Of the shearing stresses, only the component T = T is present, for
x,y
which from equation (14.13) as well as from Boussinesq's equation,
equation (14.15),there results:
T = p uvt = p du (15.2)
dy
This formula shows that IT /p equals the square of a.velocity. One
puts,therefore, for use in later calculations,
v_ = il = (15.3)
U P i
and denotes v. as shearing stress velocity. Thus this shearing stress
velocity is a measure of the momentum transfer by the turbulent
fluctuation motion.
According to Prandtl (reference 66), one may picture the turbulent
flow mechanism, particularly turbulent mixing, in the following simplified
manner: Fluid particles, each possessing a particular motion, originate
in the turbulent flow; they move for a certain distance as coherent masses
maintaining their velocity (momentum). One now assumes that such a fluid
particle which originates in the layer (yl 2) and has the velocity
u(Yl 1) moves a distance 2 = mixing length normal to the flow (fig. 77).
If this fluid particle maintains its original velocity in the xdirection
it will have, in its new location yl, a smaller velocity than its new
surroundings, the velocity difference being
u1 = u(yl) u(yl i) with v' > 0
Likewise a fluid particle coming from the layer (yl + 2) to yl has
at the new location a greater velocity than the surroundings there; the
difference is
NACA TM No. 1218
u = u(y + 7) (yl) with v' < 0
u' and u' give the turbulent velocity fluctuation in the layer yl.
One obtains for the mean value of this velocity fluctuation
u'l = (1u' I+ lu ) = 1 (15.4)
From this equation one obtains the following physical interpretation for
the mixing length 1:
The mixing length signifies the distance in the transverse direction
which a fluid particle must travel at the mean velocity of its original
layer so that the difference between its velbcity and the velocity of the
new location equals the mean velocity fluctuation of the turbulent flow.
It is left open whether the fluid particles in their transverse motion
fully maintain the velocity of their original layer, or whether they have
partly assumed the velocity of the traverse layer and then travelled
larger distances in the transverse direction. The Prandtl mixing length
which is thereby introduced has a certain analogy to the mean free path
of the kinetic theory of gases, with, however, the difference that there
one deals with microscopic motions of the molecules, here with macroscopic
motions of larger fluid particles.
One may picture the origin of the transverse fluctuation velocity v'
in the following way:
Two fluid particles flowing from the layers (yl + 2) and (yl 2)
meet in the layer yl in such a manner that one lies behind the other:
the faster (yl + 2) behind the slower (yl 2). They then collide
with the velocity 2u' and giveway laterally. Thereby originates the
transverse velocity v', directed away from the layer yl to both sides.
If, conversely, the slower of the two particles is behind the faster,
they withdraw from each other with the velocity 2u'. In this case the
space formed between them is filled up out of the surroundings. Thus
originates a transverse velocity v' directed toward the layer yl.
One concludes from this consideration that v' and u' are of the same
order of magnitude and puts
Iv'I = number lu'l = number Z d (15.5)
dy
In order to express the shearing stress according to equation (15.2) one
has to consider the mean value uv' more closely. The following
conclusions can be drawn from the previous considerations.
NACA TM No. 1218
The particles arriving in the layer yl with positive v' (from
below, fig. 77) have "mostly" a negative u' so that u'v' is negative.
For the particles arriving with negative v', u' is "mostly" positive,
so that u'v' is again negative. "Mostly" signifies that particles with
different sign are not wholly excluded, but are strongly outnumbered.
The mean value u'v' is therefore different from zero and negative. Thus
one puts
u'v' = k lu'l Iv' (15.6)
with k j 0; 0
correlation coefficient is not known more closely. According to
equation (15.5) and (15.4) one now obtains
u'v' = number 22(d (15.7)
the "number" in this equation being different from the one in
equation (15.5). If one includes the "number" in the unknown mixing
length, one can also write
uv' =  (15.8)
and thus finally obtains for the turbulent shearing stress according to
equation (15.2)
ddii
Considering that the sign of T also must change with the sign of 
it is more correct to write
T=p 2 Id du Prandtl's
dy dy formula
(15.9)
This is the famous Prandtl mixing length formula which has been very
successful for the calculation of turbulent flows.
If one compares this formula (equation (15.9)) with the equations
dii
of Boussinesq where one had put T = e (E = mixing factor = turbulent
analogue of the laminar viscosity g), one has for the mixing factor
NACA TM No. 1218
E = Pi2  (15.10)
The turbulent mixing factor e is in most cases larger than the laminar
viscosity p by several powers of ten. Moreover the mixing factor e
is dependent on the velocity and on the location and tends toward zero
near a wall, because there the mixing length goes toward zero.
If one compares Prandtl's formula equation (15.9) with Boussinesq's
equation (15.2) one could perhaps think at first that not much has been
gained, since the unknown quantity E ( = apparent viscosity) has been
replaced by the new unknown I = mixing length. Nevertheless Prandtl's
formula is considerably better than the old formula for the following
reason: It is known from tests that the drag for turbulent flow is
proportional to the square of the velocity. According to equation (15.9)
one obtains this square law for drag by assuming the mixing length to be
independent of the velocity, that is, by assuming the mixing length to
be purely a function of position. It is considerably easier to make a
plausible assumption for the length I = mixing length than for the
apparent turbulent viscosity e, and therein lies the considerable
superiority of Prandtl's formula equation (15.9) over Boussinesq's
equation (15.2).
In many cases the length 2 can be brought into a simple relation
to the characteristic lengths of the respective flows. For the flow along
a smooth wall 2 must, at the wall itself, equal zero, since all trans
verse motions are prevented at the wall. For the flow along a rough wall,
however, the limiting value of Z at the wall equals a length of the
order of magnitude of the height of the roughness.
It would be very useful to have a formula permitting the determina
tion of the dependence of the mixing length on the position for any
arbitrary flow. Such an attempt has been made by v. Karman (reference 68).
v. Karman makes the assumption that the inner mechanism of the turbulent
flow is such that the motion at various points differs only with respect
to time and lengthscale, but is otherwise similar (similarity hypothesis).
Instead of the units of time and length one may select those of velocity
and length. The velocity unit that is important for the turbulent motion
is the shearing stress velocity v, according to equation (15.3). The
corresponding unit of length is the mixing length 1.
In order to find the quantity 2 from the data of the basic flow
"(y), v. Karman applies the Taylor development* for u(y) in the neigh
borhood of the point yl.
u(y) = u(y) + (y 1) ( + (. y)2 ) +. (15.11)
\dyl 2
In the following, the bar over the mean velocity will be omitted,
for simplification.
NACA TM No. 1218
The length I cannot depend on the velocity u(yl), since according
to Newton's principle of relativity the addition of a constant velocity
has no influence on the course of motion. Thus
Mldu1
S2\
du
Id2 1
and the higher derivatives remain as characteristic data of the basic
flow. The simplest length to be formed from it is
v. Karman puts therefore
1 dy d 12
2/ p^
(15.12)
According to this formula 2 is not dependent on the amount of velocity
but only on the velocity distribution. Thus 2 is a pure position
function as required above. In equation (15.12) lK is an empirical
constant which must be determined from the experiment. To arrive further
at the turbulent shearing stress, v. Karman also maintains Prandtl's
equation (15.9).
In generalizing equation (15.9) one obtains, according to Prandtl,
the complete expression for the turbulent stress tensor of a plane flow
in the form
a (
x = p2
T ay
(XYy
6u &v
2
(15.13)
au
2 6
6u + Cv
+ 5a
The common factor on the right side signifies the turbulent mixing
factor according to equation (15.10).
NACA TM No. 1218 17
Tenth Lecture (February 9, 1942)
Flow Along a Smooth Wall
We will immediately make a first application of Prandtl's formula
(equation (15.9)) for the flow along a smooth wall. The normal distance
from the wall is denoted as y. Let the wall coincide with the xaxis.
For the velocity distribution then, u = u(y). For this case one sets
the mixing length in the neighborhood of the wall proportional to the
distance from the wall
2 = cy (15.14)
the constant K must be determined from the experiment. Moreover one
makes the assumption that the shearing stress T is constant in the
entire flow region; then the shearing stress velocity v. according to
equation (15.3) also is constant. If one further neglects the laminar
friction, one obtains from equations (15.2), (15.9), (15.14)
v2 = ,2y2(/du)2
\dy/
or
du = v
dy Ky
and by integration
v*
u = In y + constant (15.15)
In determining the constant of integration one must pay attention to the
fact that the turbulent law equation (15.9) does not apply right up to
the wall but that very near to the wall an extremely thin laminar layer
is present. From the laminar viscosity i and the turbulent shearing
stress velocity v. one can form the length V/v.. The constant of
integration in equation (15.15) is determined from the condition that
u = 0 for y = yo. Thus there results, according to equation (15.15)
u = (n y in yo) (15.16)
The as yet unknown distance from the wall yo is set proportional to
the length V/v*, thus
NACA TM No. 1218
o = (15.17)
where 3 signifies a dimensionless constant. Thus one finally obtains
for the velocity distribution at the smooth wall
u = (Zn .v 2n P) (15.18)
that is, a logarithmic velocity distribution law. It contains two
empirical constants K and p. According to measurements K = 0.4.
From equation (15.18) one can see that the dimensionless velocity u/v* = (p
can be represented as a function of the dimensionless distance from the
wall T = v*y/v. The latter is a sort of Reynolds number, formed with the
distance from the wall y and the shearing stress velocity v*. Thus one
obtains for larger Reynolds numbers from equation (15.18) the following
universal velocity distribution law
q(p() = A in q + B (15.19)
with A = 1/K = 2.5. For smaller Reynolds numbers, where the laminar
friction also has a certain influence, tests gave the velocity distribution
law
) = C n (15.20)
or
u y C (15.20a)
vw4 V
with the exponent n equalling about 1/7. These universal velocity dis
tribution laws according to measurements for pipe flow are given in
figure 78. They will be discussed in more detail in the following chapter.
CHAPTER XVI. PIPE FLOW
a. The Smooth Pipe
Among the various tlubulent flows of practical importance, pipe
flow was investigated with particular thoroughness because of its great
practical importance. We shall therefore consider the pipe flow first.
It will be noted at this point that the flow laws of the pipe flow may
be applied to other cases, as for instance the plane plate in longitudinal
flow. Consider a straight pipe of circular cross section and with a smooth
wall. Let y be the radial coordinate measured from the pipe axis. The
NACA TM No. 1218
balance of forces between the shearing stress T and the pressure
drop pl p2 on a piece of pipe of length L yields as before for the
laminar flow according to equation (2.1a), the relation:
Pl P2 y
T (16.1)
L 2
This formula applies equally to laminar and turbulent flow. In it T
now signifies the sum of the laminar shearing stresses and of the apparent
turbulent shearing stresses. Over a cross section, T is proportional
to y. The shearing stress at the wall To may be determined
experimentally by measurement of the pressure drop:
pI P2 r
TO 2 (16.2)
L 2
For the turbulent flow the connection between pressure drop and flow
volume Q = irr25 must be obtained from tests.* In the literature there
exists a very great number of pipe resistance formulas. Only those serve our
purpose which satisfy Reynolds' law of similarity. One of them is the
formula of H. Blaslus (reference 69), set up particularly carefully,
which is valid for a smooth wall and for Reynolds numbers
Re = ud/ 100 000.
If one introduces, as before in equation (2.6), the dimensionless
pipe resistance coefficient X by the equation
P1 P2 X p 2
u (16.3)
L d2
X is, according to Blasius:
X = 0.3164 () (16.4)
Comparing equations (16.2) and (16.3) one finds:
T = P 2 (16.5)
and therefore according to equation (16.4):
To = 0.03955 P u v d1 (16.6)
*In the following, U is, for the pipe flow, the mean flow velocity
at the cross section, as distinguished from the time average in the previous
sections.
NACA TM No. 1218
If one introduces, moreover, instead of the diameter d the radius r,
the numerical factor in this linear equation must be divided by
21/4= 1.19. Thus To becomes:
To = 0.03325 p 7A4 v1/4 r/4 = p v2 (16.7)
where the shearing stress velocity is defined by the wall shearing stress:
v = (16.8)
If one finally factors the quantity v 2 in equation (16.7) into v 7/ x
vl/4, one obtains:
= 0.o325 r or = 6.99 (vrl/7 (169)
This equation is very similar to equation (15.20a); however, the mean
velocity now takes the place of the local velocity and the pipe radius
takes the place of the distance from the wall. One passes first from
the mean velocity to the maximum velocity ul; based on measurements of
Nikuradse (reference 70) U = 0.8 ul, and therewith follows from
equation (16.9):
Ul = 8.74 ( 1/7
If this formula is assumed to be valid for any distance from the wall,
one obtains:
= 8.74 1 (16.10)
or
S= 8.74 1/7 (16.11)
This is the socalled 1/7power law for the velocity distribution; its
form was already given in equation (15.20a). The coefficients n and T
NACA TM No. 1218
still unknown there, have now been determined on the basis of the
resistance law of the pipe flow. Figure 78 shows, according to measure
ments of Nikuradse (reference 70) that this law is well satisfied in the
range of Reynolds numbers up to 100,000. Naturally this law of velocity
distribution can apply only to the region of Reynolds numbers where the
pipe resistance law given by equation (16.4) is valid, since it was
derived from this law.
For purposes of later calculations we shall derive from
equation (16.10) the shearing stress velocity v,. One obtains:
v, = 0.150 u ()/ (16.12)
with 8.747 = 6.65 and  = 0.150. From equation (16.12) follows:
6.65
To= p 2 = 0.0225Pu7/4 ()4 (16.13)
This relation will be needed later.
Comparing measured velocity distributions with equation (16.10)
one can state that outside of the range of validity of equation (16.10),
namely for Re >100,000, a better approximation is obtained by the
power 1/8, 1/9, or 1/10 instead of 1/7. The measurements concerning
the pipe resistance (fig. 81) show an upward deviation from the formula
of Blaslus for larger Reynolds numbers.
The logarithmic velocity distribution law, equation (15.19), derived
in the previous chapter has been verified by Nikuradse (reference 70) on
the basis of his measurements for the smooth pipe. For this purpose from
the measured pressure drop for each velocity profile one first determines
the wall shearing stress according to equation (16.2) and from that
according to equation (16.8), the shearing stress velocity v, = F p.
