On the design of airfoils in which the transition of the boundary layer is delayed


Material Information

On the design of airfoils in which the transition of the boundary layer is delayed
Series Title:
Physical Description:
74 p. : ill. ; 27 cm.
Tani, Itiro
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Laminar flow   ( lcsh )
Aerofoils   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


A method is presented for designing suitable thickness distributions and mean camber lines for airfoils permitting extensive chordwise laminar flow. Wind tunnel and flight tests confirming the existence of laminar flow; possible maintenance of laminar flow by area suction; and the effects of wind tunnel turbulence and surface roughness on the promotion of premature boundary-layer transition are discussed. In addition, estimates of profile drag and scale effect on maximum lift of the derived airfoils are made.
Includes bibliographic references (p. 40-43).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Itiro Tani.
General Note:
"Report No. NACA TM 1351."
General Note:
"Report date October 1952."
General Note:
"Translation of "Kyōkaiso no Sen̕i o okuraseru Yokugata ni tuite." Report of the Aeronautical Research Institute, Tokyo Imperial University, No. 250 (vol. 19, no. 1), Jan. 1943."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778659
oclc - 86078705
sobekcm - AA00006201_00001
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Full Text

S / "---" ,





By Itiro Tani


1. In high speed flight conditions, the drag of an airfoil is
almost exclusively due to skin friction. Therefore, if further reduc-
tion in drag is desired, it is necessary to delay as much as possible
the transition from laminar to turbulent flow in the boundary layer along
the surface, thus decreasing the extent of the turbulent boundary layer
which gives considerable skin friction. As the factors that may affect
the transition, we will consider the stream turbulence, the surface
roughness, the surface pressure distribution, and so on. In actual
flight conditions, however, the effect of turbulence seems to be unex-
pectedly small, so that, so far as smooth surfaces are concerned, there
remains only the shape of the airfoil section in relation to pressure
distribution as the most important factor affecting transition. We call
a laminar-flow airfoil that airfoil in which the shape of the section
is suitably designed so as to delay the transition of the boundary
layer flow.

2. It is evident that the laminar separation of the boundary layer
may cause the transition, as will be mentioned in the appendant part of
the paper, paragraphs 35-40. We cannot expect, therefore, to maintain
laminar flow beyond the separation point. Summarizing the results of
flight experiments on airfoils hitherto made refss. 8 to 12), we have
the conclusion that the observed transition coincides approximately with
the calculated laminar separation point at small Reynolds numbers, while
it moves upstream toward the minimum pressure point as the Reynolds
number increases. However, no example has ever yet been observed in
which the transition moves ahead of the minimum pressure point. We
therefore arrive at the supposition that the laminar-flow airfoil may be
most simply realized by designing the airfoil in which the minimum pres-
sure occurs well downstream.

'"Kyokaiso no Sen'i o okuraseru Yokugata ni tuite." Report of the
Aeronautical Research Institute, Tokyo Imperial University, No. 250
(vol. 19, no. 1), Jan. 1943.

NACA TM 1351



3. Following Professor Moriya (ref. 13), we write the coordinate
along the chord in the form x = (1 + cos t), and assign x = 0,
S= i to the leading edge, and x = 1, 5 = 0 to the trailing edge.

Expressing the ordinate of the mean camber line by M = an cos nt,

and the half-thickness measured normal to the chord by T = b sin nt 1
the pressure distribution around the airfoil in the two-dimensional
potential flow is given by

cos ia-

+ sin a -

sin g +

(1 cos 5)

nan (1 cos nt)

+ nan sin nE


; 7 nbn sin nj

n C
+ nbn cos n

- sin2 t + J;
I 1

nan sin nS +

nbn cos n 2

1 -
a 1 fAc + eB jj + sin a -x

1 + (fAs eBc)2

q= -

= 1-

1The ordinates of the upper and lower surfaces are given by M + T
and M T, respectively.

NACA TM 1351

where a 'is the angle of attack, p is the pressure acting on the air-
foil surface, measured from the static pressure of the undisturbed stream,
and q is the dynamic pressure of the undisturbed stream. We assume that

the trailing edge is sharp, so that I nbn = 0. We limit the range of
the variables t between 0 and o, and assign the upper and lower
parts of the double sign for the upper and lower surfaces, respectively.
Writing f for the maximum value of M (the maximum camber) and e for
the maximum value of 2T (the maximum thickness), we put

-* 1 cos n. r-" sin nS dM
fAc = -2 nan fAs = -2_ nan = -
1 sin t 1 sin t dx

eB sin n cos nt dT
eBs = 2 nbn eBc = 2 nbn -- -
1 sin k 1 sin t dx

The lift coefficient is given by

CL = 2r {sin a 2 nan cos Ma

We consider first only the thickness of the airfoil (the camber of
the center line will be considered in the next section, paragraphs 8
to 11). Namely, we consider the. symmetrical airfoil section set at zero
angle of attack, with a view to obtaining the minimum pressure well

4. We adopt as the typical example of the commonly used symmetrical
airfoils the NACA symmetrical airfoil (ref. 14)

T = e {1.4845E 0.6300x 1.7580x2 + 1.4215x3 0.5075x4}

The maximum thickness is located at x = 0.3, the leading-edge radius is
l.1e2, and the trailing-edge slope -(dT/dx)x=1 is 1.17e. The pressure
distribution for the case e = 0.1 is shown in figure 1. The minimum
pressure is located at x = 0.1, and the laminar separation point,
determined by the approximate method due to the author refss. 15 and 16),

NACA TM 1351

at x = 0.61. If the transition point of the boundary layer would not
move upstream beyond the laminar separation point, we might expect to
maintain a laminar boundary layer for more than half the surface of the
airfoil. The flight experiments hitherto made, however, appear to give
negative evidence for such a conjecture.

5. Now, in order to shift the minimum pressure backward, it is
required to shift the position of maximum thickness (x = m) backward.
For designing such airfoils, we represent the shapes of parts before
and after the maximum thickness by two algebraic expressions. For the
forward half (0 5 x < m)

T = e 2 + hlx + hX2

while for the rear half (m 5 x 5 1)

T = e .01 + dl(l x) l 2 + d2(- 3(1 x)3}


2 3hm 2_h 1
hl = h2 =
2m 2m2

1.47 2dl(l m) d1(l m) 0.98
d2 = d3 =
(1 m)2 (1 )3

and we assign arbitrary values for three parameters, m, h (= leading-
edge radius a- e2), and dl (= trailing-edge slope .- e). Although the
method has the drawback that the two expressions give different values
of d2T/dx2 at x = m, where dT/dx becomes zero, we nevertheless
adopt it because we are in a position to vary the forward and rearward
parts most simply and independently.

6. First, we fix the forward half with m = 0.5 and h = 0.5, and
vary the rear half by giving dI the values 1.7, 2.0, 2.5, and 3.0,
respectively. The shape of the section and the pressure distribution
for e = 0.1 are shown in figure 2. We find from this result that, as

NACA TM 1351

dl increases, the minimum pressure point moves backward and the gradient
of pressure rise following the minimum pressure steepens. We also find
that the pressure distribution in the neighborhood of the minimum pres-
sure exhibits a wavy indentation when the value of dl is too small or
too large, and that there exists a certain value of di for which the
pressure distribution is flat and smooth. Such a value of dl is about
2.5 in this case. We therefore fix the rear half with d1 = 2.5, and
vary the forward half by giving h the values 0.35, 0.50, 0.70, and
1.05, respectively. The shape of the section and the pressure distri-
bution for e = 0.1 are shown in figure 3. From this comparison, we
find that the negative pressure bump immediately behind the leading
edge decreases as h decreases, and that the maximum permissible value
of h is about 0.7.

The effect of thickness is shown in figure 4, in which curves of
pressure distribution are given for different values of e, 0.06, 0.10,
and 0.14, but for a fixed set of parameters, m = 0.5, h = 0.5, and
dl = 2.5. It is seen that the characteristics of the pressure distri-
bution do not materially change with thickness. There is, however, a
slight change in the pressure distribution, the maximum permissible
value for h slightly increasing as the thickness increases.

To see the effect of the position of maximum thickness, we give m
values ranging from 0.35 to 0.60, varying at the same time values of dl
and h so that the pressure distribution becomes flat and smooth. The
result of calculation is given in figure 5, which shows a considerable
change in the position of minimum pressure. The change is not purely
due to the effect of m, but it is at any rate to be noticed that the
value of m less than 0.4 is not sufficient for shifting backward the
minimum pressure, while increasing the value of m beyond 0.5 is of no
advantage, since the backward shift is then almost saturated, only the
adverse pressure gradient being increased.

7. From the results of calculation, we thus arrive at the conclusion
that m must be between 0.4 and 0.5 and h must be less than 0.7 in order
that the minimum pressure occurs well downstream. Smaller values of h
are desirable, but, on the other hand, we should like to make h as
large as possible, because a large value of h will be advantageous in
increasing the maximum lift coefficient and in preventing the inception
of adverse pressure gradient when the angle of attack is slightly changed.
Even if we give h the maximum permissible value 0.7, the leading-edge
radius amounts to only 60 percent of that for the conventional NACA sym-
metrical airfoil of the same thickness. In order to increase the leading-
edge radius, it is required to increase the thickness, which in turn is
accompanied by an increase in adverse pressure gradient following the
minimum pressure. The adverse pressure gradient should be kept within a

NACA TM 1351

certain limit, so that it becomes necessary to make a compromise between
conflicting requirements. Thus, we are no longer in a position to
require the farthest possible rearward location of the minimum pres-
sure. We should also use a value of dl which is somewhat smaller
than that mentioned previously.

Taking these requirements into account, we finally arrive at the
design of a series of symmetrical airfoil sections, the parameters of
which are given in the following table:

Although section I is the most ideal for delaying the transition, in
practice, its extraordinarily sharp nose and blunt tail are drawbacks.
On the other hand, section N is too much compromised. Sections K or L
seem to be suitable as laminar-flow airfoils for practical use. The
ordinates of these six sections are given in table 1, while the auxiliary
functions Bs and Bc associated with the pressure distribution (see
paragraph 3) are given in tables 2 and 3, respectively. The shapes of
the airfoil sections and the pressure distribution for e = 0.1 are
shown in figure 6.


8. A symmetrical airfoil set at zero angle of attack has no lift.
In order to obtain lift, the center line of the symmetrical airfoil must
be curved with a suitable camber. Since the effects of thickness and
camber are nearly additive with regard to the pressure distribution, the
mean camber line which maintains the nature of the pressure distribution
of the symmetrical airfoil will be such that it shall give a uniform dis-
tribution of pressure difference when the thickness is removed. Evidently,
the center of pressure is then located at x = 0.5, so that such a camber
line has the drawback that the travel of center of pressure is consider-
able. To reduce the travel of center of pressure, the uniformity of pres-
sure difference should be satisfied only in the forward part of the chord.
From the standpoint of designing the laminar-flow airfoil, however, it is

Position of
Section m h dl ii pressure
minimum pressure

I 0.500 0.35 2.384 0.63
J .500 .54 1.800 .55
K .475 .56 1.575 .51
L .450 .58 1.400 .47
M .400 .62 1.150 .37
N .350 .66 1.000 .24

NACA TM 1351

only required that the distribution of pressure difference is uniform
from the leading edge to that point corresponding to the minimum pres-
sure of the symmetrical airfoil.