Then the dimensionless velocity q = u/v, can be plotted against the
dimensionless distance from the wall = yv*/v. The measurements of
Nikuradse in a very large range of Reynolds numbers, Re = 4 x 103 up to
3240 x 103, lie very accurately on a straight line if one plots T against
log i (fig. 78). The straight line has the equation:
p = 2.5 In T + 5.5 (16.14)
This gives, by comparison with equation (15.18), the following numerical
values for the coefficients i and P
NACA TM No. 1218
K = 0.400 0 = 0.111 (16.15)
Mixing Length
From the measured velocity distribution and the measured pressure
drop the distribution of the mixing .length over the pipe cross section
can be determined according to equations (16.2), (16.1), and (15.9).
S= To Y (y = distance from the pipe axis). This determination of the
mixing length from the measurements in the pipe was made by Nikuradse
(reference 70). For large Reynolds numbers, where the influence of
viscosity is negligible, one obtains a distribution of the mixing length
I/r over y/r which is independent of the Renumber (fig. 79). The
following interpolation formula can be given for this distribution:
22 4
F = 0.14 0.08 1 O.o6(l (16.16)
rr r
In this equation y signifies the distance from the wall. The develop
ment of equation (16.16) for small y/r (neighborhood of the wall) gives
2
I = .4y .44 + (16.16a)
r
In the neighborhood of the wall the mixing length is, therefore, propor
tional to the distance from the wall. Equation (16.16) for the distri
bution of the mixing length applies not only to the smooth pipe, but,
according to the measurements of Nikuradse (reference 71) also to the
rough pipe, as can be seen from figure 79. From this fact one can derive
in a very simple manner a universal form for the law of velocity distri
bution, valid for the smooth as well as for the rough pipe. One puts
for the mixing length distribution: 1 = sy fY with ffJ 1 for y 0.
Furthermore follows from the linear distribution of the shearing stress
over the cross section:
T = To (1 !) (y = distance from the wall)
together with equation (15.9)
dn l.1 y* (16.17)
i aP lt yf(y/r)
and hence by integration:
NACA TM No. 1218
Sy/ \ d r
u (16.18;
o f (y
yo /r r r
the lower limit of integration yo where the velocity equals zero is,
according to the considerations of the previous section, proportional
to V/v*; thus: yo/r = F I v). From equation (16.18) follows:
Yr d y/ r (16.18a)
Sumaxf(.
Jo/r r a
and therefore, from equations (16.18) and (16.18a):
umax u = v. F (y/r)
(16.19)
This law, with the same function F(y/r), applies equally to smooth and
rough pipes. It states that the curves of the velocity distribution over
the pipe cross section for all Reynolds numbers and all roughnesses can
be made congruent by shifting along the velocity axis, if one plots
(umax u/v, against y/r (fig. 80). The explicit expression for the
function F(y/r) is obtained immediately from equation (16.14), according
to which
u u = 2.5v, in r = 5.75 v, log r (16.20)
max y y
Universal Resistance Law
According to their derivation the velocitydistributionlaw
(equations (16.19) and (16.20)' are to be regarded as valid for arbitrary
Reynolds number since the laminar viscosity was neglected as compared
with the turbulent viscosity. We shall now derive from the velocity
distributionlaw equation (16.20) a resistance law which, in contrast to
Blasius', applies up to Reynolds numbers of arbitrary magnitude.
NACA TM No. 1218
From equation (16.20) one may determine by integration over the
cross section the mean flow velocity u. One finds:
u = umax 3.75 vw
(16.21)
The test results of Nikuradse .(reference 70) gave
different from 3.75, namely:
a number slightly
u = ua 4.07 v7
(16.22)
According to equation (16.5):
1= 8 () 2
(16.23)
From the universal velocity distribution law of the
equation (16.14) follows:
smooth pipe
ea= v* 2.5 In r + 5.5
and hence the connection with equation (16.21):
r rv* I
S= v 2.i5 n v + 1.75 (16.24)
The Reynolds number enters into the calculation by means of the identity:
rv* 1 ld v* id
V 2V u V 4F
Thus results from equations (16.23) and (16.24):
C2.5 In ( \V
 2.5 In 4 \fl2 + 1.75
{2.035 log ( \f*)0.91
NACA TM No. 1218
or:
S2.035 log 0.91 (16.25)
Accordingly a straight line must result for the resistance law of the smooth
pipe, if one plots 1/ against log 5d ). This is very well confirmed
by Nikuradse's measurement (fig. 81). The numerical values according to the
measurements differ only slightly from those of this theoretical derivation.
From Nikluradse's measurements was found:
= 2.0 log u Vf) 0.8 (16.26)
Universal Resistance Law For Smooth Pipes
This is the final resistance law for smooth pipes. On the basis of
its derivation it may be extrapolated up to Reynolds numbers of arbitrary
magnitude. Thus measurements for larger Reynolds numbers than those of
Nikuradse's tests are not required. Up to Re = 100,000 this universal
resistance law is in good agreement with the Blasius law according to
equation (16.4). For higher Reynolds numbers the Blasius law deviates
considerably from the measurements (fig. 81).
Concerning the determination of X from equation (16.26), where it
appears on both sides, it should be added that it can be easily obtained
by successive approximation.
Eleventh Lecture (February 16, 1942)
b. The Rough Pipe
The characteristic parameter for the flow along a rough wall is the
ratio of grain size k of the roughness to the boundary layer thickness,
particularly to the thickness of the laminar sublayer 5Z which is
always present within the turbulent friction layer in the immediate
neighborhood of the wall. The thickness of the laminar sublayer is
5 = number . The effectiveness of roughness of a certain grain size
kv*
depends, therefore, on the dimensionless roughness coefficient k/52.
In the experimental investigations of the resistance of turbulent flows
over rough walls, the rough pipe has been studied very thoroughly since it
is of great practical importance. Besides depending on the Reynolds number,
the resistance of a rough pipe is a function of the relative roughness r/k.
One distinguishes for the resistance law of a rough pipe three regions.
NACA TM No. 1218
The subsequently given boundaries of these regions are valid for sand
roughness ks like those investigated by Nikuradse (reference 71).
1. Hydraulically smooth: The grain size of the roughness is so
small that all roughnesses lie within the laminar sublayer. In this
case the roughness has no drag increasing effect. This case exists for
small Reynolds numbers and for values of the characteristic roughness
< vjk <
number: 0 *ks < 5
V
2. Fully developed roughness flow: The grain size of the roughness
is so large that all roughnesses project from the laminer sublayer. The
friction drag then consists predominantly of the form drag of the single
roughness elements. A purely square drag law applies. For the pipe the
resistance coefficient X is then independent of Re and only dependent
on the relative roughness k/r. This law exists for vyry large Reynolds
numbers. For sand roughness this law applies for: > 70.
3. Intermediate region: Only a fraction of the roughness'elements
project from the laminar sublayer. The drag coefficient depends on r/k
as well as on Re. This law exists for medium Reynolds numbers and, for
< Vks <
the sand roughness, for: 5 70.
V
The dependence of the pipe resistance coefficient on the Reynolds
number and on the relative roughness according to the measurements of
Nikuradse (reference 71) can be seen from figure 82 as well as, in
particular, the three laws just given.
The velocity distribution on a rough wall is given, basically, in
the _same way as for the smooth wall by equation (15.16). One has only
to substitute for the constant of integration yo another value: yo
proportional to the roughness grain size. One puts for sand roughness
yo = 7 ks and hence obtains from equation (15.16)
u = {n k In 7 (16.27)
The constant 7 is, moreover, a function of the roughness form and the
roughness distribution. Comparison with experiments of Nikuradse
(reference 71) on pipes roughened artificially by sand yields for the
velocity distribution the general formula:
u = v* 2.5 In + B (16.28)
kes
NACA TM No. 1218
the constant B being different in each domain described above; it
depends on vks /V.
For the fully developed roughness flow B = 8.5, thus:
S= v(2.5 In + 8.5 (fully rough)
u = ,
(16.29)
whence follows:
max =v(2.5 In k + 8.5
max u = v* 2.5 In
Y
(16.30)
(16.30a)
in agreement with equation (16.20). Thus there applies also to rough
pipes, as equation (16.21) did before to the smooth pipe, the relation:
L = Umax 3.75 v.
(16.31)
From here one can, by a calculation which is perfectly analogous to the
previous one for the smooth pipe, easily arrive at the resistance law of
the rough pipe for fully developed roughness flow. By insertion of Uax
according to equation (16.30) into equation (16.31) one obtains:
S= v, 2.5 In + 4.75) (16.32)
U=V (? Y S 5)~
(v 8
= 8c(
u 1
8
( )
2.5 In + 4.75)
ks
2.0 log k +
and:
(16.33)
1.68)2
(16.34)
NACA TM No. 1218
This is the square resistance law of the fully developed roughness flow.
Comparison with the test results of Nikuradse (fig. 83) shows that one
obtains better agreement if one changes the number 1.68 to 1.74. Thus
the resistance law of the pipe flow for fully developed roughness is:
= 1 (16.35)
(2.0 log  + 1.74)
ksa
In the plots of against log r/kg, (fig. 83) the test results fall
In the plots of T
very accurately on a straight line.
For flow along a rough wall in the intermediate region the constant
B in equation (16.28) is, moreover, a function of the roughness coeffi
cient v*ks/v. For this case also one can derive the resistance law
immediately from the velocity distribution. According to equation (16.28):
U
B =u 2.5 n 2.5 no r (16.36)
v k v k
*
On the other hand, according to equations (16.31) and (16.23):
U
max + 3.75 = 2 + 3.75 (16.37)
v* t* Vx
so that one obtains from equation (16.36):
B = u 2.5 n = 2.5 In + 3.75 (16.38)
V v ks Vx kE
One can, therefore, determine the constant B as a function of v*k v
either from the velocity distribution or from the resistance law. The
plot in figure 84 shows good agreement between the values determined by
these two methods. At the same time the determination of the resistance
law from the velocity distribution is confirmed.
The formula for B includes the case of the smooth pipe. B is,
according to equation (16.14),
NACA TM No. 1218
B = 2.5 In = 2.5 n k + 5.5 (16.39)
Thus a straight line results for B in the plot against log vrk /v.
Other Roughnesses
Because of the great practical importance of the roughnessproblem
a few data concerning roughnesses other than the special sand roughness
will be given. Nikuradse's sand roughness may also be characterized by
the fact that the roughness density was at its maximum value, because
the wall was covered with sand as densely as possible. For many practical
roughnesses the roughness density is considerably smaller. In such cases
the drag then depends, for one thing, on form and height of the roughness,
and, moreover, on the roughness density. It is useful to classify any
arbitrary roughness in the scale of a standard roughness. Nikuradse's
sand roughness suggests itself as roughness reference (roughness scale)
because it was investigated for a very large range of Reynolds numbers
and relative roughnesses. Classification with respect to the roughness
scale is simplest for the region of fully developed roughness. According
to what was said previously, for this region the velocity distribution
is given by:
S= 5.75 log + Bs Bs = 8.5 (16.40)
and the resistance coefficient by:
S= 1 (16.41)
2.0 log + 1.74
One now relates to an arbitrary roughness k an equivalent sand roughness
kg by the ratio
ks = a k (16.42)
Where by equivalent sand roughness kg is meant that grain size of sand
roughness which has, according to equation (16.41) the same resistance
as the given roughness k.
Basically, of course, the equivalent sand roughness ks can be
determined by a resistance measurement on the pipe. However, such measurements
NACA TM No. 1218
for arbitrary roughnesses are difficult to perform. Measurements on
arbitrary roughnesses in a tunnel with plane walls are more convenient.
To this purpose an exchangeable wall of a tunnel with rectangular cross
section is provided with the roughness to be investigated (fig. 85).
From the measurement of the velocity distribution in such a tunnel with
a rough and a smooth longitudinal wall one obtains, for the logarithmic
plot against the distance from the wall, a triangular velocity distribu
tion (compare fig. 85). From the logarithmic plot of the velocity distri
bution over the rough wall
u = nr log y + mr (16.43)
one obtains by comparison with the universal law according to
equation (16.28) for the shearing stress velocity at the wall:
Vr nr (16.44)
5.75
Further, one determines for the roughness to be investigated the constant
B of the velocity distribution law, namely:
U. y
B =  5.75 log (16.45)
Vr k
By comparison of equation (16.45) with (16.40) one obtains for the
equivalent sand roughness:
k
5.75 Zog = 8.5 B (16.46)
k
In this way one may determine the drag for arbitrary roughnesses from a
simple measurement in the roughness tunnel. This .method may be also
carried over to the case of the intermediate region.
CHAPTER XVII. THE FRICTION DRAG OF THE FLAT PLATE
IN LONGITUDINAL FLOW
The turbulent friction drag of the plate in longitudinal flow is
of very great practical importance, for instance as friction irag of wings,
airplane fuselages, or ships. The exact measurement of the friction irag
for the large Reynolds numbers of practice is extremely difficult. Thus
NACA TM No. 1218
it is particularly important that one can, according to Prandtl
(references 73 and 7i), calculate the friction drag of surfaces from the
results of pipe flow studies. This conversion from the pipe to the plate
can be made for the smooth as well as for the rough plate.
a. The Smooth Plate
One assumes, for simplification, that the boundary layer on the
plate is turbulent from the leading edge. Let the coordinate system be
selected according to figure 86. The boundary layer thickness 5(x)
increases with the length of run x. Let b be the width of the plate.
For the transition from pipe to plate the free stream velocity Uo of
the plate corresponds to the maximum velocity umax in the pipe, and
the boundary layer thickness 5 to the pipe radius r.
One now makes the fundamental assumption that the same velocity
distribution exists in the boundary layer on the plate as in the pipe.
This is certainly not exactly correct since the velocity distribution in
the pipe is influenced by a pressure drop, whereas on the plate the
pressure gradient equals zero. However, slight differences in velocity
distribution are insignificant since it is the momentum integral which
is of fundamental importance for the drag. For the drag W(x) of one
side of the plate of length x, according to equations (10.1) and (10.2):
x 8 (x)
W(x) = b T0(x) dx = bp
Jo L
u (Uo u) dy
1 dW T x)
b dx o
The equation (17.1) can also be written in the form
W(x) = boUo2 (x)
"o
(17.2)
For the velocity distribution in the boundary layer one now assumes
the 1/7power law found for the pipe. Replacing umax by Uo and
by 51 one may write this law, according to equation (16.11):
whence
(17.1)
(17.1a)
u d
UO )
NACA TM No. 1218
Uc (h 1)/7
Hence the momentum internal becomes
Hence the momentum Integral becomes
jll
(1
o
= T'
1/7 1 7 ,/7
W(x) = bpU25(x)
72
Hence follows, according to equation (17.la)
7 2 d1
70= pU0 d
On the other hand, one had found before for the
again replacing r by 5 and u by U :
max o
(17.5)
(17.6)
smooth pipe, equation (16.13)
T0 = 0.0225 pUo (
(17.7)
By equating equations (17.6) and (17.7) results:
 pUo2 d = 0.0225 pTJU 7/4
72 o dx o 6
This is a differential equation for 5(x). The integration yields:
4 5/4
5
72 42 /4
 T 0.0225 (
0o
(17.8)
5(x) = 0.37 (l/5 x4/5
Uo
(17.3)
and thus
dr' =
72
(17.4)
u )d =
Uo 3
IACA TM No. 1218
Ux
Rex = 
V
(17.9)
For the turbulent boundary layer the boundary layer thickness, therefore,
increases with x4/5. The corresponding equation for the laminar flow
was, according to equation (9.21a), 8 = 5\vx/Uo.
By substitution of equation (17.8) into equation (17.5) one obtains
S20.036 bp2x (R 1/5
W(xJ = 0.036 bpUo x (Rex)
or for the drag coefficient c = W/ Uo2xb:
f 2 0VOXb
cf = 0.072 (Rex
Comparing this result with test results on plates one finds the numerical
value 0.072 to be somewhat too low.
1/5
cf = 0.074 (Rex)
valid for
5 105 < Re <10
corresponds better to the measurements. This law holds true only
7
for Rex < 10 corresponding to the fact that the Blasius pipe resistance
law and the 1/7power law of the velocity distribution, which form the
basis of this plate drag law, are not valid for large Reynolds numbers.
This law is represented in figure 87 together with the laminarflow law
according to equation (9.19). The initial laminar flow on the front part
of the plate can be taken into consideration by a subtraction, according
to Prandtl (reference 73):
1/5_ 1700
c Re
x
5 x 105 < Rex
< 107
The plate drag law for very large Reynolds numbers can be obtained
in essentially the same way by starting from the universal logarithmic
law for the velocity distribution equation (16.l1) which, according to
(17.10)
(17.11)
34 NACA TM No. 1218
its derivation, is valid up to Reynolds numbers of arbitrary magnitude.
Here the calculation becomes considerably more complicated. The
development of the calculation is clarified if one first introduces the
velocity distribution in a general form. We had introduced for the pipe
flow the dimensionless variables q) = u and n = n. The values at
v* V
the edge of the boundary layer are to be denoted by the index 0, thus
U
Po = VO
vV*
5T
o = V
u = v P = Uo
W=o
0= 1
(17.12)
(17.13)
(17.14)
vq0
dy = 0 dn
U
From the equation
1 dW
S= follows, with W according to equation (17.1)
o b dx
p v2 = p u(1
2 d d )
drl H
dx dio
0
Jo u) dy
u(Uo u) dy
and according to equation (17.13):
2
U0
02
t(Po
Po
VUo d
Ud
u(Uo u)dT
Then:
NACA TM No. 1218
plo
U dI
Uo o d
= Vo 10
= x dToo
a`
U
C,2
Po
d7)
=V 
dx
(1 ^>} dT]
0, (1 U0
 1 L dT
T 0 ( CPO
f
In forming the integral L dq one must
facts: The differentiation with respect to the upper
since for n = 1o, c = qcp. In the differentiation of
is to be regarded as constant, and co as a function
note the following
limit gives zero,
the integrand q
of qO. Therefore
d 'o ,, d( To dl q2
Thus follows from equation (17.15):
110
d11 dp
Uo = V ' p 2dT
o dx dTI
L 0
(17.16)
One puts, for simplification,
(po
F(no) = d
dTI a
2
q d1q
(17.17)
and obtains from equation (17.16):
(17.15)
2
cP
9o/
36 NACA TM No. 1218
dx = F(,o) d
If one assumes this law to be valid from the leading edge of the plate
(x = 0), that is, that the flow is turbulent starting from the front,
the int :,ration gives:
with
Uo
0o=o
(17.18)
(17.19)
Equation (17.18) can also be written
the length of run x appears:
so that the Renumber formed with
Ux
Re = = M (T)
I V
Equation (17.20)
layer thickness
gives the relation between the dimensionless boundary
T1 = v* /V and the Renumber U x/v.
The drag remains to be calculated. From
o2
0 "2
Tof = v. = p .V To
I
follows, because W = b To dx
o
with:
(17.20)
S=l
oJT=0
(17.19a)
p2dy d o
NACA TM No. 1218
11
W = b p Uo V
ToIO
F 2 dno = b p Uo (1)
1
*(l)
=O
(17.22)
F(Tlo)
 dn
(P_ 2
The drag coefficient
c = WA Uo2b x becomes finally
2V
f UoX (m1)
4(111)
=2
(Til)
(17.23)
Hence cf also turns out to be a function of I1. Equations (17.20)
and (17.23) together give a parametric representation of cf as a function
v,6
of Rex, where the parameter is the boundary layer thickness T1 = 
Numerical Results
In order to arrive at numerical results, one must introduce a
special function for cp(i). For the 1/7power law according to equation
(16.11) that is, with q = C1/7, one would obtain the drag law according
to equation (17.10). One uses the universal logarithmic velocity distri
bution law, equation (16.14).
S= 2.5 Zn n + 5.5
In order to make the carrying out of the integration more convenient,
one writes
p = 2.5 in (1 + 9n)
Then (p becomes, for T = 0, ( = 0. The adding of the one changes
q(p) a little, only for very small I and has only little influence
on the integrals. If one writes the law at first in the general form
where
(17.21)
SNACA TM No. 1218
(p = a In (1 + bij)
(17.24)
the calculation of the integrals equations (17.17), (17.19),
with z = 1 + bj yields
(17.22),
F(l) = a3 (n2z 2 In z + 2 
*(9) = 4 (
b(i
T1 b Z
z In2z 4z In z 2 In z + 6z 
6) > (17.25)
+ 2(z 1)
2n z
With the numerical values
a =2.493
b = 8.93
one obtains for the drag law the following table:
This table can be replaced by the following interpolation formula:
0.472
S(og Re)'58
LR e 10 c 10
103 f
0.500 0.337 5.65
1.00 0.820 4.75
2.00 1.96 4.05
3.00 3.25 3.71
5.00 6.10 3.34
12.0 17.7 2.81
20.0 32.5 2.57
50.0 96.5 2.20
100.0 217.5 1.96
500.0 1401.0 1.55
NACA TM No. 1218
Comparison with test results shows that the agreement improves if the
number 0.472 is slightly varied, by putting
S= 0.455 106 < Re < 109 (17.26)
f 2.58 x
(log Re,)
J
Prandtl Schlichting's Plate Drag Law
The laminar approach length may again be taken into consideration
by the same subtraction as before; thus:
f = 0.455 1700 0 (17.27)
(log Re)2.58 Re < Re <109
Whereas the system of formulas equation (17.25) is valid up to Renumbers
of arbitrary magnitude, the interpolation formulas, equations (17.26)
and (17.27), have the upper limit Re = 109. However, this limit takes
care of all Renumbers occurring in practice. The theoretical formula
(equation (17.27)) is also plotted in figure 87. Figure 88 gives a
comparison with test results on plates, wings, and airship bodies. The
agreement is quite good.
Very recently this plate drag law has been somewhat improved by
SchultzGrunow (reference 89). Until then, the turbulent velocity profile
measured in the pipe (1/7power law, logarithmic law) had been carried
over directly to the plate, mainly because accurate velocity distribution
measurements of the plate boundary layer did not exist. The exact
measurement of the plate boundary layer showed, however, that the plate
profile does not completely coincide with the pipe profile. The test
points show, for large distance from the wall, a slight upward deviation
from the logarithmic law found for the pipe. Thus the loss of momentum
on the plate is somewhat smaller than that calculated with the logarithmic
law. SchultzGrunow repeated the calculation of the drag law according
to the formula system given above with the velocity distribution law
for the plate measured by him. His result is represented by the
interpolation formula
c = 0.427 106< Rex < 109 (17.28)
( 0.407 + log Rex)
NACA TM No. 1218
This law is also plotted in figure 87. The differences from the Prandtl
Schlichting law are only slight.*
The corresponding rotationallysymmetrical problem, that is, the
turbulent boundary layer on a body of revolution at zero incidence, was
treated by C. B. Millikan (reference 79). The 1/7power law of the
velocity distribution was taken as basis. Application to the general
case has not yet been made.
Twelfth Lecture (February 23, 1942)
b. The Rough Plate
The conversion from pipe resistance to the plate drag may be carried
out for the rough plate in the same manner as described previously for the
smooth plate. One assumes a plate uniformly covered with the same rough
ness k. Since the boundary layer thickness 5 increases from the front
toward the rear, the ratio k/5 which is significant for the drag decreases
from the. front toward the rear. Behind the initial laminar run, therefore,
follows at first the region of the fully developed roughness flow; the so
called intermediate region follows and farthest toward the rear there is,
finally, if the plate is long enough, the region of the hydraulically
smooth flow. These regions are determined by specification of the numerical
values for the roughness coefficient vks/V. In order to obtain the drag
of the rough plate, one must perform the conversion from pipe flow to plate
flow for each of these three regions individually. This calculation was
carried out by Prandtl and Schlichting (reference 76), based on the results
of Nikuradse (reference 71) for the pipe tests with sand roughness. For
this conversion one starts from the universal velocity distribution law of
the rough pipe according to equation (16.28), the quantity B being
dependent also on the characteristic roughness value v*ks/y, according
to figure 84. The calculation takes basically the same course as described
in detail for the smooth plate in chapter XVIIa. It is, however, rather
complicated and will not be reproduced here. One obtains as final result
for the total drag coefficient of the sandrough plate a diagram (fig..89)
Which represents the drag coefficient as a function of the Reynolds
number Uol/V with the relative roughness 1/ks as parameter. Just as
for the pipe, a given relative roughness 1/ks has a drag increasing
effect not for all Renumbers, but only above a certain Renumber. This
diagram is applicable also for roughnesses other than sand roughness, if
one uses the equivalent sand roughness. In the diagram (fig. 89) the
square drag law is attained, just as for the pipe, for every relative
The tables pertaining to the plate drag formulas are given in
table 7, chapter XXII.
NACA TM No. 1218
roughness I/ks provided the Renumber is sufficiently large. The
interpolation formula
cf = (.89 + 1.62 log k (17.29)
applies to this law.
c. The Admissible Roughness
The problem of the admissible roughness of a wall in a flow is
very important in practice since it concerns the effort that might
reasonably be expended in smoothing a surface for the purpose of drag
reduction. Admissible roughness signifies that roughness above which a
drag increase would occur in the given turbulent friction layer (which,
therefore, still is in effect hydraulically smooth). The admissible
relative roughness k,/2 decreases with increasing Renumber UoZ/v
as one can see from figure 89. It is the point where the particular
curve 2/ks diverges from the curve of the smooth wall. One finds the
values for the admissible relative roughness according to the following
table; they can also be combined into the one formula
Uo k admins. = 102 (17.30)
V
k(i! 3 4 5 6 7
V
10 10 10 10 10
adamiss.
From equation (17.30) one recognizes that the admissible roughness height
is by no means a function of the plate length. This fact is significant
for instance for the admissible roughness of a wing. Equation (17.30)
states that for equal velocity the admissible roughness height is the
same for a full scale wing as for a model wing. Let us assume a numerical
example:
Wing: chord I = 2m
Velocity Uo = 300 km/h = 83 m/sec
NACA TM No. 1218
From equation (17.30) results an admissible magnitude of
roughness ks = 0.02 mm. This degree of smoothness is not always attained
by the wing surfaces manufactured in practice, so that the latter have a
certain roughness drag. In the considerations just made one deals with
an increase of the friction drag in an a priori turbulent friction layer.
However, the roughness may also change the drag by disturbing the
laminar friction layer to such an extent that the point of laminar/
turbulent transition is shifted toward the front. Thereby the drag can
be increased or reduced according to the shape of the body. The drag is
increased by this displacement of the transition point if the body in
question has predominant friction drag (for instance wing profile). The
drag might be reduced, circumstances permitting, for a body with pre
dominant pressure drag (for instance, the circular cylinder). One calls
the roughness height which causes the transition the "critical roughness
height". According to Japanese measurements (reference 77) this critical
roughness height for the laminar friction layer is given by
Vkcrit
= 15 (17.31)
A numerical example follows:
Assume, as prescribed before, a
wing 2 = 2m
Uo = 300 km/h = 83 m/sec
then Re = UoZ/V = 107. Consider the point of the wing x = 0.12, thus
Rex = U x/V = 10 Up to this point the boundary layer might remain
laminar under the effect of the pressure drop. The wall shearing stress
for the laminar boundary layer is according to equation (9.17)
u 21Y v 6900 m2
= 0.332 Uo2 = 0.332 690 2.29
P Uox 103 sec2
hence:
v* = VTo/ = 1.52 m/sec
and according to equation (17.31)
k = 15 = 15 1 104 = 0.14 mm
crit V* 1.52 7
NACA TM No. 1218
The critical roughness height causing the transition is, therefore, about
ten times as high as the roughness height admissible in the turbulent
friction layer. The laminar friction layer therefore "tolerates" a
greater roughness than the turbulent one.