9. When the angle of attack a is small, the expression for pres-
sure distribution given in paragraph 3 may be put into the form

(1 + eBs) (fAc


+ -x a) (fAs eBc)2j

1 + (fA + eBc)2

Since the effect of the term (fAs eBc) is very small, the quantity

fG = fAc + (- a

is required to be constant in order that the camber line shall not change
the nature of the pressure distribution of the symmetrical section. The
range of constancy is at least up to the position of minimum pressure of
the symmetrical section. Putting cos t = u = 2x 1, and considering
for simplicity the case when the minimum pressure is located at u = 0,
we prescribe that

G = constant = GO

G = GO(1 u2)m, m



0 : x 5 0.5, -1 6 u O 0

0.5 5 x 1, 0 5 u 1

See figure 7. Moreover, since

fG = -2 it nan

1 cos nt 1 cos t
+ a
sin t sin t

a cannot be arbitrary, but must be so chosen that the right side of the
equation does not become infinite at the leading edge, i = i. It is
given by

NACA TM 1351

a = 2 E: nan

where (') denotes that only odd integers should be taken for n. This
is the so-called ideal angle of attack due to Theodorsen (ref. 17).
Using the assumption of the thin wing theory, we neglect the terms eBs
and (fAs eBc). We then have

cL = 2(1 + a^)fGo0 = / sin2m+l di

S- l+ (l+m)om CL
S+ )(l + m),
4(1 + m)(1 + 4a)

a, CL
a = 2 T" nan + -2
1 2n

where Cm0 is the moment coefficient about the leading edge (positive
when nose up) at CL = 0. Although m = 0 corresponds to the case of
making G uniform up to the trailing edge, it seems to be impossible
to realize a finite pressure difference at the trailing edge. Moreover,
the quantity -CmO/CL (which represents the degree of center of pres-
sure travel) is as large as 0.25 in this case. If m > 0, G vanishes
at the trailing edge, and -CmO/CL decreases as m increases, tending
to 0 as m approaches m. m = o corresponds to the case when G = 0
in the rear half of the chord. Increasing the value of m, however,
steepens the pressure gradient, so the value of m from 3 to 5 seems to
be adequate.

NACA TM 1351

Now, since

S 00, CL CL
fG sin = 2 nan cos nt 2 nan + cos I + -
1 n1 2 21

the slope of the camber line having the prescribed distribution of G
is given by


0 r sin nS 1 P r dS'
-nan n =- 2 nan cos nS'
1 sin iJO 1 cos t cos '

CL 1 F
= 1 + log
2 1 1 + Cm

u +( 2)m 1g 1 u
- +u -u)log---
1 + u u

1 (1 u2)m (1 v2)m
u-v d iv + 2 nan
Ou v 1

The ordinate of the camber line is obtained by the integration

M X= M dx
M=0 dx

- nan may be determined by the condition that M = 0 at x = 1. We
call Dm the camber line thus determined. The equations for camber
lines for m = 0, 1, 3, 5, and -, namely DO, Dl 3, D, and D.,

10 NACA TM 1351

are given below, their important characteristics being summarized in the
table. It is to be noted that f is the maximum value of M, E is
the absolute value of the zero-lift angle, and E and a are measured
in radians.

1 1 (1 -
2 log 2

u) log (1 u) + (1 + u) log (1 + u)

(5 + u) log 2 -

3 (1 +

u) log (1 + u) +

1 u3 log Iu I-

(1 u)2(2 + u) log (1 u) +

(51 + 19u) log 2 -

3 (1

+ u) log (1 + u) +

1 u335 21u2 + 5u4)

log ul -

1 (1 u)4(16 + 29u + 20u2 + 5u3) log (1 u) +
(1 u2) (176 81u

-1- (1 u2)(176 81u 172u2 + 30u3 + 60u4)

m a/CL e/CL f/CL -CmO/CL

0 0 0.1592 0.0552 0.2500
1 .0380 .1211 .0711 .1750
3 .0609 .0983 .0790 .1213
5 .0703 .0888 .0816 .0979
O .1103 .0488 .0874 0

DO: M =



D3: -M =-
CL 51


- u2)

NACA TM 1351

D5: M = (949 + 437u) log 2 6r (1 + u) log (1 + u) +
CL 949 949

S-- 3(115 1386u2 + 990u 385u6 + 63u8) log ul -

-- (1 u)6(2 + 843u + 1218u2 + 938u3 + 378u4 +

63u5) log (1 u) + (1 u2) (35072 28535u 66088u2 +
113880 (

31680u3 + 68792u4 17430u5 36120u6 + 3780u7 + 7560u8)

D : 1 (1I + u) log 2 (1 + u) log (1 + u) + u log lul
0 f log 3 L J

Shapes of these camber lines are shown in figure 7, the ordinates
of them are given in table 4, and the auxiliary functions Ac and As
(see paragraph 3) and the pressure difference distribution G are given
in tables 5, 6, and 7, respectively.

10. The calculation made previously is only approximate, neglecting
the thickness. It is therefore desirable to check the result by actually
calculating the pressure distribution for the specified angle of attack
taking both camber and thickness into account. As an example, we con-
struct an airfoil by applying the thickness form K with e = 0.15 nor-
mal to the chord around the camber line D5 with f = 0.02 (the
resulting airfoil is designated as D5K 2015). We calculate the pres-
sure distribution by the formula of paragraph 3 for the optimum design
condition a = 0.990 and CL = 0.245. The result is shown in figurr 8.
The nature of the pressure distribution remains similar to that of the
symmetrical airfoil, so we may consider that the approximate determina-
tion neglecting thickness gives results sufficiently accurate for practi-
cal purposes.

11. In designing the camber line Dm, we have assumed for simplicity
that the pressure difference G is constant for u I 0. This corresponds
to the case when th- symmetrical airfoil has its minimum pressure in the

NACA TM 1351

neighborhood of u = 0. Therefore, the camber line Dm is adequate to
be combined with the symmetrical section J or K. If, however, the
symmetrical section is adopted in which the minimum pressure is located
further upstream, it is not only not necessary to maintain G constant
up to u = 0, but also of disadvantage because it makes it difficult to
reduce the value of -Cm0/CL.

To reduce the range over which
we may proceed in the following way.
should be constant from u = -1 to

S= u), we prescribe that
uI = (1 + 3u), we prescribe that

G = GO

G = G(1l u12)3

G should be maintained constant,
Assuming for instance that G
u = and using a new variable

for -1 5 u = 7,

for i 5 u 5 1,

l< <0

0 5 u, 5 1

The calculation may be performed similarly to the case of Dm. The
resulting camber line is designated as F3. The camber line lying in
the middle between F3 and D5 is also designed, and designated as E4.
Their important characteristics are given in the following table together
with those of D5. Other numerical data for these camber lines are given
in tables 4 to 7.

Camber line a/CL E/CL f/CL -Cm0/CL

D5 0.0703 0.0888 0.0816 0.0979
E4 .0752 .0840 .0813 .0859
F3 .0764 .0827 .0795 .0833

Since the camber lines E4 and F3 enable us to maintain G con-
stant up to the point x = 0.42 and x = 0.33, respectively, they are
adequate to be combined with the symmetrical sections L and M, respec-
tively. The pressure distribution is shown in figure 10 for the airfoil
obtained by applying the thickness form M with e = 0.15 around the
camber line F3 with f = 0.02. The optimum design condition corresponds
to a = 1.100 and CL = 0.252.

NACA TM 1351


12. In order to ascertain whether it is possible to prevent the
forward movement of the boundary layer transition by shifting the mini-
mum pressure on the airfoil surface, we have to perform experiments in
a low turbulence wind tunnel or on the actual airplane in flight. When
the Reynolds number is not too large, however, we can still use a con-
ventional wind tunnel in which the stream turbulence is relatively small.
So we made at first comparative measurements on two symmetrical airfoils,
NACA 0010 and L.B. 24 in the 1.5 m wind tunnel of the Aeronautical
Research Institute. L.B. 24 is a laminar-flow airfoil of 10 percent
thickness, already shown in figure 3. The theoretical pressure distri-
bution is also given in figure 11. The minimum pressure is located at
x = 0.64, and the laminar separation at x = 0.77. The wind tunnel was
of the lowest turbulence level available for the author, the critical
Reynolds number of a sphere being 3.66 X 105 and the transition Reynolds
number of a flat plate 1.05 X 103 (see paragraph 28). In order to raise
the Reynolds number as high as possible, unusually large models were
used. They were made of laminated mahogany, of highly polished surface,
of 0.8 m span, of 1.2 m chord, and fitted with end plates 1.3 m X 0.6 m.
Since the model was large compared with the size of the tunnel and the
end plates were not sufficiently large, the results for a given airfoil
may not correspond even approximately with those for the same airfoil in
an undisturbed two-dimensional flow. Our object, however, was merely
to ascertain the relation between pressure distribution and transition,
and it seemed reasonable to expect that the relation will not be seriously
affected by limitations in the conditions of the experiments. As a matter
of fact, marked difference was found in the calculated and measured dis-
tributions of pressure, the latter of which was measured along the median
section of the model with a static tube of 1 mm diameter'(fig. 12).2
This discrepancy, however, is immaterial, since our object was merely to
compare the two airfoils, both of which are affected quite similarly by
experimental limitations.

13. The angle of attack of the model was zero, and the wind speed
was varied from 6 to 40 m/s. The local drag of the median section was
determined from wake measurements, that were made in the section 11 cm
behind the trailing edge. Measurements of static and total pressures in
the wake were made, respectively, with a static tube of 2.5 mm external
diameter and a pitot tube with a flattened mouth of 0.65 mm external
depth and 2.6 mm width. The profile drag coefficient CDO was obtained

2The measured values are those for a Reynolds number of about 2 X 106.
The distribution of pressure changes but little with the Reynolds number.

13 .

NACA TM 1351

from the measured pressures by Jones' formula (ref. 18). Figure 13 3
presents CDO plotted against Reynolds number R referred to chord
length. For a lower range of R, the drag of L.B. 24 is higher than
that of NACA 0010, the reason probably being that a turbulent boundary
layer associated with the laminar separation is established at a higher
Reynolds number for the former airfoil than for the latter. For a higher
range of R, however, the condition is reversed, L.B. 24 giving a drag
less than half that of NACA 0010 for R higher than 2 x 100. This is
probably due to the fact that the transition may occur much later for
L.B. 24 than it does for NACA 0010, as also observed from the compari-
son of wake conditions for the two airfoils (fig. 14).4

14. In order to verify the aforementioned supposition, a pitot tube
with a flattened mouth of external depth 0.9 mm and width 2.7 mm was
placed in contact with the airfoil surface, and the wind speed, and con-
sequently the Reynolds number R, were determined at which the indicated
total pressure G* divided by the dynamical pressure q of the undis-
turbed stream begins to rise suddenly. The results are shown in fig-
ure 15. From this figure, the dependence of the transition point on
Reynolds number as shown in figure 16 is obtained. At the same Reynolds
number, the transition occurs much farther from the leading edge for
L.B. 24 than for NACA 0010. Even at the highest Reynolds number reached,
L.B. 24 hss a transition as far back as x = 0.80. This is somewhat
beyond the laminar separation point, x = 0.77, which is calculated from
the theoretical pressure distribution. However, this is not contradictory,
because the actual pressure distribution differs from the theoretical one
in a manner to delay the transition (fig. 12).