The following can be said about the influence of the roughness on
the form drag: Sharpedged bodies are indifferent to surface roughnesses
because for them the transition point is fixed by the edges, as for
instance for the plate normal to the flow. Short curved bodies, on the
other hand, as for instance the circular cylinder, are sensitive. For
the circular cylinder the critical Reynolds number, for which the known
large pressure drag reduction occurs, is largely dependent on the
roughness. With increasing relative roughness k/R (R = radius of the
circular cylinder) Recrit decreases. According to British measurements
(reference 90) the drag curves for a circular cylinder with different
relative roughnesses have a course as indicated in figure 90. The
boundary layer is so disturbed by the roughness that the laminar/turbulent
transition occurs for a considerably smaller Renumber than for the
smooth cylinder. The roughness has here the same effect as Prandtl's
trip wire, that is, in a certain region of Renumbers It decreases the
drag. It is true, however, that the supercritical drag coefficient is
then always larger for the rough circular cylinder than for the smooth
one.
CHAPTER XVIII. THE TURBULENT FRICTION LAYER IN
ACCELERATED AND RETARDED FLOW
The cases of turbulent friction layer treated so far are relatively
simple insofar as the velocity outside of the friction layer along the
wall is constant. Here as for the laminar flow the case of special
interest is where the velocity of the potential flow is variable along
the wall (pressure drop and pressure rise). As for laminar flow, the
form of the boundary layer profile along the wall is variable. In
practice this case exists for instance for the friction layer on the
wing, on the turbine blade, and in the diffuser. Of special interest is
the question of whether separation of the boundary layer occurs and,
if so, where the separation point is located. The problem consists
therefore for a prescribed potential flow in following the turbulent
friction layer by calculation. The calculation of the turbulent friction
drag is of importance. The corresponding problem for the laminar friction
layer was solved by the Pohlhausen method (chapter X).
For the turbulent friction layer the method of Gruschwitz
(reference 78) proved best. Gruschwitz makes the assumption that the
velocity profiles of the turbulent boundary layer for pressure drop and
NACA TM No. 1218
rise can be represented as a oneparameter family, if one plots u/U
against y/l. a signifies the momentum thickness which is, according
to equation (6.32), defined by:
U2 = u(U u) dy (18.1)
As form parameter one selects
S= 1 ( (18.2)
u(3) denoting the velocity in the friction layer at the distance from
the wall y = . That n actually is a serviceable form parameter can
be recognized from figure 91 where a family of turbulent boundary layer
profiles is plotted according to Gruschwitz. Gruschwitz found from his
measurements that the turbulent separation point is given by
S= 0.8 (Separation) (18.3)
The form parameter n is analogous to the Pohlhausenparameter of
the laminar friction layer. However, a considerable difference exists
between T and X: for the laminar friction layer an analytical relation
exists between X and the pressure gradient and the boundary layer
thickness, namely according to equation (10.41)
S 2 P du (18.4)
v dx
Such a relation is thus far lacking for the turbulent boundary layer,
since one does not yet possess an analytical expression for the turbulent
velocity profiles*. One needs therefore an empirical equivalent for
equation (18.4).
For the special case of the turbulent friction layer without
pressure gradient where the 1/7power law u/U = (y/5)1/7 applies for
the velocity profile, one finds from equations (18.1) and (18.2)
Compare, however, chapter XXIIb, where under certain assumptions
such an analytical connection is. indicated.
NACA TM No. 1218
S 7 = 0.487 (18.5)
5 72
Since in the case of the turbulent boundary layer, an analytical
expression for the velocity distribution does not exist, the calculation
is limited to the determination of the four characteristics of the friction
layer: form parameter n, wall shearing stress To, displacement
thickness 8*, momentum thickness . Four equations are required for
their calculation.
As for the laminar boundary layer, the momentum theorem yields
the first equation; the momentum theorem may, according to equation (10.36)
be written in the form:
o d d + 1 84 8 dU2
2 2 ) U2
pU
(I) (18.6)
The second equation is yielded by the function
8 = H() (II) (18.7)
obtained by Gruschwitz by evaluation of the measured velocity profiles
(fig. 92), and regarded as generally valid. It can be derived also by
calculation from the form of the velocity profile (compare appendix
chapter XXII) and yields:
S= 1 B (18.8)
H(H + 1)
The third equation is empirically derived by Gruschwitz from his
measurements. He considers that the energy variation of a particle moving
parallel to the wall at the distance y = 0 is a function of uj, U, a, V.
Dimension considerations suggest the following relation:
Idgl = F(, Re) (18.9)
q dx
q = U2 = p + 2 signify the total pressure in the layer
S= The evaluation of the test relts showed that a dependence on
y = 3. The evaluation of the test results showed that a dependence on
NACA TM No. 1218
the Renumber is practically nonexistent, and that one can represent
equation (18.9) in the following manner:
dg
= 0.00894 o 0.00461
q dx
(18.10)
Furthermore, the identity
p+ pp2
P2(2
is valid. One puts
(18.11)
and has therefore
dg1
dx
(18.12)
dx
dx
Now equation (18.10) can be written:
S = 0.00894 t + 0.00461 q
The fourth equation is still missing and is replaced by the following
estimation of To: According to the calculations for the plate in
longitudinal flow, equation (17.7) was:
0 = 0.0225
7P (
1/4 0.0225 (Re
= 0.0225 (Re1) (18.14)
If one takes into consideration that for the 1/7power law of the velocity
distribution:
go 81g
uU2 )~q
(111) (18.13)
IIACA TM Ho. 1218
8
one can write equation (18.14) also:
T 01/4
S= 0.01338 (Re ,)
PI
= 0.01256 (Re.,
1/4
) (IV)
For calculation of a and I (( = q 1, respectively)
the following system of equations:
one must now solve
(18.16)
.q = U2 s a given function of x; H and . are given functions
2 pU2
of n = (/q or 3, respectively. This system of equations is to be
solved downstream from the transition point.
Initial values: As initial value for 1 one takes the value from
the laminar friction layer at the transition point:
oturb. o Zam.
(18.17)
This is based on the consideration that the loss of momentum does not
vary at the transition point since it gives the drag. The initial value
of j is somewhat arbitrary. Gruschwitz takes
o0 = 0.1
and states that a different choice has little influence on the result.
With these initial values the system of equations (18.16) may be
solved graphically, according to a method of Czuber (compare appendix,
Chapter XXII, where an example is given). A first approximation for B
is obtained by first solving the second equation with constant values
for TO/p2 and E;
72
72
(18.15)
NACA TM No. 1218
9= 0.002; H = 1.5 (18.18)
pU
are appropriate. Thereby the second equation is a differential equation
of the first order for .6. This first approximation .l1(x) is then
substituted into the first equation, which then becomes a differential
equation of the first order for t(x); let its solution be denoted
by L(x). Thus one has also a first approximation for q: l(X).
With ~l(x) one determines the course of H(T) according to figure 92
and is now able to improve T according to equation (18.15). These
O
values of both H and T are now inserted in the second equation, and
a second approximation 92(x) is obtained. By substitution of 82(x)
into the first equation one obtains the second approximation (2(x), etc.
The method converges so well that the answer is essentially attained in
the second approximation.
The separation point is given by
7 =.0.8
Incidental to the boundary layer calculation one obtains the following
characteristic values of the friction layer as functions of the arc
length x:
3(x), 8*(x), q(x), To(X).
The boundary layer calculation for the profile J 015, ca = 0 is given
as example in figure 93. The transition point was assumed at the velocity
maximum. The calculation of the laminar boundary layer for the same case
was indicated in chapter XII. The details of this example are compiled
in the appendix, chapter XXII.
It should be mentioned that the calculation for the turbulent
boundary layer must be performed anew for every Renumber UoZ/V, whereas
only one calculation was necessary for the laminar boundary layer. The
reasons are, first, that the transition point travels with the Renumber,
and second, that the initial value of /t varies with Re, since for
the laminar boundary layer \ at the transition point is fixed.
It must be noted that the values obtained for To become incorrect
in the neighborhood of the separation point: At the separation point T
must equal zero, whereas equation (18.15) gives everywhere To 0.
IACA TM No. 1218
Boundary Layer Without Pressure Gradient
In this case q(x) = Constant. Equation (18.13) can be written:
Sd = o.0089h4 + 0.00461 (18.19)
q dx
or, because q(x) = constant;
= d( q) (18.20)
dix ix dx
Thus equation (18.19) becomes:
~ = 0.00894n + 0.00461 (18.21)
A solution of this equation is:
0.00461
T = 0 6 = 0.516 (18.22)
0.00894
Since at the beginning of the turbulent friction layer n is smaller
than this value (transition point n = 0.1) and since according to
equation (18.21) dT1/dx > 0, n must in this case approach'the value
n = 0.516 asymptotically from below. For the velocity profile of the
1/7power law, I = 0.487 (compare equation (18.15)). The profile
attained asymptotically for uniform pressure (p = constant) therefore
almost agrees with the 1/7power law that was previously applied to the
plate in longitudinal flow.
A great many boundary layer calculations according to this method
are performed in the dissertation by Pretsch (reference 80).
Thirteenth Lecture (March 2, 1942)
CHAPTER XIX. FREE TURBULENCE
a. General Remarks; Estimations
After considering so far almost exclusively the turbulent flow
along solid walls, we shall now treat a few cases of the socalled free
turbulence. By that one understands turbulent flows where no solid walls
NACA TM No. 1218
are present. Examples are the spreading of a jet and its mixing with the
surrounding fluid at rest; or the wake flow behind a body towed through
the fluid at rest (fig. 94). Qualitatively these turbulent flows take a
course similar to that for the laminar case (compare chapter IX); quanti
tatively, however, considerable differences exist, since the turbulent
friction is very much larger than the laminar friction. In a certain way,
the cases of free turbulence are, with respect to calculations, simpler
than turbulent flows along a wall, since the laminar sublayer is not
present and the laminar friction as compared with the turbulent one can
therefore be neglected for the entire flow domain. The free turbulence
may be treated satisfactorily with Prandtl's concept of the turbulent
shearing stress according to equation (15.9):
2 bu au
T = p 2 (19.1)
the mixing length 1 being assumed a pure position function. The
turbulent friction has the effect of making the jet width increase and
the velocity at its center decrease with increasing distance along the
jet.
We now perform rough calculations, according to Prandtl (reference 2,
Part I), for a few cases of free turbulence which give information about
the laws governing the increase of width and the decrease of "depth" with
the distance x.
It has proved useful for such turbulent jet problems to set the
mixing length 2 proportional to the jet width b:
= = constant (19.2)
b
Furthermore, the following rule has held true: The increase of the
width b of the mixing zone with time is proportional to the fluctuation
of the transverse velocity v':
Db vt (19.3)
Dt
D/Dt signifies the substantial derivative; thus: u + v
Dt ax cx
According to our previous estimation, equation (15.5): v' = 2 1u
Therefore:
Therefore:
NACA TM No. 1218
Db ou
Dt oy
Flu'thi r'rire, the rVIarn value of u equals approximately:
CV.
O i,
7 = nlnMbtr
Oz,' b
and thus:
Db 2
 = number u = number 0 u
Ut b max max
Jet (Plane and Circular'
We shall estimate, by means of these relations, how the width
increases with the distance x and the velocity at the center decreases.
At first, for the circular as well as for the plane jet:
Db db
 = number u
Dt max dx
(19.6)
It follows, by comparison with equation (19.5):
db 2
= number = number
dx b
b = number x + constant
If the origin of coordinates is suitably selected
with the orifice) one has therefore:
b = number x
(it need not coincide
(plane and circular jet)
(19.7)
The relation between uax and x is obtained from the momentum
theorem. Since the pressure is constant, in the xdirection, the x
momentum must be independent of x, this:
J = p 1 u2 d F = constant
(19.4)
(19.5)
NACA TM No. 1218
whence follows for the circular jet:
2 2
J = number pu b
*^ Tna
1 U n
u = number
max b p
and because of equation (19.7):
max = number ~ 1 (circular jet)
(19.8)
For the plane jet, if
jet:
J' signifies the momentum per unit length of the
2
J' = number p u b
max
Umax = number 
and because of equation (19.7):
umax = number 1
(plane jet)
(19.9)
Wake (plane and circular)
The calculation for the wake is somewhat different, since the
momentum which gives directly the drag of the body must be calculated in
a slightly different way. The momentum integral is now (compare
equation (9.40)):
w = J = pu (Uo u) d F
(19.10)
At large distance from the body u' = Uo u is small compared with Uo
(fig. 95) so that u' < Ug and
NACA TM No. 1218
u(Uo u) = (Uo u') .u' Uo u'
Thus one obtains for the circular wake:
Scw F U2 p Uo u' I b2
u2 w
U 0
o a b2
Instead of equation (19.6) now applies:
Db d!b
Dt dx
and instead of equation (19.5):
Db 2
= number u'
Equating of equations (19.12) and (19.13) gives:
U db ,L u' = 0 u'
o dx b
By comparison with equation (19.11) one obtains:
b2 db P cw F
dx it
b a'I Cw F x (Wake Circular)
By insertion in equation (19.11) results:
(Wake circular)
(19.11)
(19.12)
(19.13)
(19.14)
(19.15)
(19.16)
54 NACA TM No. 1218
For the plane wake behind a long rod, wing, or the like with the
diameter d and the length L, W = c p Uo2 Ld and
W= J, Uo ut bL
and hence:
cd
U W (19.17)
Uo 2 b
and from this in combination with equation (19.14):
db
2 b ' c d
b ~ P c d x (wake plane) (19.18)
By substitution in equation (19.17) results:
c l d1/1 2
1 w (Wake plane) (19.19)
Uo 2 0 x
Thus, for the circular wake, the width increases with \3 and
2/3
the velocity decreases with x /; for the plane wake, the width
increases with \{il, and the velocity decreases with x1/2.
The power laws for the width and the velocity at the center are
compiled once more in the following table. The corresponding laminar
cases which were partially treated in chapter IX, are included. More
over, the case of the free jet boundary is given, that is the mixing of
a homogeneous air flow with the adjoining air at rest. (Compare
figure 97.)
NACA TM No. 1218
Laminar Turbulent
Velocity at Center Velocity at Center
Width u Width u
max max
Sor u' or u'
respectively respectively
Plane Jet x2/3 x1/3
Circular jet x x x x1
Plane wake x1/2 ,1/2 x1/2 x~1/2
Circular wake x1/2 x1 xl/3 x2/3
Free jet boundary x1/2 xO x zo
POWER LAWS FOR TEE INCREASE WITH WIDTH AND THE DECREASE OF VELOCITY
WITH THE DISTANCE x
For a few of the cases treated here the velocity distribution will
be calculated explicitly below. The calculation on the basis of the
Prandtl mixing length theorem was performed for the free jet boundary,
the plane jet, and the circular jet by W. Tollmien (reference 81), for
the plane wake by H. Schlichting (reference 82) and for the circular
wake by L. M. Swain (reference 83).
The equations of motion for the plane stationary case are, according
to equation (14.14), if the laminar friction terms are completely
neglected:
bu _u 1P
u + v , 
^CTy P ;T
1 ap
P y
u I +
ox
+ +
P ox p oy
I 7Txy 1 _
+ +
p ax p 6y
1u av
u + T = 0
SC) v
S(19.20)
NACA TM No. 1218
b. The Plane Wake Flow
We shall now calculate the velocity distribution for the plane wake
flow. A cylindrical body of diameter d and sTan h is considered.
Further, let
U' = U u
o
(19.21)
be the wake velocity. One applies the momentum theorem to a control
area according to figure 95, the rear boundary B C of which lies at
such a large distance from the body that the static pressure there has
the undisturbed value. As shown in detail in chapter DI, equation (9.40),
one obtains:
W=hp / u (Uo u) dy
BC
(19.22)
W = h p (Uo u) u' dy
BC
For large distances behind the body
2
approximately neglect the term u'
equation (19.22). Hence:
x>oo: W = h
Since, on the other hand W = cw d h
+b dy
/ u' dy =
Ub
u't Uo so that one may
in comparison with Uo a'
p Uo u' dy
rBC
U 2, there becomes:
2 o
1 d U
2 Cw dU
(19.23)
This problem can be treated with the boundary layer differential equati ons;
they read according to equation (19.20) with the Prandtl expression fo the
iLrb)1lnt shearing stress accori'r. to equation (15.9), with p and ax
no lectedl:
in
(19.22a)
NACA TM No. 1218
u + v 
ax OJ
S = 22
QNo
u + = 0
ox by
For th mixing length one puts, according to equation (19.2):
I = b (
Further:
u' = u1 + u2 + *
y
For the wake velocity ul and the width b the power laws for the
decrease and increase, respectively, with x were already found in
equations (19.18) and (19.19). One therefore writes:
U
0o
x 1/2
c d
w
(19.24)
1. 25)
19.26)
19.27)
(19.28)
(19.29)
1/2
b = B (cwd x)1/2
According to equations (19.27) and (19.29)
S) 1
ay B(cwd x)1/2
N 12
ax 2 x
The estimation of the terms in equation (19.24) with respect
order of magnitude in x gives:
Lu 3/2.
Ie
oy 3/2 bu 1
~X ; ~ ;
oy cy
u N .1
Oa ,3j'
to their
a8u 3/2
. 2
oy
1
V 1 X
*The terms u2 signify additional terms of higher approximation,
which disappear according to a higher power of x than does u1.
58 NACA TM No. 1218
bu 2
Hence the term v x whereas the largest terms
equation (19.24) is simplified to:
ull
O I
3/2
~ x
2 oul ul
22 y 2
y ay2
. Thus
(19.30)
The neglected terms are taken into account only in the next approximation.
The further calculation gives:
Sx 1/2
x \1/2
= U0
(,d
\w /
:o w
1
2
a f, _1
x 2x
1
f'
1/2
B(cwd x)/2
f" 1
f2
2
B cdx
2 ul 2ul
y2 y2
oy
insertion
S1/2
(W
= 2 2 ( 1
o twd)
ff" ,, 1
f I 1 /2
B(w^d)
in equation (19.30) and eliminating the factor
the following differential equation results for f(T)
1
(f +
2
2
i ft) = 23 f'f"
B
(19.31)
The boundary conditions are:
bul
y = b: u = 0;  = 0
dy
bul
1
by
2
u
2
After
2 1
Uo x
IACA TM No. 1218
That is:
n = 1: f = f' = 0
(19.32)
The differentialequation (10.30) may immediately be integrated once
and gives:
2
1 f f' + Constant
2 B
Because of the boundary conditions, the
zero; thus:
integration constant mast equal
2
2f B
This may be integrated in closed form:
f1/2 1/2 =
S2
SB
df B
Ff = 20
2fl /2 B
F21
B 3,
21 2
Because f = 0 for T = 1, C
31 B
3 F20
+ Constant
+ 
and hence:
f = L 13/2)
9 202(
(19.33)
3/2
60 NACA TM No. 1218
The condition f' = 0 for T1 = 1 is simultaneously satisfied according
to equation (19.33). In f", that is u, a singularity results at the
y2
center (T = 0) and on the edge. For q = 0, f" = o; the velocity
profile there has zero radius of curvature. At the edge there exists
a discontinuity in curvature. In contrast to the laminar boundary
layer solutions, where the velocity asymptotically approaches the value
of the potential flow, one obtains here velocity profiles which adjoin
the potential flow at a finite distance from the center.
The constant B remains to be determined:
u' dy =*2 u' dy = 2 ul dy = 2Uo Cwd B f () dT
b 0 Jo
From equation (19.33) one finds: 2f O
(1 3/2)2
and hence:
1
2 f ff(TI) dq = B
0O 2032
and, by comparison with equation (19.23):
u' dy = cvd
B2 =1,
2 2
20P
B2 1
 U 1d Uo
202 o
B = 0iO 3
Thus the final result for the width of the wake and the velocity
distribution from equations (19.28) and (19.29) is:
':.iewer's note: Integrating from 1 to +1, as was done in the
original German version, results in an imaginary term, which was avoided
in the translation by integrating from 0 to +1 and doubling the result.
dT = 9
2
S0
or:
(19.34)
NACA TM No. 1218 61
b = o 10 (ci :1/2
S (19.35)
I ,10 1/2 r 2
Uo 180 wcd by 
The constant 0 = 2/b is the only empirical constant of this theory; it
must be determined from the measurements.
Comparison with the tests of Schlichting (reference 82) shows that
the two power laws (equations (19.28) and (19.29)) are well satisfied,
and also that the. form of the velocity distribution shows good agreement
with equation (19.35), figure 96. The constant P is determined as
0 = 0.207
b
The solution found is a first approximation for large distances; according
to the measurements it is valid for x/Cwd > O. For smaller distances
one may calculate additional terms which ire proportional to x, x3/2
for the wake velocity in equation (19.26).
The rotationallysymmetrical wake problem was trecatel by
Miss L. M. Swain (reference 83). For the first approximation results
exactly the same function for the velocity distribution; only the power
laws for the width b ard the velocity at the center um' are different,
1/3 2/3
namely b /x and um' x2/, as already indicated in
equations (19.15) and (19.16).
Fourteenth Lecture (March 9, 1942)
c. The Free Jet Boundary
The plane problem of the mixing of a homogeneois air stream with
the adjoining air at rest shall also be treated somewhat more accurately
(fig. 97). It is approximately present for instance at the edg', of the
fru. jet of a wind tuunel. The problem was solved by TolLmien
referencee 81).
Th. velocity profiles at variot.3 iistances x arx affine. One
C'ZL'
u = U f(r() = U F'(.)
(1Q0. )
NACA TM No. 1218
where
S=x; (b~ x)
(19.37)
and
Furthermore set
(19.38)
I = c x = Cb
The equation of motion reads:
6U 1 T
+ V =
by p ay
(19.39)
= 2 6 u 2
= 2 U
a" oy2
One integrates the continuity equation by the stream function:
= u dy = Uo x
f(n) d1 = Uo x F(q)
2U
by2
ay2
__ U
Ix x F
U
SF'"
2
x
v = = Uo (F IF') (19.40a)
Substitution into the equation of motion (19.39) gives, after division
by Uo2/;
FF" + 2c2 F"F'" = 0
au
U 
ax
Then:
(19.40)
F(T) = f(l) dy
u U
= 2 F"*
ay =x P
(19.41)
NACA TM No. 1218
The boundary conditions are:
at the inner edge:
T = 1:
at the outer edge:
T = I2:
Since the boundary points i, and
conditions can be satisfied by the
equation (19.41). By introduction
12 are still free, these five boundary
differential equation of the third order
of the new variable:
11* = 1
3J2
\Y2
(19.43)
the differential equation (19.41) is transformed into (' = differentiation
with respect to q*)
F F" + F" F'" = 0
(19.44)
The solution F"
general solution
= 0, which gives u = Constant, is eliminated.
of the linear differential equation
F + F'" = 0
(19.45)
F = e *
with X signifying the roots of the equation X3 + 1 = 0,
u = U:
u = 0:
v =0:
F' = 1
F" = 0
F = l
(19.42)
u = 0:
au 6:
as
F' = 0
F" = 0
The
thus:
NACA TM No. 1218
X1 = 1;
1 i
k2, 3 =+ V
Hence the general solution is:
F = C1 e*
+ C2 e
cos( I*)
(19.46)
If, moreover, one measures the icoordinate from the inner boundary
point, thus puts:
t* = T*n ri
the solution (equation (19.46)) can also be written:
F = dI e
+ d2
2
e
cos ( T
*+
+ d3e 2 sin
From the boundary conditions (equation (19.42)) result
the values:
for the constants
0.981 ;
dl = 0.0062;
Tj = 2.04 ;
d2 = 0.987;
4 = 3.02
d3 =
0.577
For the width of the mixing region one obtains:
b = X'(l 2)= x'2c2 (7l*
b = 3.02 2c2 x
The constant c must be determined from experiments. From measurements
it is found that
b = 0.255 x (19.47)
( )
' =
_
+ C3 e sin( I*)
NACA TM No. 1218
Hence
2
S2c = 0.0845; c = 0.0174
and
S= 0.0682 (19.48)
b
It is striking that here the ratio Z/b is essentially smaller than for
the wake.
The distribution of the velocity components u and v over the
width of the mixing zone is represented in figure 98.
From the second equation of motion one may calculate the pressure
difference between the air at rest p and the homogeneous air stream pl.
One finds:
p p = 0.0048 U 2 (19.49)
1 o 2 o
Thus an excess pressure of onehalf percent is present in the Jet. For
the inflow velocity of the entrained air one finds according to
equation (19.40a):
v_ = F(n2) Uo = + 0.379 2 Uo
and with the measured value of c;
v_ = 0.032 Uo (19.49a)
d. The Plane Jet
In a similar manner one may also calculate the plane turbulent Jet
flowing from a long narrow slot (compare fig. 94). The laws for the
increase of the width and the decrease of the center velocity have
already been given in equations (19.7) and (19.9): b x; um .
The calculation of the velocity distribution was carried out by Tollmien
(reference 81); it leads to a nonlinear differential equation of the
second order the integration of which is rather troublesome. Measurements
for this case were performed by Forthmann (reference 91). In figure 99
NACA TM No. 1218
the measurements are compared with the theoretical curve. The agreement
is rather good. Only in the neighborhood of the velocity maximum is
there a slight systematic deviation. There the theoretical curve is
more pointed than the measured curve; the theoretical curve, namely,
again has at the maximum a vanishing radius of curvature.
According to the Prandtl formula, equation (15.9), the exchange
becomes zero at the velocity maximum, whereas actually a small exchange
is still taking place.
e. Connection Between Exchange of Momentum,
Heat and Material
In concluding the chapter on turbulent flows I should like to
make a few remarks about the connection between the turbulent exchange
and the heat and material transfer in a turbulent flow.
In the Prandtl theorem equation (15.9) for the apparent turbulent
stress:
2 u Au ou
T = p 2 A (19.50)
one can interpret:
A 2 u kg seci
A = p e I 2
Zm 2
as a mixing factor. It has the same dimension as the laminar viscosity i.
Furthermore, the shearing stress T may be interpreted as a momentum flow:
momentum
T = mom = momentum flow (19.51)
2
m sec
Momentum = mass ) velocity = kg sec].
Another effect of the turbulent mixing phenomena, besides the
increased apparent viscosity by transport of momentum,is the transport
of all properties inherent in flowing matter, as heat, concentration of
impurities, etc. If this concentration is not uniform, more heat or
impurity is carried away by the turbulent exchange from the places of
higher concentration than is brought back from the places of lower concen
tration. Thus there results, on the average, transfer from the places of
higher to those of lower concentration.
NACA TM No. 1218 67
This results, for temperature differences, in a turbulent heat
transfer; for concentration differences (for instance, of salt), in a
turbulent diffusion. They can, in analogy to equation (19.50) be expressed
as follows:
Momentum flow = momentum transport A d momentum
m sec
d2 8, Tl.. \ i t mhtm
Heat flow = heat transport = A ( heat
m2 sec dy \unit mass/
m sec
Transport of material _d (material
Flow of material
2 M dy unit mass
m sec
The heat content of the unit mass is 'cp 0 (8 = temperature,
c = specific heat =calm For chemical or mechanical
p 2
g sec degree
concentrations the concentration of material per unit mass is called the
concentration c; it is therefore the ratio of two masses and therefore
dimensionless. Thus the above equations may also be written in the
following forms:
du
T = A
7 dy
Q = AQ (19.52)
de
M=AM dy
The question arises as to whether AT, AQ, AM are numerically the same
or different. If the momentum is transported exactly like heat or material
concentration Prandtl's theorem is based on this assumption it would
follow that AT = AQ = AM and, for instance, the velocity and temperature
distributions in a turbulent mixing region would have to be equal.
However, measurements show partially different behavior.
One has to distinguish between wall turbulence and free turbulence.
Concerning free turbulence, calculations of G. I. Taylor (reference 92)
and measurements of Fage and Falkner (reference 93) showed for the velocity
and temperature profile of the plane wake flow
(free turbulence)
(19.53)
NACA TM No. 1218
The heat exchange is, therefore, larger than the momentum exchange.
Consequently the temperature profile is wider than the velocity profile.
The theory given for that phenomenon by G. I. Taylor operates with the
conception that the particles, in their turbulent exchange movements,
do not maintain their momentum (Prandtl), but their vortex strength 6.