15. With further increase in Reynolds number, the transition may
move toward the leading cdgp, but it seems improbable that the transition
moves forward beyond the minimum pressure. It is highly desirable to
check this point also by wind tunnel experiments, but all the wind tunnels
now available to the author are of no use for making measurements at
sufficiently high Reynoids numbers, because the transition is prematurely

,In this figure, th. curves L and T represent the drag of a
flat plate whe -n the boundiiry layr i entirely laminar and entirely
turbulent, re3spT:tively. The curves r O and FF represent the drag
of airfoil NACA 0009 mpi siured in the INACA Variable-Density Wind Tunnel
and NIACA Full-ScaIl Wind Tunnel, respectively.
41n this figure -G and p are the total and static pressures in
th( wake, respectively, and G0 is the total pressure outside the wake,
all being measured from the static pressure of the undisturbed stream.
y is the distance across the wake, and t is the chord of the model.

NACA TM 1351

induced by the turbulence of the stream (see paragraphs 27 to 29). It
seems urgent to build a special wind tunnel of low turbulence level.
For the present, however, it is simplest to rely upon experiments in
actual flight. Such a hope of the author was fortunately realized by
the specially planned flight experiment, which was performed at the
Navy Aeronautical Technical Arsenal (ref. 19).

16. The airplane used for the experiment was a biplane; two portions
of the lower wing, each of 1.1 m span, were covered with the airfoil to
be tested. The test portions were of chord 2.4 m, made of Japanese Hinoki,
highly polished, and fitted with a partition fence of small'height at
both ends. Two test portions were placed symmetrically, pressure distri-
bution and wake measurements being performed on the starboard portion,
while the boundary layer was observed on the port portion. The airfoil
section was not one of the most appropriate design now considered, because
it was required to put it on the original section of the airplane, and,
moreover, to determine the section before completion of the final design
calculation. It has the following characteristics:

Mean camber line: M = 0.0667x(1 x)(l x + x2), f = 0.0125

Thickness distribution: e = 0.12, m = 0.45, h = 0.56, dl = 1.60

The camber line is similar to DO of paragraph 9, but there exists a
slight lack of uniformity of G in the neighborhood of leading and
trailing edges. The thickness distribution is similar to L of para-
graph 7, but the trailing edge slope is somewhat larger than L.

17. Results of flight experiments are summarized in figure 17
and 18. In figure 17, the section lift coefficient CL, obtained by
integrating the pressure distribution curve, is shown by.a broken line
plotted against the Reynolds number R referred to the flight speed
and chord length, and CL is again shown by a solid line plotted against
the profile drag coefficient CDO determined from the wake measurements.
In figure 18, the measured pressure distribution is shown in comparison
with the theoretical one (two-dimensional potential flow) having the
same value of CL. The transition points estimated from the change in
boundary layer velocity profiles are also marked. Generally speaking,
the measured pressure distribution agrees fairly well with the theo-
retical one, although a slight difference appears when CL becomes
large. An adverse pressure gradient is found on the lower surface when
CL is small, thus resulting in the transition point being observed
unexpectedly far forward. Such a discrepancy in pressure distribution
as compared with the theoretical one seems to be probably due to the
fact that the span of the test portion was not sufficiently large. As
a result, the profile drag coefficient CDO has the minimum value 0.0042

NACA TM 1351

at about CL = 0.26, which is larger than the value CL = 0.18 theo-
retically estimated on the assumption that the transition occurs far
back on both upper and lower surfaces. Therefore, the observed value
of CDO, although much smaller than that of the conventional airfoils,
seems to be still somewhat large when compared with the optimum case.
At any rate, however, no transition was found to occur upstream of the
minimum pressure. It is important to note that such experimental evi-
dence was obtained on an airfoil section in which the minimum pressure
is located further downstream than on the conventional one. This finding
will give valuable data to establish a basis for design of the laminar-
flow airfoils.


18. As mentioned previously, the results of flight experiments
seem to support the basis for the design of laminar-flow airfoils,
namely, the possibility of maintaining the boundary layer laminar at
least up to the minimum pressure point. It is interesting, therefore,
to estimate the drag of laminar-flow airfoils by assuming a laminar
boundary layer from the leading edge to the minimum pressure point and
a turbulent boundary layer downstream to the trailing edge.

For the laminar boundary layer, the momentum thickness is given by

o2 .-v [4 4s
S= ul ds

with a sufficient approximation (ref. 16), where ul is the velocity
outside the boundary layer and s is the distance measured along the
airfoil surface from the forward stagnation point. Writing t for the
chord length and V for the velocity of the undisturbed flow (velocity
of flight), and putting

Ul = UV a = t R = Vt

we have the nondimensional expression

NACA TM 1351


=0.44 a
RUa 0o


where the subscript a refers to the point of minimum pressure.
Applying then the solution due to Buri (ref. 20) for a turbulent
boundary layer assumed to extend from the minimum pressure point to
the trailing edge, we have the result

l_ ] T a

= 0.0162R-1/4 f U4 do

where the subscript b refers to the trailing edge.
values originally given by Buri are slightly modified
with measurements when applied to the flat plate.

The numerical
so as to agree

According to Squire and Young (ref. 21), the profile drag coeffi-
cient is given by

CD = (u )b3

where the subscripts u and I refer to the upper and lower surfaces,
respectively. The exponent 3.2 of Ub has been obtained by assuming
the ratio of displacement and momentum thicknesses equal to 1.4. But the
ratio seems to exceed 1.4 near the trailing edge, so we replace 3.2 by
3.4 with a view to improving the accuracy and at the same time to
simplifying the algebra. Since

Ob 3.4
T "b

-1/4 l b u 4/5
+ 0.0162R f U do

[(La)/4 17/4
t U

NACA TM 1351

we have

CDO = 0.07-1/5 R-3/8 + T)/5 + (LR-3/8 + T+ /d


L = 37Ua a U d 5/8 T = b U4 do

If the velocity distribution ul = UV is calculated by assuming the
potential flow of an ideal fluid, it is desirable to modify the distri-
bution to take account of the effect of separation near the trailing
edge. We tentatively modified the distribution of U such that the
Buri parameter r = (8/ul)(dul/ds)(ulo/V)1/ at the trailing edge for
the case when the boundary layer is assumed turbulent from the leading
edge, namely

r 0.0081 IdU2 b U4 do
Ub6 \do Jo

shall not become smaller than -0.06. In almost all the cases, values
of Ub thus modified are found in the range between 0.95 and 1.00.

19. Applying this method of calculation, the profile drag coeffi-
cient C O is estimated first for a series of symmetrical airfoils set
at zero angle of attack. The series consists of the six symmetrical
airfoils, I, J, K, L, M, N, as given in paragraph 7 and the NACA con-
ventional airfoil. Values of CDO at R = 2 x 107 for three different
thicknesses (maximum thickness in terms of chord e = 0.10, 0.15, 0.20)
are shown in figure 19 plotted against the position of minimum pressure.
CDO seems to decrease almost linearly as the minimum pressure is shifted
backward, the most ideal airfoil I giving a value about half of that of
the NACA conventional airfoil. If it is desired to realize a profile
drag of two-thirds of the conventional airfoil, it will be required to
use the symmetrical airfoil L with the maximum thickness located at
45 percent chord from the leading edge.

NACA TM 1351

Effect of camber is relatively small. If, for instance, the
center line of the symmetrical airfoil K with e = 0.15 is curved
into the camber line D5 with f = 0.02 (see paragraph 9), the esti-
mated increase in profile drag at the optimum angle of attack is only
0.0001. For f = 0.04, it is 0.0003.

Finally, we compare the laminar-flow airfoil with the most exten-
sively used airfoil, NACA 23012, for which the leading-edge radius is
0.0158, and the optimum lift coefficient corresponding to the minimum
profile drag coefficient is about 0.15. If we consider the'symmetrical
airfoil section K (h = 0.56) combined with the camber line D5, it is
necessary to use the thickness e = 0.15 in order to obtain the same
magnitude of leading-edge radius, and the camber f = 0.012 in order to
realize the optimum lift coefficient 0.15.5 Therefore we construct an
airfoil by applying the symmetrical form K with e = 0.15 normal to
the chord around the camber line D5 with f = 0.012. We call it
D5K 1215. The angle of attack corresponding to CL = 0.15 is 1.560
for NACA 23012 and 0.60 for D.K 1215. The pressure distribution for
that condition is shown in figure 20.

We then estimate CDO for the two airfoils by the method explained
previously. The results are shown by broken lines in figure 21. In
order to check the results, measured values taken from various sources
for the two airfoils and similar airfoils are also plotted in the same
figure by different marks. The mark o refers to the value obtained
by flight experiments on a smooth surface, and refers to that
obtained by wind tunnel experiments where the stream turbulence has no
effect on transition. The mark + refers to the flight experiment on
a rough surface, while X refers to the wind tunnel experiment where
the stream turbulence causes the transition to occur prematurely. There-
fore, only o and are adequate for our present purpose. Drawing
curves through these points and extrapolating to higher Reynolds numbers,
we find that the result agrees fairly well with the estimated values.
Therefore, we may consider that the method of estimating CDO is suffi-
ciently accurate at the Reynolds numbers corresponding to actual flight

5The calculation developed in paragraphs 8 to 11 refers to the poten-
tial flow of an ideal fluid, so that it gives the slope of lift curve
--= 2n. In real fluids, however, the slope of lift curve amounts to
only 80 to 90 percent of the theoretical value. If we take this effect
into account, we have to increase the necessary amount of f by 10 to
20 percent in order to realize the given lift coefficient. However,
such a slight change in the value of f will scarcely affect the esti-
mation of CDo.

NACA TM 1351

conditions. Comparison of two airfoils, laminar-flow and conventional,
also suggests the possibility of 40 percent reduction in profile drag
by using a fairly practical laminar-flow airfoil.


20. The fact that, so far as flight experiments with smooth wings
are concerned, the boundary layer transition occurs only in the region
of rising pressure, not only warrants the principle of designing the
laminar-flow airfoil by shifting the minimum pressure backward, but
also suggests the possibility of delaying the transition by using an
airfoil with uniform distribution of pressure. Therefore, in para-
graphs 21 to 23, the shape of such a symmetrical airfoil is determined
by a method similar to that used for designing the camber line of laminar-
flow airfoils, and the airfoil was examined by wind tunnel experiments.
In paragraphs 24 to 26, a calculation is made to inquire about the method
of sucking away the boundary layer over the region of rising pressure in
such a way that the boundary layer velocity profile shall remain the same
as that for the point of minimum pressure.