(Prandtl's momentum exchange theory Taylor's vorticity transfer
theory). However, there are cases not satisfied by the Taylor theory
(for instance the case of the rotationally symmetrical wake). That the
heat exchange for free turbulence is considerably larger than the momentum
exchange is also shown by experiments of Gran Olsson (reference 88)
concerning the smoothing out of the temperature and velocity distributions
behind grids of heated rods. With increasing distance behind the grid
the temperature differences even out much more rapidly than'the differ
ences in velocity.
For wall turbulence the difference between the mixing factors for
momentum and temperature is smaller. H. Reichardt (reference 87) was
able to show, from measurements of the temperature distribution in the
boundary layer on plates in longitudinal flow by Elias (reference 86)
and in pipes by H. Lorenz (reference 14), that here
AQ = 1.4 to 1.5 (wall turbulence) (19.54)
A
T
Herewith we shall conclude the considerations of free turbulence.
CHAPTER XX: DETERMINATION OF THE PROFILE DRAG FROM
THE LOSS OF MOMENTUM
The method, previously discussed in chapter IX, of determining
the profile drag from the velocity distribution in the wake is rather
important for wind tunnel measurements as well as for flight tests; we
shall therefore treat it in somewhat more detail. The determination of
the drag by force measurements is too inaccurate for many cases, in the
wind tunnel for instance due to the large additional drag of the wire
suspension; in some cases (flight test) it is altogether impossible. In
these cases the determination of the drag from the wake offers the only
serviceable possibility.
The formula derived before in chapter IX, equation (9.41) for
determination of the drag from the velocity distribution in the wake is
valid only for relatively large distances behind the body. It had been
assumed that in the rear control plane (test plane) the static pressure
equals the pressure of the undisturbed flow. However, in practically
carrying out such tests in the wind tunnel or in flight tests one is
NACA TM No. 1218
forced to approach the body more closely. Then the static pressure gives
rise to an additional term in the formula for the drag. For measurements
close behind the body (for instance, for the wing, for x < t) this term
is of considerable importance, so that it.must be known rather accurately.
A formula was indicated, first by Betz (reference 84), later by B. M. Jones
(reference 85) which takes this correction into consideration. Although
at present most measurements are evaluated according to the simpler Jones
formula, we shall also discuss Betz' formula since its derivation In
particular is very interesting.
a. The Method of Betz
One imagines a control surface surrounding the body as shown in
figure 100. In the entrance plane I ahead of the body there is flow
with freestream total pressure go, behind the body in plane II, the
total pressure g2 < g. The lateral boundaries are to lie at so large
a distance from the body that the flow there is undisturbed. In order
to satisfy the continuity condition for the control surface the velocity
u2 in plane II must be partially greater than the undisturbed velocity
Uo. Consider the plane problem; let the body have the height h.
Application of the momentum theorem to the control surface gives:
W = h
+ pu1 2dy 
(20.1)
In order to make this formula useful
must be transformed in such a manner
only over the "wake". For the total
for test evaluation the
that the integral need
pressures
integral
to be extended
at infinity:
in plane I:
in plane II:
go =Po + Uo2
S 2 o
p 2
go = pl + ul
P 2
g2 = 2 + f u 2
Outside of the wake the total pressure everywhere equals go. Hence
equation (20.1) becomes
(20.2)
70 NACA TM No. 1218
+00 +O
W h g 2)dy + 2 u dy (20.3)
Thus the first integral already has the desired form, since the integrand
differs from zero only within the wake. In order to give the same form
to the second integral, one introduces a hypothetical substitute flow
u2'(y) in plane II which agrees with u2 everywhere outside of the wake,
but differs from u2 within the wake by the fact that the total pressure
for u2' equals go. Thus
go = p + '2 (20.4)
0 2 22
Since the actual flow ul, u2 satisfies the continuity equation the
flow volume across section II for the hypothetical flow ul, u2' is too
large. It shows a source essentially at the location of the body which
has the strength
Q = h u2 u2 dy (20.5)
A source in a frictionless parallel flow experiences a forward thrust
R = p Uo Q (20.6)
One now again applies the momentum theorem according to equation (20.3)
for the hypothetical flow with the velocity u1 at the cross section I
and the velocity u'2 at the cross section II. Since g2 = go and the
resultant force, according to equation (20.6), equals R, one obtains
p TTo Q = h (2u u2 dy (20.7)
By subtraction of equation (20.7) from equation (20.3) there results
w + p U Q = h g + u2 u2 dy (20.8)
NACA TM No. 1218 71
or because of equations (20.5) and (20.6):
W = h [ g gdy + (u122 u p u2)d
One may now perform each of these integration only across the wake, since
outside of the wake u2 = u Due to u'22 u22 = (u2 2)(u2 + u2)
a transformation gives the following formula:
W = h go ) + 0 (u'2 u2)(u'2 + u2 2U) d (20.9)
Betz' Formula
In order to determine W according to this equation, one has to measure
in the test cross section behind the body the following values:
1. Total pressure g2 (therewith go is the value of g2 outside
of the wake).
2. Static pressure p2.
Furthermore, p = static pressure at infinity.
Hence one obtains all quantities required for the evaluation of
equation (20.9).
It is useful for the evaluation of wind tunnel tests to introduce
dimensionless quantities. With F = ht as area of reference for the
drag:
p 2
W = c, h t U
w 2 o
and hence from equation (20.9):
w 7g\ \2 4_ 2p oP ______
o2 o 2 o 2
(20.10)
For the case in which p2 = p = 0, at the test cross section one can
write this equation, because go = qo:
NACA TM No. 1218
(1 1q
V 2 Xv )d
c = 2
c = 2
W I
(1 V d
U d
0 l
This agrees with equation (9.41).* Thus in this case Betz' formula
changes, as was to be expected, into the previous simple formula.
b. The Method of Jones
Later B. M. Jones (reference 85) indicated a similar method which
in its derivation and final formula is somewhat simpler than Betz' method.
Let cross section II (fig. 101) (the test cross section) lie close
behind the body; there the static pressure p2 is still noticeably
different from the static pressure po. Let cross section I be located
so far behind the body that the static pressure there equals the undis
turbed static pressure. Then there applies for cross section I according
to equation (9.41)*
W =h p u
Uo l) d7
(20.12)
In order to relate the value of u, back to measurements at cross
section II, continuity for a stream filament is first applied:
p ul dyl = p u.2 dy2
82
q10
d 
t
2 \f
CIO
(20.11)
In chapter IX the total drag of the body (both sides of the
plates) was designated by 2 W; here the entire drag equals d!
(20.13)
c
+ d7y
1
U2
U
O
I\
NACA TM No. 1218
Jones makes the further assumption that the flow from cross section II
to cross section I is without loss, that is, that the total pressure is
constant along each stream line from II to I:
(20.14)
First, according to equations (20.12) and (20.13):
W = hp u2 Uo ul) dy
1 2
Furthermore:
P 2
S+U = =
o 2 0o o o
p 2
Po + E U1
= g = g2
with p = 0
o
(20.16)
p 2
S+ u 2= g2
2 2 2 2
and hence
U2
U
01
g2 P2
I ; o
V 90
U1
0
U q
t h u
From equation (20.12) follows, with W = c
c u =2
(u lo Go U
(20.17)
P 2
thR U
2
dt
t
and because of equation (20.17):
cW = 2 2 I 2 1 
(20.15)
(20.18)
g2 = gl
NACA TM No. 1218
Formula of Jones
Thus all quantities may be measured in cross section II close to
the body. This formula is simpler for the evaluation than Betz' formula,
equation (20.10).
In the limit, when the static pressure in the test cross section
becomes p2 = Po, this formula, of course, must also transform into the
simple formula equation (20.11). One obtains for p2 = Po = 0 from
equation (20.18):
cy = 2 1 d = 2 21 U2d
This is in agreement with equation (9.41).
Fifteenth Lecture (March 16, 1942)
CHAPTER XXI: ORIGIN OF TURBULENCE
a. General Remarks
In this section a short summary of the theory of the origin of
turbulence will be given. The experimental facts concerning laminar/
turbulent transition for the pipe flow and for the boundary layer on the
flat plate have been discussed in chapter XIII. The position of the tran
sition point is extremely important for the drag problem, for instance
for the friction drag of a wing, since the friction drag depends to a
great extent on the position of the transition point.
The socalled critical Reynolds number determines transition. For
the pipe (id/V)crit = 2300, and for the boundary layer on the plate
(UoX/V)crit = 3 to 5 X 105. However, experimental investigations show
the value of the critical Reynolds number is very dependent on the
initial disturbance. The value of Recrit is the higher the smaller the
initial disturbance. For the pipe flow the magnitude of the initial
disturbance is given by the shape of the inlet, for the plate flow by
the degree of turbulence of the oncoming flow. For the pipe, for instance,
a critical Reynolds number (id/V)crit = 40,000 can be attained with very
special precautionary measures.
According to today's conception regarding the origin of turbulence,
transition is a stability phenomenon. The laminar flow in itself is a
solution of the NavierStokes differential equations up to arbitrarily
high Reynolds numbers. However, for large Renumbers the laminar flow
NACA TM No. 1218 75
becomes unstable, in the sense that small chance disturbances (fluctu
ations .in velocity) present in the flow increase with time and then alter
the entire character of the flow. This conception stems from Reynolds
(reference 101). Accordingly, it ought to be possible to obtain the
critical Reynolds number from a stability investigation of the laminar
flow.
Theoretical efforts to substantiate these assumptions of Reynolds
mathematically reach rather far back. Besides Reynolds, Rayleigh
(reference 102) in particular worked on the problem. These theoretical
attempts did not meet with success for a long time, that is, no instability
could be established in the investigated laminar flows. Only very
recently has success been attained, for certain cases, in the theoretical
calculation of a critical Reynolds number.
One assumes for the theoretical investigations that upon the basic
flow which satisfies the NavierStokes differential equations a disturbance
motion is superimposed. One then investigates whether the disturbance
movement vanishes again under the influence of friction or whether it
Increases with time and thus leads to ever growing deviations from the
basic flow. The following relations will be intorduced for the plane
case:
basic flows: U(x, y); V(x, y); P(x, y)
disturbance movement: u'(x, y); v'(x, y); p'(x, y) (21.1)
resultant movement: U + u'; V + v'; P + p'
P, p' signify pressure. The investigation of the stability of such a
disturbed movement was carried out essentially according to two different
methods:
1. Calculation of the energy of the disturbance movement.
2. Calculation of the development of the disturbance movement with
time according to the method of small oscillations.
I am going to say only very little about the first method since it
was rather unsuccessful. The second method was considerably more
successful and will therefore be treated in more detail later.
The first method was elaborated mainly by H. A. Lorenz
(reference 103). The following integral expression may be derived for
the energy balanceof the disturbance movement:
SEdV = p MdV NdV (21.2)
Dt f
NACA TM No. 1218
In it E = u (2 + ,2) signifies the kinetic energy of the disturbance
movement. The integration is performed over a space which participates
in the movement of the basic flow and at the boundaries of which the
velocity equals zero. signifies the substantial derivative. Thus
one finds on the left side of equation (21.2) the increase with time of
the energy of the disturbance movement. On the right side,
2 oU 2 7 U ,o Y
M = u* + v2 + u'v' + 
(21.3)
N = v' ut) 2
The first integral signifies the energy transfer from the main to the
secondary movement, the second the dissipation of the energy of the
secondary movement. If the right side is greater than zero, the Intensity
of the secondary movement increases with time, and the basic flow is thus
unstable. An assumed disturbance movement u', v' satisfies merely the
continuity equation, but no heed is paid to its compatibility with the
equations of motion. If one could prove that the right side is negative
for any arbitrary disturbance movement u', vt this would serve as proof
of the stability of the basic flow. On the other hand, the instability
would be proved as soon as the right side is positive for a possible
disturbance. Unfortunately general investigations in this direction
are very difficult and have not led to much success. H. A. Lorenz
(reference 103) treated as an example the Couetteflow (fig. 102), assuming
an elliptical vortex as a superimposed disturbance movement. He found for
this case \U/ t = 288, whereas Couette's measurements for this case
Kkrit
gave the value 1900.
b. The Method of Small Oscillations
For the second method (method of small oscillations) the disturbance
movement is actually calculated, that is, its dependance on the spatial
coordinates x, y and the time t is developed on the basis of the
hydrodynamic equations of motion. We shall explain this method of small
oscillations in the case of a plane flow. In view of the applications
of this method we shall immediately assume a special basic flow: the
component U, namely, is to be dependent only on y and t and V7 0.
Such basic flows had been previously called "layer flows". They exist
for instance in tunnel flow and pipe flow, approximately, however, also
in the boundary layer since here the dependence of the velocity component
U on the longitudinal coordinate x is very much smaller than the
dependence on the transverse coordinate y. One now assumes a basic flow
NACA TM No. 1218
U(y, t); Vn 0; P(x, y)
(21.4)
This basic flow, by itself, then satisfies the NavierStokes equations,
thus
U 1 aP a U
 +  = 2
at P ax 2
(21.5)
0
A disturbance movement which is also twodimensional is superimposed
upon this basic flow:
disturbance motion: u'(x,y,t); v'(x,y,t);
p'(x,y,t)
(21.6)
One then has as the
resultant motion: u = U + u'; v = 0 + v'; p = P +p' (21.7)
This resultant motion is required to satisfy the NavierStokes differential
equations and one investigates whether the disturbance motion dies away
or increases with time. The selection of the initial values of the dis
turbance motion is rather arbitrary, but it must of course satisfy the
continuity equation. The superimposed disturbances are assumed as "small",
in the sense that all quadratic terms of the disturbance components are
neglected relative to the linear terms. According to whether the dis
turbance motion fades away or increases with time, the basic flow is called
stable or unstable.
By insertion in the NavierStokes differential equations (3.18) one
obtains, neglecting the quadratic terms in the disturbance velocities
a+6+U 6u au + 1 aP
at5U+v. + +
t at a6x ay + x
av' i P
at Ox P ay
1 axp'
P 6x
vay
1 a V A v
p op
+ v
dy
Au )
>(21.8)
78 NACA TM No. 1218
If one now notes the fact that the basic flow by itself satisfies the
NavierStokes differential equations, equation (21.5), equation (21.8)
is simplified to:
6u' 2u' t U 1 6pt
+ U + v' + = VA u'
ft ax dy P dx
+ U vl i = + VA > (21.9)
6t dx P 6y
'x Py
+ =0
The pertinent boundary conditions are: Vanishing of the disturbance
components u' and v' on the bounding walls. From the system
equation (21.9) of three equations with three unknown quantities u',
v', p' one may at first eliminate p' by differentiating the first
equation with respect to y and the second with respect to x and then
subtracting the second from the first. This gives, with continuity taken
into consideration:
a2 +U a2u', + a2u a, aU
dty gxy2 atxx
> (21.10)
a3u + a3u' a3v' 63v'
+1 2 3 2 x 3/
In addition to this there is the continuity equation (21.9). There are
now two equations with two unknown quantities u', v'.
Form of the Disturbance Movement
For cases where the basic flow predominantly flows in one direction
as for instance boundarylayer or pipe flow, the disturbance motion is
assumed to be a wave progressing in the xdirection (= main flow direction),
the amplitude of which depends solely on y. The continuity equation of
the disturbance motion may in general be integrated by a disturbance
function for which the following expression may be used:
NACA TM No. 1218
lai(xct)
S(x,y,t) = cp(y)e( (21.11)
Where:
X = 2y/a the wave length of the disturbance (a = real)
C = C + c.
r 1
c = velocity of wave propagation
c = amplification factor; ci < 0: stable; c. > 0: unstable
m(y) = r (y) + 1 q.(y) = amplitude of the disturbance movement
From equation (21.11) one obtains for the components of the disturbance
movement
u cpl(y)e i(xct) 1
v ia.( (xct)
v = iap(.y)e
By substitution into equation (21.10) one obtains the following differ
ential equation for the disturbance amplitude c:
2 2\ V/ 2 4\
ia(Uc" cC" pU" + a ccp Ua c = vp"" 2a cp" + a m}
or
(U c)(p" a2%) U"cp = (cp"" 2a2p" + am) (21.14)
One introduces dimensionless quantities into this equation by referring
all velocities to the maximum velocity U of the basic flow (that is
m
for the friction layer the potential flow outside of the boundary layer
and all lengths to a suitable reference length ) (for instance, for
*The convenient complex formulation is used here. The real part of
the flow function, which alone has physical significance, is therefore
Re(l) = eCit cos a (x ct ip sin a (x crt (21.12)
NACA TM No. 1218
the boundary layer flow, the boundary layer thickness). Furthermore,
differentiation with respect to the dimensionless quantity y/S will
be designated by a prime mark (').
One then obtains from equation (21.14)
(U c)(q" a 2q) U" i ((,"" 2a2cp + am4) (21.15)
aR
DISTURBANCE DIFFERENTIAL EQUATION
where R = Um. This is the disturbance differential equation for the
amplitude c of the disturbance movement. The boundary conditions are,
for instance, for a boundary layer flow
y = 0 (wall): u' = v' = 0: p = c' = 0
(21.16)
y = : u' = v' = 0: p = 9' = 0
The stability investigation is an eigenvalue problem of this differential
equation for the disturbance amplitude p(y) in the following sense: A
basic flow U(y) is prescribed which satisfies the NavierStokes differ
ential equations. Also prescribed is the Reynolds number R of the basic
flow and the reciprocal wave length a = 2ir/ of the disturbance movement.
From the differential equation (21.15) with the boundary conditions
equation (21.16) the eigenvalue c = cr + i ci is to be determined. The
sign of the imaginary part of this characteristic value determines the
stability of the basic flow. For ci < 0 the particular flow (U, R)
is, for the particular disturbance a, stable; for ci > 0, unstable.
The case ci = 0 gives the neutrally stable disturbances. One can
represent the result of the stability calculation for an assumed basic
flow U(y) in an a, Rplane in such a manner that a pair of values
cr, ci belongs to each point of the a, Rplane. In particular the
curve ci = 0 in the a, Rplane separates the stable from the unstable
disturbances. It is called the neutral stability curve (fig. 103). In
view of the test results one expects only stable disturbances to be present
at small Reynolds numbers for all wave lengths a, unstable disturbances,
however, for at least a few a at large Reynolds numbers. The tangent
to the neutral stability curve parallel to the aaxis gives the critical
Reynolds number of the respective basic flow (fig. 103).
NACA TM No. 1218 81
Methods of Solution and General Properties of the Disturbance
Differential Equation
Since the stability limit (c, = 0) is expected to occur at large
Reynolds numbers, it suggests itself to suppress the friction terms in
the general disturbance differential equation and to obtain approximate
solutions from the socalled frictionless disturbance differential
equation which reads
(U c) (p" a 2) U"p = 0 (21.17)
Only two of the four boundary conditionsequation (21.16),of the complete
disturbance differential equation can now be satisfied since the friction
less disturbance differential equation is of the second order. The
remaining boundary conditions are:
y = 0: v' = 0, p = 0; y = m: v' = 0: q = 0 (21.18)
The cancellation of the friction terms in the disturbance differential
equation is very serious, because the order of the differential equation
is thereby lowered from 4 to 2 and thus important properties of the
general solution of the disturbance differential equation of the fourth
order possibly are lost. (Compare the previous considerations in
chapter IV concerning the transition from the NavierStokes differential
equations to potential flow.)
An important special solution of equation (21.17) is the one for
a constant basic flow, U = constant, which is needed for instance for
the stability investigation of a boundary layer flow as a joining
solution for an outer potential flow. One obtains from equation (21.17)
for
U = constant: cp = e
However, due to the boundary conditions for ( at y = w, the only
permissible solution is
p = e (21.19)
We shall prove at first two general theorems of Rayleigh on the
neutral and unstable oscillations of the frictionless disturbance
differential equation.
NACA TM No. 1218
Theorem I: The wave velocity cr for a velocity profile with
U"(y) 0 must, for a neutral oscillation (ci = 0, c = Cr), equal
the basic velocity at a point so that there exists within the flow a
point U c = 0.
Proof: (indirect) One makes the assumption c > U (= maximum
velocity of the basic flow). One then forms from equation (21.17) the
following differential expressions:
2 U"
L(m) = p" a = 0
U c
Uc
2 U" 
L(p) = a p  cp = 0
U c
(21.20a)
(21.20b)
L(cp) signifies the expression obtained from L(C), if one inserts
everywhere the conjugate complex quantities. Because of the boundary.
conditions
y = 0: p = = 0
y = : = = 0
One forms further the expression
it between the limits y = 0 and y = oo.
y = co, since for large y, p N e'aY.
Ji must then be
qpL(p) + p L(p) and integrates
The integrals may be taken up to
Because of equation (21.20a, b)
(21.21)
Jl = .L(cp) + cp L((p) dy = 0
y=o
After insertion of equation (21.20a and b) results, because c = c,
=1
y=o
(P 'p" + q,"p 2ami 2 U" dy = 0
U  c
and
NACA TM No. 1218
J = q p + q'qP 2 qP'p'dj 2 a2 + u ) dy = 0
1 L Jo 1 U c)
o o
The first term vanishes due to the boundary conditions, hence there
remains
Jl = 2 q'p'' + U U" C) dy = 0 (21.22)
y=o 0 
p5'c as well as q are positive throughout; if U" 0 and c > U ,
U"/U c 0 and hence the integrand in equation (21.22) is positive
throughout. Thus the integral cannot become zero. The assumption made
at the beginning c > Um therefore leads to a contradiction.
For basic flows with U" O, as for instance boundarylayer flows
in a pressure drop, the wave propagation velocity therefore must be smaller
than Um for neutral disturbances. Hence a point U c = 0 exists
within the flow. This point is a singular point of the frictionless
disturbance differential equation (21.17) and plays as such a special role
for the investigation of this differential equation. The wall distance y
at which U c = 0 is called y = yK = critical layer.
This first Rayleigh theorem proved above applies as shall be noted
here without proof in the same manner to flows with U" > 0.
Sixteenth Lecture (March 23, 1942)
Theorem II: A necessary condition for the presence of amplified
oscillations (ci >0) is the presence of an inflection point within the
basic flow (U" = 0).
Proof: (indirect) According to assumption, ci / 0; thus
U c / 0 for all y. With L(qp) and L(q() one forms, according to
equation (21.20a) and (21.20b), a similar expression as before. This
latter, integrated from y = 0 to y = o, must again give 0, thus
J2 = [~ L(p) q L(p) ] dy = 0 (21.23)
Ly=o
NACA TM No. 1218
By substitution according to equation (21.20a and b) results with
S= c c.
r 1
0o
jC U
J2 TM = dy = 0
c U c
y=O
or
C U"
J2= T 2 c.  (p dy 0 (21.24)
o Uc2
The first term again vanishes because of the boundary conditions. Since q
is positive throughout and IU cl j 0, the integral can only vanish if
U" 'changes its sign, that is, an inflection point of the velocity profile
U" = 0 must be present within the flow. It has, therefore, been proved:
In order to make the presence of amplified oscillations possible, an 
inflection point must exist in the velocity profile of the basic flow, or,
expressed briefly, such oscillations are possible only for inflection
point profiles.
Later on Tollmien (reference 110) proved that the presence of an
inflection point is not only a necessary but also a sufficient condition
for the existence of amplified oscillations. Hence the following simple
statement is valid: Inflection point profiles are unstable. It must be
mentioned that all these considerations apply in the limiting case R~o
since the proofs were obtained from the frictionless disturbance differ
ential equation.
We know from our previous considerations about the laminar boundary
layer that inflection point profiles always exist in the region of pressure
rise, whereas in the pressure drop region the boundary layer profiles are
always without an inflection point, (fig. 104). Hence we recognize that
the pressure rise or pressure drop is of decisive significance for the
stability of a boundary layer flow.
The converse of the theorem just set up is also valid, namely, that
for R velocity profiles without inflection point are always stable.
From this, however,'one must not conclude that profiles without inflection
point are stable for all Reynolds numbers. A closer investigation for
Reynolds numbers of finite magnitude shows that there profiles without an
inflection point also become unstable. One is faced with the peculiar
fact that the transition from Re = to Renumber of finite magnitude,
that is,the addition of a small viscosity to a frictionless flow,has a
NACA TM No. 1218
destabilizing effect, whereas one intuitively expects the opposite. As
later considerations will show in more detail, the typical difference
between the neutral stability curves of a basic flow with and without
inflection point appears as represented in figure 105. For the velocity
profile without an inflection point the lower and the upper branch of the
neutral curve have, for Roo, the same asymptote a = 0. For the
velocity profile with inflection point the lower and upper branch of the
neutral curve have, for R>o, different asymptotes so that for R =c
a certain wave length region of unstable disturbances exists. Furthermore
the critical Reynolds number is smaller for velocity profiles with an
inflection point than for those without an inflection point.
Hence it is to be expected for very large Renumber, to a first,
very rough approximation, that the transition point in the boundary layer
of a body lies at the pressure minimum. Figure 106 shows schematically
the pressure distribution for a rather strongly cambered wing profile at
a small lift coefficient. The transition point would be expected in this
case just behind the nose on the pressure side, slightly more toward the
rear on the suction side.
Solution of the Disturbance Differential Equation
In order to perform the actual calculation for the boundaryvalue
problem just formulated, one needs at first a fundamental system
(pl,' ...... p of the general disturbance differential equation (21.15).
One imagines the basic flow U(y) given in the form of a power series
development:
U"
U(y) = U' y + o y2 + (21.24)
S2!
If one introduces this expression into equation (21.15) and then wants
to construct a solution from the complete differential equation which
satisfies the boundary conditions (equation (21.16)), one encounters
extreme difficulties of calculation, due to the two conditions to be
satisfied for y = oo. In order to obtain any solution at all, one has
to make various simplifications. The simplifications concern:
1. The basic flow: Instead of the general Taylorseries
equation (21.24) one takes only a few terms, thus for instance a linear
or a quadratic velocity distribution.
2. The disturbance differential equation: For calculation of the
particular solutions the disturbance differential equation is considerably
simplified.
Regarding 1, it should be noted that linear velocity distributions
frequently have been investigated with respect to stability, as for
NACA TM No. 1218
instance the Couette flow according to figure 102 or a polygonal approxi
mation for curved velocity profiles according to figure 107. This facili
tates the calculation due to the fact that then the singular point
U c = 0 is avoided in the frictionless disturbance differential
equation (21.17) for neutral disturbances. However, all investigations
with linear velocity distributions (references 104, 105, 106) were unsuc
cessful with the frictionless as well as with the complete differential
equation. No critical Reynolds number resulted. When one later took
for a basis parabolic profiles, these negative results became intelligible.
One must, therefore, take at least a parabolic distribution as a basis
for the basic flow.
Regarding 2, it should be noted that one can provide approximate
solutions for the solutions of the complete differential equation (21.15)
from the frictionless differential equation (21.17) since the solutions
are required only for large Renumber R. The frictionless differential
equation however can yield no more than two particular solutions; two
more have to be calculated, taking the largest friction terms in
equation (21.15) into consideration.
The course of the calculation for the particular solutions will be
briefly indicated. One limits oneself to neutral disturbances, assumes
a parabolic velocity distribution, and imagines the latter developed in
the neighborhood of the critical layer.
y = YK: U c =UCr =0
U" 2
U c = U' ( 7) ( K) (21.25)
The first pair of solutions P1 P2 is then obtained from the friction
less disturbance differential equation (21.17) by substitution of
equation (21.25). According to known theorems about linear differential
equations with a singular point a linearly independent pair of solutions
has the form
U1 (21.26)
S2 =2 + (y Y ) og (y YK) Pl(
P1 and P2 are power series with a constant term different from zero.
The particular solution T2 is especially interesting.
NACA TM No. 1218
P'2 for y = yK
That is, the u'component of the disturbance velocity becomes infinitely
large in the critical layer. This can also be understood directly from
the frictionless disturbance differential equation (21.17). According
to equation (21.17)
U 
Uc
or
1
IY
P' ~ log (y YK), If U"K 1 0
This singular behavior of the solution (p in the critical layer stems
of course from neglecting the friction. The frictionless differential
equation here no longer gives a serviceable approximation. In the
neighborhood of the critical layer the friction must be taken into
consideration. Moreover, there is another inconvenience connected with
the c2. For fulfillment of the boundary conditions one requires the
solution for y yK > 0 as well as for y yK < 0. However, for cp2
it is at first undetermined what branch of the logarithm should be chosen
at transition from y yK > 0 to y yK < 0. This also can be clarified
only if in the neighborhood of. y yK, at least, the large friction terms
of the complete differential equation (21.15) are taken into consideration.