21. Consider the symmetrical airfoil set at zero angle of attack.
According to the formula of paragraph 3, the pressure distribution is
given by

00 sin n]
+ 2 nbn----
1 sin j
q 2

1 + L nbn cos nt
11 isin t

where p is the pressure acting on the airfoil surface, measured from
the static pressure of the undisturbed stream, q is the dynamic pres-
sure of the undisturbed stream, x = 1 (1 + cos S) is the coordinate
along the chord, and the half-thickness of the airfoil is expressed in
the form

T = bn sin nt

NACA TM 1351

If the thickness is sufficiently small, the square of

dT C cos n
= -2 nbn
dx 1 sin t

may be neglected, so that the condition of uniform distribution of pres-
sure is satisfied by putting all the coefficients bn, other than bl,
equal to zero, 'namely, by an elliptic section. In order to take the
thickness into account approximately, we substitute the value of dT/dx
for the elliptic section into the denominator of the expression for p/q.
Then, writing e for the maximum thickness in terms of the chord, we
0 2
sin +2 nbn sin n
p 1
q sin2t + e2cos2t


2 y nbn sin nt = sin t + B.sin2t + e2cos2t
S(- /2
where B is the constant value of (l p/q) We have therefore

NACA TM 1351

2 nbn sin nW'

sin k' dt'
cosB cos '

sin d' dt'
coB t COs '

1 x js -

sin B

SU B 2ku cos-1 e +
6t i

1 k2u2 log

(i + u) (1

- k2u + e 1- ku2)

+ k2u + e 1 k2u )

u = cos = 2x 1

k = 1 e2

Upon integrating we get

T = k cosle -

1- k2u2
-- log
\1 u2

( + u)(l

(1 u)(l

- k2u + e~l k2u2)
+ k2u + e\1 k2u2)

The integral is evaluated by a numerical method, and the value of the
constant B determined from the condition that T = le when u = 0.
The numerical results for three values of e are given in table 8, T
and dT/dx being expressed in terms of those for the elliptic section.

sin d = -


(l u)(l

2 -i1

HACA TM 1351

The shapes of the airfoils are shown in figure 22. The shape resembles
an elliptic section, although it is somewhat fuller at the ends. It will
also be seen that the constant B, as shown in the following table, is
slightly smaller than 1 + e, the maximum value of (1 p!q)1/2 for the
elliptic section. Since the values of (dT/dx) (-eu/l u2) are not far
different from 1, it seems to be sufficiently accurate to substitute
the value of dT/dx for the elliptic section into the denominator
of 1 p/q.

e B

0.1 1.097
.2 1.188
.3. 1.273

22. The uniform distribution of pressure requires, however, an
infinite pressure gradient at both leading and trailing edges. In order
to see to what degree such a sharp pressure gradient may be realized in
actual fluids, measurements were made on a model of the airfoil section
with uniform distribution of pressure with e = 0.1 (we call it U.P. 0010)
in the 1.5 m wind tunnel of the Aeronautical Research Institute. The
model was made of laminated mahogany, of 0.8 m span, of 0.8 m chord, and
fitted with end plates 1.3 m x 0.6 m. Measurements of pressure distribu-
tion, wake traverse and boundary layer transition were similar to those
already mentioned in paragraphs 12 to 14.

The pressure distribution along the chord is shown in figure 23
for three values of R, the Reynolds number referred to chord length.
The observed value is somewhat high compare to the theoretical
value E = -0.203, the discrepancy probably being due to the excessive
size of the model in proportion to that of the wind tunnel. At any rate,
however, the pressure distribution is nearly uniform. The lack of uni-
formity exists at both edges due to the impossibility of realizing the
infinite pressure gradient. The boundary layer separates near the
trailing edge, but the effect of separation becomes small as the Reynolds
number increases. This scale effect seems to be of the same nature as
that responsible for the sudden drop in sphere drag; the boundary layer
separates in a laminar state when the Reynolds number is low, while it
becomes turbulent before separation when the Reynolds number is high,
thus being able to proceed against a larger pressure gradient. This is
also seen from the measurements in the wake, where the indentation of the
curve of total pressure distribution is shallow and wide for low Reynolds
numbers, while it becomes deep and narrow as the Reynolds number increases.
As a result, the profile drag coefficient CDO decreases considerably

NACA TM 1351

as the Reynolds number increases, as shown in figure 24. Transition to
turbulence was found very near to the trailing edge, occurring downstream
of x = 0.9 in the range of measurements. Measured values of the drag
of the model when a piano wire of 0.5 mm diameter was placed at x = 0.8
and x = 0.9, respectively, are also plotted in the same figure. The
drop in drag occurs at a lower value of the Reynolds number when the
surface is roughened by the wire.

23. The profile drag coefficient of the airfoil U.P. 0010 is shown
below in comparison with other symmetrical airfoils at R = 2.2 X 106:

NACA 001 CDO = 0.0064 (fig. 13)
L.B. 24 0.0032 (fig. 13)
U.P. 0010 0.0059
U.P. 0010, a wire at x = 0.9 0.0044

All the airfoils are of 10 percent thickness. NACA 0010 is a conventional
airfoil, and L.B. 24 is a laminar-flow airfoil with far back minimum
pressure. The drag of U.P. 0010 is between that of these two airfoils,
the drag when a wire is placed being nearly the mean of the two. This
result seems to be interesting in that the drag of an airfoil with a
blunt tail is smaller than commonly considered.

The airfoil with uniform distribution of pressure will also probably
be favorable when used at high subsonic speeds. Even if the shock wave
occurs at high subsonic speeds, the increase in drag will remain small
when the boundary layer does not separate. This expectation was really
verified by the experiment due to Kawada and Kawamura (ref. 22), the
drag of the airfoil U.P. 0010 being smaller at high Mach numbers as
compared with other airfoils.

24. From the fact that the boundary layer transition occurs only in
the region of rising pressure, we may also expect suction of the boundary
layer to delay transition. For example, if the boundary layer is sucked
into a slot, there is a well-known sink effect (ref. 23) which relieves
the adverse pressure gradient somewhat upstream of the slot. We may con-
sider an alternative possibility. That is, we assume that the boundary
layer is laminar in the region of falling pressure, and that it remains
laminar also in the region of rising pressure provided that the boundary
layer profile is the same as that at the minimum pressure point. We
then ask what suction arrangement must be applied in order to realize
such a condition.

25. We denote by s the coordinate measured along the surface,
y perpendicular to the surface, 6 the boundary layer thickness, u
the velocity in the boundary layer, ul the velocity outside the boundary

NACA TM 1351

layer, = -pu (dul/ds) the pressure gradient, and To = -i 2u/4y y=0

the skin friction at the surface.
porous, through which the fluid is
the equation of continuity

Assuming the surface (y =
sucked with the velocity

0) is made
c, we have

w c = -- u dy
ds 0

and the equation of momentum

pulw P -
d J

u2 dy = TO +

where w is the velocity of fluid entering the
y = 8. The equation of motion reduces to


= Uld

boundary layer through


for y = 0.

Now, the velocity profile
by the Pohlhausen polynomial

in the boundary layer may be approximated

u = u 1


when neither pressure gradient nor suction exists.

For this profile we

TO = 2

pu1W = 7- T7

NACA TM 1351

To simplify the calculation, we assume that the expressions (d), (e),
and (f) still hold when both pressure gradient and suction exist. Then
we have from (b)

d 630 v 104 5 du1 (g)
ds 37 u18 367 U1 ds

Integrating we have

m 52 m 52 1260 -s m-1
Ul = ul -) + -- u d h)
vs=so 37 sO

where m = -, and so is the initial position of suction, which is
the minimum pressure point in the present case. We have also from (a)

1841 dul
c =- (i)
3670 ds

If we substitute (d) and (i) into both sides of the equation (c) however,
the left and right sides become 1.003ul(dul/ds) and ul(dul/ds), respec-
tively. This contradiction is evidently due to the crude assumption of
using (f) in spite of the presence of pressure gradient and suction, but
we may overlook the error because it is small.

26. We apply the calculation to the symmetrical laminar-flow air-
foil of 10 percent maximum thickness, L.B. 24, set at zero angle of
attack. The velocity distribution ul/V calculated from the potential
flow of ideal fluids is used,6 the maximum velocity (minimum pressure)
being located at 64 percent of the chord from the leading edge (so = 0.65t).
Applying a distributed suction downstream of the minimum pressure point
so as to maintain the velocity profile in the boundary layer the same as
at this point, we have the boundary layer thickness B and the required

6It is assumed that the velocity distribution is not affected by
the suction. Theoretically ul should be 0 at the trailing edge, but
the distribution was somewhat modified so as to give ul =. 0.85V there.
The effects of these assumptions appear to be too small to affect the
result materially.

NACA TM 1351

suction velocity c as shown in figure 25. It is to be noted that s
is measured along the surface from the leading edge, and R is the
Reynolds number based on the chord length t and the velocity of the
undisturbed stream V. Integrating the area under the curve of c, we
have the total amount of suction 1.3v'R per unit span of the two sur-
faces. If we assume span = 35 m, t = 5 m, V = 200 m/s, v = 0.15 cm2/s,
the total amount of suction amounts to 5.6 m3/s, which will require an
exit area of only 0.028 m2 when discharged with the velocity equal to V.
Integration of TO gives the drag coefficient CDO = 0.0005. This value
may be compared with CDO = 0.0003 for the flat plate with laminar
boundary layer, CDO = 0.0044 for the flat plate with turbulent boundary
layer, and CDO = 0.0025 for L.B. 24 without suction. If the thickness
of the airfoil is doubled (20 percent chord), then the amount of suction
will be nearly doubled; the drag is however almost unchanged.

It should be noticed again that the calculation is based on the
assumption that no transition occurs if the velocity profile in the
boundary layer maintains the form at the minimum pressure.' It is the
purpose of the calculation to show that extraordinarily low profile drag
may be expected with a relatively small amount of suction under such a


27. Although the transition of the boundary layer occurs only down-
stream of the minimum pressure, so far as flight experiments on smooth
airfoil surfaces are concerned, there are many examples of wind tunnel
experiments in which the transition moves upstream of the minimum pres-
sure. This seems to be due to the premature transition caused by the
turbulence in the wind tunnel stream. For example, the transition on
the airfoil L.B. 24 was found only downstream of the minimum pressure
in the range of Reynolds numbers covered by the author's wind tunnel
experiments (the Reynolds number based on chord length up to 3 x 106;
see paragraphs 12 to 14); as a result very low values of the profile drag
coefficient CDO were observed. The same airfoil, however, when tested
with a larger model of 2 m chord in the 2.5 m wind tunnel of Kawasaki
Aircraft Company, Gihu, gave the result as shown in figure 26, in which
CDO increases considerably when the Reynolds number exceeds 5 X 106
refss. 24 and 25). There is reason to believe that the increase in drag
is due to the effect of stream turbulence. The boundary layer observa-
tion at the Kawasaki wind tunnel shows that the transition is found at
50 percent chord (x = 0.5) for the Reynolds number 6 X 106 and moves

NACA TM 1351

further forward as the Reynolds number increases. The boundary layer
velocity profile observed at transition has a form factor -= 2.6
(56 and e are the displacement and momentum thicknesses of the
boundary layer), which is very near to the value for the case of zero
pressure gradient. This result seems to suggest that the transition is
at least not correlated with the laminar separation (ref. 1).

28. In order to verify this conjecture, it is desirable to show
that the transition in the boundary layer along a flat plate occurs
under the same condition, because that transition may be considered to
be independent of the laminar separation. Unfortunately, however, no
flat plate was measured in the Kawasaki wind tunnel. Therefore we pro-
ceed in a somewhat indirect way. We assume that the degree of stream
turbulence is represented by the conventional critical Reynolds number
of the sphere, RC, and the condition of transition due to turbulence
represented by the local Reynolds number, Re = --, at transition on a

flat plate, where ul is the velocity outside the boundary layer and
e is the momentum thickness of the boundary layer. It is generally
accepted that the turbulence in the wind tunnel stream will give a
fluctuation of pressure gradient, as a result of which an instantaneous
and intermittent separation will occur. Such an instantaneous and inter-
mittent separation, however, does not necessarily lead to the transition
into turbulence; for the transition really to occur, it seems probably
necessary that the Reynolds number Re which represents the ratio of
inertia pul2 to viscous stress pvul/O exceed a certain critical value.
It is also expected that the critical value depends on the degree of tur-
bulence; it must increase as RC increases. This is really shown by
the experimental data hitherto published, which are given in the following
table and also by white circles (o) in figure 27. The available data are
scanty, especially because the experiment on a flat plate is very diffi-
cult. It was necessary for the author to perform a new experiment (ref. 30)
with a view to adding one point in the range of high RC.