The details of the calculation will not be discussed here. The calcu
lation leads, as Tollmlen (reference 109) has shown, to the result that
one obtains for the solution T2 a socalled transitionsubstitution in
the critical layer which appears as follows:
y yK >0:
y yK <0:
U"
92= p2( K) + U (y K) 1(y lo) l y yK)
K
2 = P2( )+ e (y PK(y K l y YK l i
K /
(21.27)
If one writes, according to this, the complete u'component, then in the
neighborhood of y yK:
88 NACA TM No. 1218
U"
y yK > 0; ut = .. + K Zog C (Y cos(a3 3t)
U"
< 0: u' = ... .. + log ly y cos (a t) (21.28)
U"
K
+  sin (ax Ot)
U'
One obtains therefore in the critical layer a phase discontinuity for the
u'component. This is retained even in going to the limit, R o It
is lost, however, if one neglects the curvature of the basic flow U"
or if one operates only with the frictionless differential equation. This
phase discontinuity is very significant for the development of the motion.
The loss of the phase discontinuity is the reason that stability investi
gations neglecting the curvature U" or operating only with the frictionless
differential equation remain unsuccessful.
With this friction correction in the critical layer the pair of
solutions (p, p2 is sufficiently determined. By taking the friction
terms in equation (21.15) into consideration, one then obtains a second
pair of solutions (p3, q which can be represented by Hankel and Bessel
functions. Of these'two solutions (4 tends very strongly towards
infinity and is therefore not used because of the boundary conditions,
equation (21.16). c3 tends, for large y, towards zero.
The Boundary Value Problem
The general solution as a linear combination of the four particular
solutions is:
'P = C(1 + C2q2 + C3( + C4 (21.29)
Let us consider in particular the case where a boundary layer profile is
investigated with respect to stability. For this case the boundary value
problem can be somewhat simplified. The previous considerations showed
that in the disturbance differential equation the friction essentially
needs to be taken into consideration only in the neighborhood of the
critical layer; also, of course, at the wall, because of nonslip. The
critical layer is always rather close to the wall; hence for y > 6,
NACA TM No. 1218 89
where U = U = constant, one may use the frictionless solution which is
according to equation (21.19) 9= ey. Thus the condition that the
solution for y = 5 joins the solution for U = constant is
9p' + a 9P = 05 = 0 (21.30)
This mixed boundary condition is therefore to be set up on the outer edge.
Furthermore, the particular solution 94 is a priori eliminated in the
general solution (equation (21.29)), since it grows, for positive y yK,
beyond all limits; thus C4 = 0. Hence there remains for the boundary
value according to equation (21.16)
C 9CPP C 92 + C 9 = 0
1 lo 2 2o 3 30
C '1o + C29' C' 3 = 0 (21.31)
l15 + C228 + C3 =
A further simplification takes place because of the fact that because of
the rapid fading away of the solution 93 on the outer edge y = 5, the
solution 015 already practically equals zero. In the third equation of
equation (21.31) 03. may therefore be cancelled. The boundary value
problem actually to be solved is, therefore,
Slo 2o 30
s'lo '2o 9V2o = 0 (21.32)
15 25 0
This determinant gives the eigenvalue problem indicated above, which
requires as has been said before the solution of the following problem:
Given
NACA TM No. 1218
1. basic flow U(y)
2. Reynolds number Re = Um6/v
3. wave length of the disturbance a = 2x/X
One seeks from equation (21.32) the pertinent complex eigenvalue
c = cr + i ci. Therein cr gives the velocity of wave propagation and
ci the amplification or damping.
Equation (21.32) may formally be written in the form:
U' U" ... .) = 0
O O
(21.33)
where equation (21.33) signifies a complex equation, hence is
to two real equations
f a, c c., R;
U' U" =
o 0 /
equivalent
(21.34)
f2(a, r, c, R; U' U"o, .) =
If one imagines for instance cr
one obtains one equation between
eliminated from these two equations,
a, R, ci:
g8(m, ci, R; U'o, U", .) = 0
From this equation
The constants U'o,
if equation (21.35)
ci can be calculated as a function of a and R.
U"o, are parameters of the basic flow. Thus,
is assumed solved with respect to ci,
i= 2 (a, R; U'o, U"o, .
(21.36)
Finally one obtains from it, for the neutral disturbances ci = O, a
curve in the a, Rplane, given by the equation
(21.37)
g2a, R; U' U .
(21.35)
F(a, c r, ci, R;
NACA TM No. 1218
This is the sought for neutral stability curve (compare figure 103), which
separates the unstable from the stable disturbances and also yields
the theoretical stability limit, that is, the critical Reynolds number
Recrit'
The performance of the calculation, here only indicated, is analyt
ically not possible since the quantities a, cr, R enter into the
determinant, equation (21.32), in a very complicated mariner. One has
therefore to resort to numerical and graphical methods. The critical
Reynolds number is very largely dependent on the form of the velocity
profile of the basic flow, in particular on whether the velocity profile
of the basic flow has an inflection point, thus on U"(y).
The critical Reynolds number found from such a calculation gives
exactly the boundary between stability and instability, hence the first
occurrence of a neutrally stable disturbance. In comparison with the
transition point of test results it is therefore to be expected that the
experimental transition point appears only for larger Reynolds numbers
where an amplification of the unstable disturbance has already occurred.
c. Results
A few results of such stability calculations will be given. The
completely calculated example concerns the boundary layer on the flat
plate in longitudinal flow with the laminar velocity profile according
to Blasius (compare chapter IXa). In figure 108 the streamline pattern
of this plate boundary layer with the superimposed disturbance movement
is given for a special neutral disturbance. Figure 109 shows, for the
same neutral disturbance, the amplitude distribution and the energy
balance. Since the disturbance in question is neutral, the energy
transfer from the main to the secondary movement is of exactly the same
magnitude as the dissipation of the energy of the secondary movement.
Figure 110 shows the neutralstability curve as result of the stability
calculation according to which the critical Reynolds number is referred
to the displacement thickness 8* of the boundary layer
(Um5*/V)crit = 575. The connection between displacement thickness 5*
and length of run x is for the laminar boundary layer according to
equation (9.21)
m 1.73 \f
Thus a critical Reynolds number formed with the length of run x
(Umx/v)crit = 1.1 X 105 corresponds to the critical Renumber
(UmS*/V) = 575. The critical number for this case observed in tests
crit
NACA TM No. 1218
was 3 to 5 105. It was explained above that it must be larger than
the theoretical number. Furthermore, figure 110 shows that at the
stability limit the unstable wave lengths are of the order of magnitude
k = 55. The unstable disturbances thus have rather long wave lengths.
This calculation, carried out by Tollmien (reference 109) for the
flow without pressure gradient was later applied by Schlichting
(reference 114, 115) to boundary layer flows with pressure drop and
pressure rise. The boundary layer profiles with pressure rise and
pressure drop can be represented in a manner appropriate for the stability
calculation as a oneparameter family with the form parameter Xp 4
according to Pohlhausen's approximate calculation. One then obtains for
each profile of this family a neutralstability curve as indicated in
figure 111. Hence the critical Reynolds number (Umb*/v)crit is a
function of the form parameter Xp 4 according to figure 112. In
retarded flow (Xk 4 <0) the critical Renumber is smaller than for the
plate flow (X = 0), for accelerated flow (XP4 > 0) it is larger.
With this result of a universal stability calculation the position of the
theoretical transition point may be determined conveniently for an
arbitrary body shape (plane problem) in the following manner: At first,
one has to calculate for this body the potential flow along the contour,
furthermore one has to carry out, with this potential flow, a boundary
layer calculation according to the Pohlhausen method. This calculation
yields the displacement thickness and the form parameter Xp4 as
functions of the arc length along the contour, in the form
S =_ fl(s) and p f2(s)
Since in general there exists, accelerated flow at the front of the body
and retarded flow at the rear Xp4 decreases from the front toward the
rear. By means of the universal stability calculation according to
figure 112 one may determine a critical Reynolds number (Um*/V)crlt
for each point of the contour. The position of the transition point
for a prescribed Ren.umber Uot/v is then given by the condition
UmS* (Um*"
crit v v crit
Figure 113 shows, for the example of an elliptic cylinder*, how to find
the transition point. The curve (UmS*/V)crlt decreases from the front
*The boundary layer calculation for this elliptic cylinder was given
in figUre ,2.
NACA TM No. 1218
toward the rear; the curve Um6*/V for a fixed Renumber Uot/v increases
from the front toward the rear. The intersection of the two curves gives
the theoretical transition point for the respective Renumber Uot/v.
By determining this point of intersection for various Uot/v one obtains
the transition point as a function of Uot/V. The result is represented
in figure 114. The transition point travels with increasing Renumber
from the rear toward the front; however, the travel is considerably
smaller than for the plate in longitudinal flow which is represented in
figure 114 for comparison. Finally figure 115 shows the result of such a
stability calculation for four different elliptic cylinders in flow
parallel to the major axis. The shifting of the transition point with
the Renumber increases with the slenderness of the cylinder. For the
circular cylinder the shifting is very slight, which is caused by the
strongly marked velocity maximum. As a last result, figure 116 shows the
travel with Renumber of the transition point on a wing profile for various
lift coefficients. The profile in question is a symmetrical Joukowsky
profile with lift coefficients ca = 0 to 1. With increasing angle of
attack the transition point travels, for fixed Renumber toward the front
on the suction side,toward the rear on the pressure side. (compare the
velocity distirbutions for this profile, given in figure 54.) One recog
nizes that the shift of the transition point with the lift coefficient
is essentially determined by the shift of the velocity maximum.
The last examples have shown that it is possible to calculate
beforehand the position of the transition point as a function of the
Renumber and the lift coefficient for the plane problem of an arbitrary
body immersed in a flow (particularly a wing). Regarding the comparison
with test results it was determined that the experimental transition
point always lies somewhat further downstream than the theoretical tran
sition point. The reason is that between the theoretical and the experi
mental transition points lies the region of amplification of the unstable
disturbances. This amplification also can be calculated on principle
according to methods similar to those previously described. (Compare
Schlichting (reference 112) where this was done for the special case of
the plate in longitudinal flow.) Presumably one can obtain a still closer
connection between the theoretical instability point and the experimental
transition point by applying such calculations to the accelerated and
retarded flow.
CHAPTER XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION
LAYER ACCORDING TO THE METHOD OF GRUSCHWITZ (REFERENCE 78)
a. Integration of the Differential Equation of the
Turbulent Boundary Layer
In order to integrate the system of equations (18.16), one
first introduces dimensionless variables. One refers the lengths to the
NACA TM No. 1218
wing chord t and the velocity to the free stream velocity Uo,
x 9t = t*. = 2
S= x; i = = I
t t p 2'
e te 2 o
Hence the system of equations (equation (18.16)) may be written:
thus:
(22.1)
 + 0.00894 =
dx* 9*
+ 2 1 + H) 7
dx* 2
First, the second
and H, namely
0.00461 \r I__
1
x pU2
U dx* pU2
equation is solved with constant values for
S= 0.002; H = 1.5
p U2
The first approximation S *(x*) obtained from that is then substituted
in the first differential equation. From the latter one obtains a first
approximation tl*(x*) and from that, according to equation (22.1),
nl(x*). With Tl(x*) one determines according to figure 92 the course
of H(() and corrects T according to equation (18.15). Then one
obtains from the second differential equation a second approximation
2*(2),. etc.
For the solution of the differential equations one uses the isocline
method which can be applied for the present case, according to Czuber,
in the following manner: Both differential equations have the form:
d + f(x)y = g(x)
dx
(22.3)
As can be easily shown, this differential equation has the property that
all line elements on a straight line x = constant radiate from one point.
The coordinates of this point (= pole) are:
(22.2)
NACA TM No. 1218
=x + 1 x X) (22.4)
f(x) f(x)
Thus one has only to calculate a sufficient number of these poles and
can then easily draw the integral curve.
Figure 93 indicates the result of such a calculation for the profile
J 015; ca = 0. The calculation of the laminar boundary layer for the
same profile was performed in chapter XII, figure 49, table 6.
Initial values: The transition point was placed somewhat arbitrarily
at the velocity maximum of the potential flow (x/t = 0.141). It was
assumed that:
Ut 6
Re = 0 = 10
V
For the laminar boundary layer was found:
= 0.141: V = 1.56 (table 6)
t t
Hence there results, with 5*/S = 2.55;
x = 0.141: = 0.611 x 13 (table 8)
The corresponding I value was assumed to be
n = 0.1 (table 8)
Calculation to the second approximation suffices. The result is compiled
in table 8 and figure 93. A turbulent separation point does not exist
since I remains below 0.8. From the variation of the shearing
stress T along the wing chord the drag coefficient of the surface
friction may be determined:
W =2 b To dx (x = measured along chord)
NACA TM No. 1218
2
or cw = W/2 b t j Uo
W =j o 0
cw = d x
U 2 t
oo
The evaluation of the integral gives
c. = 0.0090
b. Connection Between the Form Parameters sT and
(22.5)
H = 3/6*
of the Turbulent Boundary Layer
According to Pretsch (reference 80) one may also represent analyti
cally the relation between the form parameters = 1 (u ~/UT)2 and
H = 8*/8 which was found empirically by Gruschwitz, compare figure 92.
A power law is set up for the velocity distribution, of the form:
(22.6)
with n = 1/6,
Hence results:
1/7, 1/8 according to the experiments so far.
1 z dz =
z n +1
8*
5
y/&=0
Furthermore:
(22.7)
p1
p1
4 ^
Ly/&=0
*f 1
z=0
/:=o1
n) dz
z) dz
(22.8)
n z 2ndz = 1= 1 n
z n + 1 2n + 1 (n + 1)(2n + 1)
U = =
u y
nj s
zn
u 5 1 0
(, O