RC Re Wind Tunnel Reference

1.40 x 105 0.21 x 103 National Bureau of Standards 26
2.75 X 105 .70 x 103 National Bureau of Standards 26
2.20 X 105 .42 X 103 N.P.L. Compressed Air Tunnel 27, 28
3.66 X 105 1.05 x 103 Aero. Res. Inst. 1.5 m Tunnel 29, 30

NACA TM 1351

29. Now, we calculate the value of Re at transition of the air-
foil L.B. 24 tested in the Kawasaki wind tunnel, and correlate it with
the critical Reynolds number RC of that tunnel. We analyze similarly
the other available data, and summarize the result in the following
table. The values of the form factor &8/9, not shown, were all found
in the range from 2.1 to 2.7. We then plot the data also in figure 27
by black circles (e). The black circles are seen to define a single
curve together with the white circles already mentioned. This result
seems to support the supposition that the transition under consideration
is mainly caused by the stream turbulence, but not correlated with the
laminar separation.

RC Re Model Wind Tunnel Reference

1.85 x 100 0.41 x 103 Symmetrical airfoil N.P.L. 7 ft 31, 32
2.10 x 105 .56 x 103 Airship model M.I.T. 7 ft 33, 34
3.50 x 105 .95 x 103 Airfoil N-22 NACA Full-Scale 12, 35
3.65 x 105 1.08 x 103 Airfoil L.B. 24 Kawasaki 2.5 m 24, 25

In reference 12 (the third line in the preceding table), the same
airfoil was examined both by the full-scale wind tunnel and by the flight
tests. We calculate the form factor 6*/e from these tests and plot the
values against s/t in figure 28, where t is the chord length and s
is the length measured along the surface from the leading.edge. The
value of 8*/e at transition is 2.6 in the wind tunnel, while it reaches
as high as 3.1 and drops sharply in the flight test. The minimum pres-
sure is located at = 0.18, and the laminar separation'calculated from

the measured distribution of pressure at = 0.36. This example is very
interesting because the cause of transition is quite different in the two
cases (namely, it is due to the stream turbulence in the wind tunnel,
while it is related to laminar separation in the flight test), although
the positions of transition are almost the same.


30. Up to this point, we have only considered the case when the sur-
face of the airfoil is smooth. If the surface is rough, however, there
is a possibility that the transition may also be caused prematurely by
surface roughness. So, it is important in practice to estimate the

IACA TM 1351

approximate order of magnitude of the permissible roughness in the lami-
nar boundary layer. Nothing has been known concerning this problem,
except a mere conjecture or fragmentary data. Schiller (ref. 36) sug-
gested that a local separation occurs and hence leads to transition
when the Reynolds number kuk/V exceeds a certain critical value Rcrit,
where k is the height of projection and uk is the velocity at the
top of projection. The exact value of Rcrit is not known, but it is
not likely to differ much from the critical value of the Reynolds number,
above which vortices are shed from the obstacle of the same shape as that
of the projection placed in a uniform stream. The experimental result
due to Wieselsberger (ref. 37) shows that such a critical Reynolds number
is roughly 50 for a circular cylinder. Assuming that the height of pro-
jection k is small, and that the presence of the projection in no way
alters the character of the flow, we have the shearing stress at the
surface TO = pV(uk/k). Using the so-called friction velocity v. = o/P

instead of uk, we have then kv./V = kuk/v. The permissible roughness
is therefore given by kv*/v = Rcrit, or, with Rcrit = 50, kv/V = 7.
On the other hand, according to Nikuradse's experiments on roughened
pipes (ref. 38), the critical Reynolds number is = 4, above which
the roughness projections disturb the laminar sublayer of the turbulent
boundary layer and hence increase the pressure drop. It appears there-
fore that the permissible roughness is smaller in the turbulent boundary
layer as compared with the laminar boundary layer. This is confined by
a British flight experiment (ref. 39) on the airfoil section of 10 feet
chord (Reynolds number 1.8 x 107), because the effect of camouflage paint
of 0.001 inch thickness increased the drag by about 6 percent without
moving the transition forward. At any rate, however, such a estimate is
nothing but mere conjecture. With a view to making the estimate more
definite, we performed wind tunnel experiments, although of small scale,
of quantitative character (paragraphs 31 to 32).

31. A polished aluminum plate, 80 cm long, 60 cm wide, and 3 mm thick,
was held horizontally in the 1.5 m wind tunnel of the Aeronautical Research
Institute. So that the flow at entry would not be disturbed, the leading
edge of the plate was rounded, and the plate slightly tilted so that the
forward stagnation point was on the same surface as that where the observa-
tion was made. The tilting, however, was so slight that the static pres-
sure was observed to be practically uniform along the plate. The plate
was roughened by a wire, which was stretched across the flow, in contact
with the plate. The diameters k of the wire were 0.25, 0.4, and 0.7 mm,
respectively, and the distances x of the wire from the leading edge were
15, 30, 45, and 60 cm, respectively. When the wind speed V was low, the
boundary layer was laminar all along the plate, but from a certain speed
upward, the transition to turbulent flow was observed at that point where

NACA TM 1351

the wire was placed. Transition was detected by a sudden change in the
value of the total pressure G* in terms of the dynamic pressure q of
the undisturbed stream, G* being indicated by a small pitot tube with
a flattened mouth of 1 mm external width and 0.3 mm width, which was
placed in contact with the plate at a point 70 cm behind the leading
edge. A sample record of measurements for k = 0.4 mm is shown in
figure 29.

When a flat plate is placed along a uniform stream of velocity V,
the Blasius solution (ref. 40) of the laminar boundary layer equation

= 0.576v( 3/4

for a point of distance x from the leading edge. Writing K for the
critical value of kv*/v, the permissible height of projection k is
given by

0.576 = K( -3/4

We determine V from the wind speed corresponding to the kink of the
curve as shown in figure 29, plot 0.576(k/x) in a logarithmic scale
against Vx/V, and draw a straight line of the slope -3/4 through the
points (fig. 30). We thus obtain K = 13, which is far greater than
the value K = 7 estimated previously.

32. Similar measurements were also performed on an airfoil section
L.B. 24. The model was of 0.8 m span, of 1.2 m chord, fitted with end
plates 1.3 m x 0.6 m, and set at zero angle of attack in the same wind
tunnel. Wires of various diameters (k = 0.25, 0.4, 0.7 mm) were attached
parallel to the span, in contact with the surface, at 10 percent of the
chord from the leading edge (x = 0.1). Transition was detected by the
sudden change in total pressure as indicated by a pitot tube with a
flattened mouth of 2.7 mm external width and 0.9 mm depth, which was
placed in contact with the surface at 50 percent of the chord from the
leading edge. Results of measurements are shown in figure 31, where
R = t, and t is the chord length.

The friction velocity may be generally expressed in the form

v, = AVR-1/4

NACA TM 1351

where A is a function of s/t (a is the length measured along the
surface from the forward stagnation point). We can calculate A by
applying either the Pohlhausen approximate solution (ref. 41) or the
simplified method due to the author refss. 15 and 16). The permissible
height of projection is then determined by

A =KR-3/4

Applying the Pohlhausen method to the theoretically calculated
distribution of pressure, we get the values of function A, as shown
in figure 32. Since A is 1.23 at the position of wire (x = 0.1),
the value 1.23(k/t) is plotted in a logarithmic scale against R in
figure 33, R being the Reynolds number corresponding to the kink of
the curve as given in figure 31. It will be seen that although the
measured points are on a straight line of the slope -3/4, they give
K = 15, which is somewhat higher than the value found for the flat plate.

33. Now we apply the preceding result to the fragmentary data
hitherto known in order to check the adequacy of the estimate. First,
we examine the results of wind tunnel experiment on a symmetrical
laminar-flow airfoil L.B. 27,7 on which various projections are attached
at 3 percent of the chord from the leading edge (x = 0.03). The model
was of 0.8 m span, of 1.2 m chord, and set at zero angle of attack.
The profile drag was measured by the method similar to that for L.B. 24
(see paragraphs 12 to 13). The results are shown in figure 34, from
which we find that the rubber tape of 0.07 mm thickness gives no effect
over the range of Reynolds numbers R covered by the experiment, while
the piano wire of 0.5 mm diameter gives a completely turbulent friction.
The effect of the wire of 0.25 mm diameter begins to appear at
R = 1,3 X 106. Inserting the values k = 0.25 mm, t = 1.2 m and
A = 1.95 in the formula A = KR3/4, we get K = 15.5, which is in
good agreement with the result in paragraph 32. The value A = 1.95
was read from figure 32, since the leading edge portion of L.B. 27
almost coincides with that of L.B. 24. Profile drag coefficients of
both airfoils are also the same over the range of Reynolds numbers

34. In connection with the determination of boundary layer transi-
tion on airfoils in the NACA full-scale wind tunnel (ref. 42), an aux-
iliary measurement has been reported, in which the effect was examined

7L.B. 27 has a maximum thickness of 10 percent of the chord at
60 percent of the chord from the leading edge. See figure 5 of paragraph 6.

NACA TM 1351

of rubber tapes attached at 5 percent of the chord (x = 0.05) from the
leading edge of the model. The airfoil section was NACA 0012, the chord
was 72 inches, and the Reynolds number R was 4.18 X 106 No effect
was found when the tape was 0.003 inch thick, some effect began to
appear when it was 0.006 inch thick, and the transition moved right to
that position where the tape of 0.009 inch thickness was attached.
Assuming A = 1.6 and K = 15, we estimate from the preceding formula
the value 0.007 inch for the permissible thickness, which seems to agree
well with the observation.

If we further assume that the value A = 1.6 is also applicable
to the case of the British flight experiment mentioned in paragraph 30,
we find 0.004 inch for the critical height for transition with
t = 10 feet, R = 1.8 X 10, and K = 15. On the other hand, we esti-
mate the permissible limit in turbulent boundary layer by k-- = 4,
which may be written in the form

kt= l2 V 4
Scf ul R

by the relation v2 = 2 cful2, where ul is the local wind speed and
cf is the coefficient of local skin friction. If we assume ul = 1.2V
and cf = 0.003 (which is equal to the coefficient of mean skin friction
for a flat plate at R = 1.8 X 107), we obtain 0.0006 inch for the per-
missible roughness thickness. Since the thickness of thecamouflage
paint is reported to be about 0.001 inch, it may be concluded that the
paint increases only the friction in turbulent boundary layer, without
affecting, however, the transition to turbulent flow. This is in good
agreement with the experimental results.

NACA TM 1351



35. As is well-known, the phenomenon of sudden decrease in drag of
a sphere at a certain value of the Reynolds number R = Vd (V is the
speed of undisturbed stream and d is the diameter of sphere) is
explained by supposing that the boundary layer separates while it is
laminar when R is low, but it separates after transition to turbulence
when R is high, thus resulting in diminishing the so-called dead water
region. Probably the transformation from laminar separation to turbulent
separation may proceed as follows:

When the laminar boundary layer separates from the surface, the
detached layer remains also laminar at first, but it is so unstable that
it becomes turbulent at a short distance. This transition from laminar
to turbulent flow is considered to occur when the local Reynolds number
based on the width of the detached layer and the velocity outside the
layer exceeds a certain value, so that the transition moves upstream
toward the separation point as R increases. When the transition
approaches sufficiently near the separation point, it becomes possible
for the detached layer to come back again to the downstream surface,
because the turbulence produced will drive the flow forward. The layer
reattaches to the surface as a turbulent layer, and accordingly the drag
coefficient begins to decrease. The distance between the separation and
the first turbulent boundary layer decreases as R increases, and finally
the fully developed turbulent boundary layer commences just downstream
of the separation point. The drag coefficient then ceases to decrease.

36. Now, in order that the separated layer reattach to the surface,
it seems necessary for the local Reynolds number R = at separation

to exceed a certain critical value, where uI is the velocity outside
the boundary layer, and 9 is the momentum thickness of the boundary
layer. This may be explained as follows: According to the laminar
boundary layer theory, the separation occurs when the quantity

82 dul dp ul
Sds ds *

exceeds a certain value, suggesting that the pressure rise (dp/ds)8
becomes too large in proportion to the shearing stress at the sur-
face pv(ul/0). Assuming analogically that the separated layer leaves

NACA TM 1351

the surface when the pressure rise becomes too large in proportion to
the momentum pu2, we then find that Re at the separation point must
exceed a certain critical value in order that the separated layer
reattach to the surface.

37. In order to determine the critical value of Re, we consider
in detail the condition where the coefficient of sphere drag begins to
decrease. This condition corresponds to the point B of the curve of
figure 35, which represents an idealized variation of the drag coeffi-
cient CD[= drag (pV2/2) (d2/4 or the pressure difference coeffi-

cient Ap/q = difference of pressures at the forward stagnation point

and the point corresponding to the central angle 157.50 s- (pV2/2) with
the Reynolds number R. Within the range AB, the pressure distribution
around the sphere is approximately independent of R; the typical example
may be found from the experiments due to Fage (ref. 43). Fortunately,
the boundary layer calculation has also been performed for that distri-
bution of pressure by Tomotika and Imai (ref. 44), so that the local
Reynolds number Re is given by

Re = o.40o

at the separation point. Although the calculation has originally been
made for a particular Reynolds number, R = 1.57 X 105, the preceding
relation may be applied for any value of R in the range of AB. Putting
the value of R at B, and writing

Rcrit = 0.40 4

we have Recrit as the critical value of Re above which the separated
laminar layer reattaches to the surface. Conventionally the Reynolds
number RC corresponding to CD = 0.3 (or Ap/q = 1.22) has been used,
instead of RB, for representing the degree of stream turbulence, but it
is not so difficult to estimate the value of RB from the measured curve
of CD (or Ap/q) against R. For example, we have from the experiments
of towing spheres in the free atmosphere (ref. 35)

RC = 3.85 x 105

Recrit = 240

RB = 3.6 x 105

NACA TM 1351

Since these values refer to the case of very low turbulence, it will be
seen that crit = 240 represents the highest possible value. It is
also to be noted that RB = 0.94 in this example and that almost the

same value has been obtained by the author's experiments on spheres of
various diameters (ref. 29).

38. If the stream turbulence is not low so that RC is less than
3.85 X 105, then R crit will be less than 240. Assuming the ratio RB/RC
to be constant, we can estimate the corresponding value by

RB = 240
crit 3.85 x 10

On the other hand, we can also estimate the value of Recrit directly
from the boundary layer measurements. The results of the analysis for a
sphere as well as circular and elliptic cylinders are summarized in the
following table, where Rcrit is the critical value R% estimated
from RC by the preceding formula, Ree is the value of Re observed
at the separation point when the boundary layer really separates while
it is laminar, and trans is the value of Rl observed at the calcu-
lated laminar separation point when the boundary layer separates after
transition. The fact that RBcrit lies between Rsep and trans
R~crit .sep trans
seems to suggest the adequacy of the preceding consideration.

Body RC Recrit R1sep Retrans Reference

Sphere 2.5 X 105 190 160 220 43
Circular cylinder 1.5 x 105 150 140 225 45
Elliptical cylinder 2.7 x 105 200 160 400 46, 47

39. We now proceed to apply our result to interpreting the effect of
Reynolds number on maximum lift of airfoils. For the angle of attack
near the stall, the flow separates shortly downstream of the leading edge
while the boundary layer is laminar. If the flow fails to reattach to
the surface as a turbulent layer, the maximum lift coefficient CLmax of

NACA TM 1351

the airfoil will be almost independent of the Reynolds number R. At
the separation point, similar to.the case of a sphere, the relation of
the form

R, = kfR

Rt 8
holds, where R = is the Reynolds number referred to chord length t,8

and k is a constant depending on the shape of airfoil and the value of
Cmax. If R is low so that Re is less than the critical value Recrit
then CLmax will be independent of R. Assuming the same value of

R crit for the sphere as for the airfoil, we obtain

R 0.16
R = -l Rg

for the Reynolds number above which Cmax begins to increase with the
Reynolds number. Therefore, the ratio of the Reynolds number corresponding
to a certain value of Cax of an airfoil and the critical Reynolds
number of sphere in the same stream, RB or RC, becomes independent of
the stream turbulence. Denoting the values for a reference tunnel with
asterisk, we have


which in turn means that the ratio of Reynolds numbers corresponding to
a certain value of CLax is equal to the ratio of critical Reynolds
numbers of a sphere. This is useful for comparing the values of CLmax
obtained in two different wind tunnels. Considering the reference condi-
tion to be the free flight in the atmosphere, we find

R* =RX C

8It is to be noted that R is referred to t, while RB and RC
are referred to d.

NACA TM 1351

as the free flight Reynolds number which will give the same value of
CLax as that observed in a wind tunnel. This is just what is called
the effective Reynolds number. Strictly speaking, such an argument as
mentioned before should apply only to the Reynolds numbers near the
critical value, but there are many experimental evidences showing the
usefulness of the concept of effective Reynolds number for most practi-
cal purposes, as far as the commonly used airfoils and range of Reynolds
numbers of both wind tunnel and free flight are concerned.

40. Finally, we consider a more quantitative example to show the
adequacy of the preceding argument. In figure 36, CLmax for various
NACA symmetrical airfoils are plotted against the effective Reynolds
number R*, the experimental data being taken from the results of the
NACA variable-density wind tunnel (ref. 48). As already mentioned, up
to a certain value of R*, CLmax is almost independent of R*. This
corresponds to the condition in which the laminar separation just behind
the leading edge fails to reattach to the surface, resulting in a con-
siderable dead water region above the airfoil surface. The value of
CLmax is approximately 0.9, irrespective of the thickness; it is almost
equal to the value for a flat plate of vanishing thickness. Theory of
discontinous flow, when applied to the flat plate, seems to give a lift
coefficient close to 0.9 (ref. 49). We idealize, therefore, the experi-
mental curve as shown by dotted lines in figure 36. Then, the point
where the dotted line meets the line CLmax = 0.9 will be considered to
correspond to crit = 240. In order to determine this point, we calcu-
late the value of Re/J at the laminar separation point for a lift
coefficient CL = 0.9, and the value of R which gives Re = 240. We
first calculate the pressure distribution by the formula of paragraph 3
for the two-dimensional potential flow around the airfoil section.
Although the formula may be applied to any arbitrary airfoil section,
we have determined the pressure distribution only for the airfoil

T = 0.287e x(l x)(5 4x)

in order to simplify the calculation, because no great exactitude is
required in the present problem. x is the coordinate along the chord,
x = 0 and x = 1 corresponding to the leading and trailing edges,
respectively, T is the half-thickness, and e is the maximum thick-
ness in terms of chord length. The airfoil represented by the preceding
expression coincides with sufficient accuracy with the true NACA symmetri-
cal airfoil except near the trailing edge. The value of Re/~ at the
laminar separation point was then determined for the calculated pressure
distribution by applying the approximate method due to the author refss. 15
and 16).

NACA TM 1351

The critical values R* thus calculated are shown by a solid
line in figure 37, while the corresponding values taken from figure 36
are shown by white circles. The agreement is fairly good, and especially
satisfactory when the thickness of the airfoil is small. In general, the
thickness of the boundary layer near the trailing edge increases as CL
increases. If CL is further increased, however, a laminar separation
suddenly occurs near the leading edge when the thickness is small, while
the trailing-edge turbulent separation moves a considerable extent for-
ward before the leading-edge laminar separation occurs when the thickness
is moderate. Therefore, the assumption of the analysis is more satis-
factorily realized in the case of small thickness, thus bringing the
calculated and observed values in close agreement.

In conclusion, the author wishes to acknowledge his indebtedness
for the assistance given by Messrs. C. Noda, S. Mituisi, I. Shinra,
S. Asaka, R. Hama, and K. Takeda.

Translation by Itigo Tani
University of Tokyo
Tokyo, Japan

NACA TM 1351


1. Tani, I.: On the Interplay Between the Laminar Separation and the
Transition to Turbulent in the Boundary Layer (in Japanese). Jour.
Soc. Aero. Sci. Japan, vol. 6, 1939.

2. Tani, I.: On the Transition Caused by Laminar Separation of the
Boundary Layer (in Japanese). Jour. Soc. Aero. Sci. Japan, vol. 7,

3. Tani, I., and Noda, C.: Design of Symmetrical Airfoils in Which the
Minimum Pressure Occurs Well Downstream (in Japanese). Jour. Aero.
Res. Inst., Tokyo Imperial Univ., nos. 190 and 193, 1940.

4. Tani, I., and Mituisi, S.: Contributions to the Design of Aerofoils
Suitable for High Speeds. Rep. Aero. Res. Inst., Tokyo Imperial
Univ., no. 198, 1940.

5. Tani, Itiro, Hama, Ryosuke, and Mituisi, Satosi: On the Permissible
Roughness in the Laminar Boundary Layer. Rep. no. 199 (vol. XV, 13),
Aero. Res. Inst., Tokyo Imperial Univ., Oct. 1940.

6. Tani, I., and Mituisi, S.; Wind Tunnel Experiments on the Airfoil
U.P. 0010 (in Japanese). Jour. Aero. Res. Inst., Tokyo Imperial
Univ. no. 205, 1941.

7. Tani, I.: A Simple Calculation on Boundary Layer Suction (in Japanese).
Jour. Soc. Aero. Sci. Japan, vol. 8, 1941.

8. Stuper, J.: Untersuchung von Reibungsschichten am fliegenden Flugzeug.
Luftfahrtforschung, vol. 11, 1934, p. 26.

9. Jones, B. Melvill: Flight Experiments on the Boundary Layer. Jour.
Aero. Sci., vol. 5, no. 3, Jan. 1938, pp. 81-94.

10. Serby, J. E., Morgan, M. B., and Cooper, E. R.: Flight Tests on the
Profile Drag of 14 Percent and 25 Percent Thick Wings. R. & M.
No. 1826, British A.R.C., 1937.

11. Bicknell, Joseph: Determination of the Profile Drag of an Airplane
Wing in Flight at High Reynolds Numbers. NACA Rep. 667, 1939.

12. Goett, Harry J., and Bicknell, Joseph: Comparison of Profile-Drag
and Boundary-Layer Measurements Obtained in Flight and in the Full-
Scale Wind Tunnel. NACA TN 693, 1939.

NACA TM 1351

13. Moriya, T.: A Simple Method of Calculating the Aerodynamic Charac-
teristics of an Arbitrary Wing Section (in Japanese). Jour. Soc.
Aero. Sci. Japan, vol. 5, 1938, p. 7.

14. Jacobs, Eastman N., Ward, Kenneth E., and Pinkerton, Robert M.: The
Characteristics of 78 Related Airfoil Sections From Tests in the
Variable-Density Wind Tunnel. NACA Rep. 460, 1933.

15. Tani, I.: A Simplified Boundary-Layer Calculation for Laminar Separa-
tion (in Japanese). Jour. Aero. Res. Inst., Tokyo Imperial Univ.,
no. 199, 1941.

16. Tani, I.: On the Laminar Boundary Layer in Compressible Fluids (in
Japanese). Rep. Aero. Res. Inst., Tokyo Imperial Univ., no. 251,

17. Theodorsen, Theodore: On the Theory of Wing Sections With Particular
Reference to the Lift Distribution. NACA Rep. 383, 1931.

18. The Cambridge University Aeronautics Laboratory: The Measurement of
Profile Drag by the Pitot-Traverse Method. R. & M. No. 1688,
British A.R.C., 1936.

19. Shinra, I.: Flight Experiments on Laminar-Flow Airfoil (in Japanese).
Paper presented at the Eighth Conference of Wind Tunnel and Water
Basin Researches, Navy Aeronautical Technical Arsenal, 1940.

20. Buri, A.: Eine Berechnungsgrundlage fur die turbulente Grenzschicht
bei beschleunigter und verz6gerter Grundstrimung. Dissertation,
Zurich 1931.

21. Squire, H. B., and Young, A. D.: The Calculation of the Profile Drag
of Aerofoils. R. & M. No. 1838, British A.R.C., 1938.

22. Kawada, S., and Kawamura, R.: High-Speed Wind Tunnel Experiments on
Airfoils U.P. 0010 and NACA 0009 (in Japanese). Jour. Aero. Res.
Inst., Tokyo Imperial Univ., no. 205, 1941.

23. Schrenk, 0.: Versuche mit Absaugefligeln. Luftfahrtforschung,
vol. 12, 1935, P. 10.

24. Yamashita, H.: On the Kawasaki 2.5 m Wind Tunnel (in Japanese).
Jour. Soc. Aero. Sci. Japan, vol. 6, 1938, p. 29.

25. Yamashita, H., Asaka, S., Morioka, K., and Oue, M.: Wind Tunnel
Experiments on the Airfoil L.B. 24 (in Japanese). Research Report
of the Kawasaki Aircraft Company, vol. 2, 1941, p. 193.

NACA TM 1351

26. Dryden, Hugh L.: Air Flow in the Boundary Layer Near a Plate.
NACA Rep. 562, 1936.

27. Jones, R., and Williams, D. H.: The Profile Drag of Aerofoils at
High Reynolds Numbers in the Compressed Air Tunnel. R. A M.
No. 1804, British A.R.C., 1937.

28. Fage, A., and Williams, D. H.: Critical Reynolds Numbers of Spheres
in the Compressed Air Tunnel. R. & M. No. 1832, British A.R.C., 1938.

29. Tani, I., and Hama, R.: Experiments on Pressure Spheres in the 1.5 m
Wind Tunnel (in Japanese). Jour. Aero. Res. Inst., Tokyo Imperial
Univ., no. 188, 1940.

30. Tani, I., Hama, R., Mituisi, S., and Iriyama, T.: Boundary Layer
Measurements on a Flat Plate in the 1.5 m Wind Tunnel (in Japanese).
Jour. Aero. Res. Inst., Tokyo Imperial Univ., no. 189, 1940.

31. Fage, A., and Falkner, V. M.: An Experimental Determination of the
Intensity of Friction on the Surface of an Aerofoil. R. & M.
No. 1315, British A.R.C., 1931.

32. Fage, A.: On Reynolds Numbers of Transition. R. & M. No. 1765,
British A.R.C., 1937.

33. Lyon, Hilda M.: Effect of Turbulence on Drag of Airship Models.
R. & M. No. 1511, British A.R.C., 1933.

34. Lyon, H. M.: Flow in the Boundary Layer of Streamline Bodies.
R. & M. No. 1622, British A.R.C., 1934.

35. Platt, Robert C.: Turbulence Factors of NACA Wind Tunnels As
Determined by Sphere Tests. NACA Rep. 558, 1936.

36. Schiller, L.: Handbuch der Experimentalphysik, vol. 4, part 4, 1932,
p. 191.

37. Wieselsberger, C.: Neuere Feststellung uber die Gesetze des
Flissigkeitsund Luftwiderstandes. Phys. Zeit., vol. 22, 1921,
pp. 321-328.

38. Nikuradse, J.: Stromungsgesetze in rauhen Rohren. Forechungsheft 361,
Beilage zu Forschung auf dem Gebiete des Ingenieurwesens, Ausg. B,
Bd. 4, July-Aug. 1933.

39. Young, A. D.: Surface Finish and Performance. Aircraft Engng.,
vol. 11, 1939, p. 339.

NACA TM 1351

40. Blasius, H.: Grenzschichten in Flissigkeiten mit kleiner Reibung.
Z.f. Math. u. Phys., vol. 56, no. 1, 1908, pp. 1-37.

41. Pohlhausen, K.: Zur naherungsweisen Integration der Differential
gleichung der laminaren Grenzschicht. Z.f.a.M.M., vol. 1, 1921,
pp. 252-268.

42. Silverstein, Abe, and Becker, John V.: Determination of Boundary-
Layer Transition on Three Symmetrical Airfoils in the NACA Full-
Scale Wind Tunnel. NACA Rep. 637, 1939.

43. Fage, A.: Experiments on a Sphere at Critical Reynolds Numbers.
R. & M. No. 1766, British A.R.C., 1936.

44. Tomotika, S., and Imai, I.: On the Transition From Laminar to
Turbulent in the Flow of the Boundary Layer of a Sphere. Rep.
Aero. Res. Inst., Tokyo Imperial Univ., no. 167, 1938.

45. Fage, A., and Falkner, V. M.: Further Experiments on the Flow Around
a Circular Cylinder. R. & M. No. 1369, British A.R.C., 1931.

46. Schubauer, G. B.: Air Flow in a Separating Laminar Boundary Layer.
NACA Rep. 527, 1935.

47. Schubauer, G. B.: Air Flow in the Boundary Layer of an Elliptic
Cylinder. NACA Rep. 652, 1939.

48. Jacobs, Eastman N., and Sherman, Albert: Airfoil Section Character-
istics As Affected by Variations of the Reynolds Number. NACA
Rep. 586, 1937.

49. Muller, W.: Einfuhrung in die Theorie der iAhen Flissigkeiten.
Leipzig 1932, section 79.

NACA TM 1351

0x K L M
0 0 0 0 0
,003 .0465 .0763 o57- .058 .0606 .o627
.006 .0661 0o793 .0808 .0o823 .084 .0886
.025 .0962 .1135 .1158 .118o 1226 .17
.025 .1374 1589 .1620 .1652 .1720 1297
05 .1963 .2208 .222 .1-2396 9408
.075 .2414 .2663 .216 .21 .2892 .3031
.10 .2789 .3029 .3089 3151 3288 .3447
.15 .3394 3599 .3667 3738 .3893 3.447
.20 .386 .4o026 .4097 .4170 .4328 .4503
25 .45236 .4353 .4422 .4492 .4638 .4788
.30 .4523 .4601 .4663 .4724 .4844 .4949
.35 .4737 4783 4833 .4881 .4962 000
.40 .4906 4942 .4971 5000 .953000
.45 .4972 4977 .4994 5000 4948 48853
55 4 *0? 4 948 .4818
.50 5 000 .49 O0 .4994 .4931 4797 .46o4
55 .0 44944 .4873 .4781 : .558 .4320
.60 .4871 4778 .4656 .4522 .4241
.65 .4691 .4509 .4342 175 -3856 -3578
70 .4418 4139 3939 3751 3413 3138
.8075 .438 3675 3455 .3258 .2921 .2664
.83538 .3121 .2899 .2706 .239321265
.85 .2908 .2481 .2277 .2105 .9 .165
.90 .2133 1762 1598 .1464 .1262 .1128
95 .1201 .0966 .0870 0793 .0680 .
1.00 .01 00 0100 0 .0100 .0609
--- L- -010)0 0100 .0100

NACA TM 1351



x I J K L M N

0.0125 0.59 1.10 1.13 1.16 1.20 1.24
.025 .66 1.06 1.09 1.13 1.20 1.26
.05 .75 1.04 1.07 1.11 1.20 1.29
.10 .86 1.03 1.07 1.11 1.21 1.33
.20 .96 1.03 1.08 1.12 1.22 1.36
.30 1.02 1.04 1.09 1.14 1.23 1.33
.40 1.05 1.05 1.10 1.15 1.23 1.22
.50 1.08 1.17 1.19 1.19 1.14 1.04
.60 1.12 1.18 1.11 1.05 .92 .80
.70 1.09 .98 .88 .79 .63 .51
.80 .89 .65 .52 .42 .28 .19
.90 .33 .06 -.03 -.10 -.18 -.22
.95 -.32 -.47 -.50 -.52 -.53 -.52
.975 -.95 -.93 -.91 -.89 -.84 -.78

46 NACA TM 1351




NACA TM 1351


x DO DI D3 D5 Dm E4 F3

O 0 0 0 0 0 0 0
.003 .0295 .0282 .0292 .0300 .0371 .0313 .0327
.006 .0529 .0509 .0526 .0540 .0666 .0564 .0589
.0125 .0969 .0914 .0969 .0995 .1220 .1039 .1081
.025 .1687 .1636 .1695 .1740 .2119 .1814 .1887
.050 .2864 .2797 .2897 .2973 .3585 .3093 .3212
.075 .3843 .3772 .3907 .4006 .4787 .4161 .4315
.10 .4690 .4620 .4784 .4902 .5808 .5084 .5266
.15 .6098 .6045 .6252 .6396 .7443 .6615 .6831
.20 .7219 .7191 .7424 .7579 .8638 .7812 .8041
.25 .8113 .8112 .8353 .8507 .9451 .8733 .8952
.30 .8813 .8838 .9067 .9207 .9898 .9387 .9586
.35 .9341 .9384 .9579 .9688 .9963 .9826 .9939
.40 .9710 .9758 .9892 .9953 .9589 .9998 .9973
.45 .9928 .9962 1.0000 .9988 .8625 .9872 .9679
.50 1.0000 .9988 .9881 .9761 .6300 .9387 .9085
.55 .9928 .9813 9 .9472 .9178 .3885 .8561 .8238
.60 .9710 .9414 .8735 .8192 .2646 .7473 .7204
.65 .9341 .8794 .7704 .6903 .1811 .6259 .6056
.70 .8813 .7878 .6459 .5474 .1207 .4918 .4871
.75 .8113 .6898 .5108 .4078 .0772 .3682 .3721
.80 .7219 .5673 .3766 .2847 .0459 .2590 .2669
.85 .6098 .4305 .2535 .1846 .0242 .1686 .1761
.90 .4690 .2845 .1493 .1071 .0101 .0973 .1020
.95 .2864 .1366 .0651 .0472 .0024 .0425 .0442
1.00 0 0 0 0 0 0 0
x for M/f = 1 .500 .482 .450 .433 .333 .406 .381

NACA TM 1351



x DO Di D3 D5 D, E E4 F3

0.0125 4.53 -0.53 -2.50 -3.17 -5.49 -3.51 -3.62
.025 4.53 .88 -.47 -.91 -2.17 -1.07 -1.08
.05 4.53 1.89 .99 .71 .22 .67 .74
.10 4.53 2.61 2.03 1.88 1.94 1.93 2.05
.20 4.53 3.15 2.80 2.75 3.20 2.85 3.01
.30 4.53 3.40 3.17 3.15 3.79 3.29 3.46
.40 4.53 3.56 3.40 3.41 4.18 3.57 3.61
.50 4.53 3.68 3.57 3.61 4.46 3.50 3.10
.60 4.53 3.61 3.21 2.95 -1.03 2.35 2.14
.70 4.53 3.19 2.07 1.31 -.83 1.00 1.04
.80 4.53 2.43 .75 .05 -.63 .03 .17
.90 4.53 1.34 -.05 -.26 -.42 -.26 -.22
.90 4.53 .68 -.15 -.20 -.29 -.21 -.21
.975 4.53 .33 -.08 -.14 -.20 -.15 -.15

NACA TM 1351



x DO D1 D3 D5 D. E4 F3

0.0125 -6.31 -6.13 -6.35 -6.52 -7.92 -6.79 -7.07
.025 -5.28 -5.18 -5.37 -5.50 -6.61 -5.73 .-5.94
.05 -4.25 -4.21 -4.36 -4.46 -5.25 -4.63 -4.79
.10 -3.17 -3.19 -3.29 -3.36 -3.78 -3.46 -3.55
.20 -1.99 -2.05 -2.09 -2.10 -2.00 -2.11 -2.11
.30 -1.21 -1.27 -1.22 -1.18 -.52 1.09 -.99
.40 -.58 -.58 -.43 -.31 1.26 -.07 .26
.50 o .14 .50 .76 o 1.33 1.46
.60 .58 1.02 1.79 2.32 2.00 2.38 2.21
.70 1.21 1.90 2.63 2.87 1.02 2.30 2.36
.80 1.99 2.61 2.60 2.24 .52 2.00 1.97
.90 3.17 2.97 1.88 1.35 .21 1.25 1.31
.95 4.25 2.91 1.63 1.05 .10 .91 1.01
.975 5.28 2.75 1.59 .94 .05 .82 .88

NACA TM 1351



x DO D1 D3 D5 D, E4 F3

0.0125 4.53 4.22 4.34 4.47 5.72 4.70 4.93
.025 4.53 4.22 4.34 4.47 5.72 4.70 4.93
.05 4.53 4.22 4.34 4.47 5.72 4.70 4.93
.10 4.53 4.22 4.34 4.47 5.72 4.70 4.93
.20 4.53 4.22 4.34 4.47 5.72 4.70 4.93
.30 4.53 4.22 4.34 4.47 5.72 4.70 4.93
.40 4.53 4.22 4.34 4.47 5.72 4.70 4.78
.50 4.53 4.22 4.34 4.47 5.72 4.42 4.06
.60 4.53 4.05 3.84 3.65 o 3.10 2.92
.70 4.53 3.54 2.57 1.87 0 1.60 1.67
.80 4.53 2.70 1.14 .48 0 .49 .65
.90 4.53 1.52 .20 .03 0 .05 .11
.95 4.53 .80 .03 .00 0 .00 .02
.975 4.53 .41 .00 .oo o .00 .00

NACA TM 1351

_________ 0 OLcO t*FN

o o\ o\ co 0O0 o0 ccD ,-..- 0

_I :' 0'''_'''*'o

H CI r--
O c V ) wO O 8 c) mlk 0 m m m m
3 (u
D 00 C 00000
1Y cO rl F- m n L U _: ot

Ii 9 r r4 r 4 A 9 .4 9

E|-- co O Cu i 0-4r -l Yi -lHr
oo0ot 4C cnou 000 00
O OC r-;0 \I r; 0 H Ht CU 0 H O

CP\ c 0 n C) cu H O O O
*. ,- L tN 04 0 O 0 p- 0
: 0 C CU 0- 0 \O-) CU r 0

0. D.4-flCH0H00

00000 000000

II a 9 1 0 09 *


NACA TM 1351


NACA 0010



Figure 1

Figure 2

NACA TM 1351

U I.- I I 1-
LB No. e m h d,
0.1 -24---- 0.1 0.5 035 2.5
31 --- 0.1 0.5 0.50 2.5
0 32 0.1 0.5 0.70 2.5
33--- 0.1 0.5 1.05 2.5
I---- I





Figure 3




* I ___

0.8 09 1.



05 0.6

NACA TM 1351




Figure 4

S.- 0 --05-. 0.70- -
03- ---- ---N_


Q2 0---- 0 -0

0 x
0 0.1 0.2 03 04 0.5 0.6 Q0. 0.8 0.9 1.1

LB No. e m Ih d,
-- 27--- 0.1 0.600.35 2.5
39-- 0.1 0.50 0.50 .0 /
-- 52- 0.1 0.4
156 --- 0.1 035 0.63 1.0 1 .0




Figure 5




NACA TM 1351

Figure 6

NACA TM 1351



5 \

W:0 V7 1.

" s
ol- --_ _-:=-

- -- Do

D5\D3 \

02 0 0 _

0- I---- -- I-
-1.0 -Q8 -0.6 -0.4 -Q2 0 0.2 0.4 0.6 0.8 1.0

Figure 7

M oC

S 0.7 0.8 09 1.0
x I I


In it* *__




I 0.2 0.3 0.4

NACA TM 1351

a s 0.990

0.81 I I I--I I I I I I

D5 K-2015

0.6 _


O 0.1 0.2 0.3 04 0.5 o. 0.7 0 .9 I.C

Lower surface
-0. ____
-o.6 --- -- -- 7 -~- 'J -- -- -

Upper surface
-08 I 1 1

Figure 8

n _


NACA TM 1351

In i-._ u~u~.. r

U. n --, F #-- I t





OA ---- -- --- --- __ --- --- ---_


0 -.- 0
0 0.1 0.2 0.3 OA 0.5 0.6 0.7 0.8 0.9 IX)

Figure 9

f .

/\ r


NACA TM 1351

0 =1100

p F3M-2015
q O.6 -- -


o 0.1 0,2 0.3 04 0.5 Q6 O. 0.9 1(

-04 Lower surface
-0.6 -
-0.6 __ s___ __- -- -- -

Upper surface

Figure 10

Figure 11

NACA TM 1351

Figure 12

0NACA 0010 -

0.005 Ik --" "
..oLB 2


105 2 4 6 8 106 2

Reynolds number R

Figure 13

NACA TM 1351

G x
as \

-X- NACA010 R = 2.31X106
-o- LB 24 R=2.22X106

p ---o X-
"-o -X X x-
n --------------

0.03 0.02

Figure 14


NACA 0010 50 50
S 0% chord



-0.2 -
A i a i6 o A


Reynolds number, R

Figure 15

0.01 0 0.01 0.02


LB 24

90% chord



o--- a If

4 6

w IW-

A c0 a IrM,

rIACA TM 1351

Reynolds number, R

Figure 16

4 6



04 _
o \

0.3 0
Q2 ---


0 0,002 0.004 0.006 0.008

Figure 17



--- -. 1.0

C- = 0.1-


--- -0.4
0 0.2 0.4 0.6 0.8 x 1.0

C ,=0.2 -
L q

-----T7 T

I I I 0.6

Measured o* (o upper surface lower surface)

Figure 18

0 0.2 Transition 0.6

0.8 x 1.0


C D NACA-O0 I I-''h t1
0 0.004
0003 -; -^


0 0.1 0.2 0.3 0.4 0.5 0.6 0.1
x for minimum pressure point

Figure 19

IJ -
NACA 23012 vs LB DSK 1215
0 -- -

q o4t_

Figure 20

NACA TM 1351


Id ^

Q &


e =0.1

e = 0.2

e =0.3

Figure 22

NACA TM 1351




p R=

-0.1 -


0 0.1 02 0.3 0.4

0.5 0.6 0.7 0.8 0.9 1.0

Figure 23

HACA TM 1351

0.0 12
Bore model

o.ooe --

0.5mm wire at X 0.9

logl R

Figure 24

Figure 25



- -


NACA TM 1351

Z 4

6 8 10o

2 4 6 8107

Figure 26

Re 0.6

Figure 27

0.015 --- --- -- -- | -- I
o 1.5m Tunnel Komaba

2.5m Tunnel Gihu

Y \'"-

005 -- ^ ---__ _

J L I -







NACA TM 1351

_ 2

Figure 28

k= 0.4 mm X 150 mmJ
450 mm

G_* 600mm
I smooth

0 L
0.05 ---- x

0 10 20 30 40 51

Figure 29

NACA TM 1351

+ k= 0.25mm
-2.6 '- o 0.4mm
A 0.7 m m

-3.0 --


- 3L

5.4 56
log vx-

5.8 6.0 6.2

4.8 5.0 5.2

Figure 30



Figure 31


NACA TM 1351

Figure 32





-3.8 -- -- -- ---- -- ---


Figure 33


5.6 5.8

NACA TM 1351


Figure 34


0.47 -1.40

Q30- -1.22




Figure 35

NACA TM 1351

Figure 36

Figure 37

NACA-Langley 10-24-52 1000

L max


=-i 9-i g1- 0'
*L C, u C-j Z
*" C 4l Jbfl c urN
0 ~e .m:.' -NC fl-
Iu C Ii CQ

-1 r' ma; "- 3 ^
2 cu .
CI w a, m -' i?
a e 0. i i .. 0
iz 0 : 0 t >L 0 W ~ Co a: n 1

;eIg SiSlll?
.- CO0L -4

ga ai* rtaii- 3 _i
vi 4"C; ui e i

I 1
s23- ^ 0....s o C c
ed : Cl3 Z

S, .. .g iS < 'g0ll^l5f
o a)

3-l- cCll3T )
I,. w c: *S

irIi s h
1- 4) 0 ro r IB I
~0 to l0. ~ C

cm 6 o T-agk fl
.0 I s 3
am o 'o"
ootB so. 0, V 0,
1 C 0
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S. Z PL, Z, 0 0-1 '00

m~ .

4 0
o 'm

UU 0
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u zo .2 ;MD Eq
ZO H INZ co 0 o: -.4 r. 0,4 o,

V; W .I, z~o -
02 020 ow c!
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0 Z N ~X
to ~
0 0S
H z

ird 6

a 1 a pp

-l N E q 3 t

2 .0 !

u .. bL. ., i0 U -
o o a o- -

ri N Eq ui :z w iB

4 a
in 0. C;I

- a a
S Oa R at0

40 0 0
I3 I3

<-1 0 C; < 3>SN*

z cz


S S'


0 ,e

il Ul
-a 8a.'i
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io al,
0 5 0

o a

1 ; 1


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4: -4: a

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0. 0 0 m ,
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-4 vi C ;l13 6 0
'^*< ~ Ci ML ~ Et 3 I j

1 c E-4 C

r4 0
i1ri s E4
(E i 3 l l ';r ii
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L )C oi CU4'aE oo
c .4 gj..i- -4i d

-u --i0 1I _, c -*- -
ai :3 3i 0 T'So t 3 o 0tS ..- l
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MU = :1 l to iUll
0 Co V< 0 rt :S 4
0 c e a
cd m ,% z 4
~S~ 4%
PI -
0 ow 0 m o a k w 0
lu~ m, 1.3^s)r~ s4< 0, zog<-i t '
.< if w~ 5


012 0
0 g R w-,- E g ;; p -I ;00! [,w0 d 0

S0 =) ow o A v it 0'o' m E4M-a q. f- w 20
d -0 0 s .
0i o C)
I!0^ o'l.-^: 0i^f 5 .|M 0 0
oEi j B S a -n' s ., E' as

P4 OR (.0, |, g.0, g -Va c ta ^^ ^ l l a s -s
B Saoagi a, o 3| Ias
1,04 -vu To I 1'.s r I
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0 Qom- -u Or
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uJ uu
t o

Sf 5 3

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U, U
4) t )-
a 25
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r-< U


= .- M pa

i I v i
Sg 4 .2 -; 6 0

-3 F V -

- E. t -6
2~r > .2 z

-a =
3 *-5S



0 o S4
o 2

S0 2 N







s w o t0 g
iI o I UC

0 n N i

Ue y
S n

i: : F-.I I a
au Ma g u- g) o- u

r an V a -?4
94 C4 4 l it; l. cc; /




- .
z z

0 F- *
a) ., ~ mV

w 0 0 CD o
S. a | |I |g

0 S

g.-S "C
* 0 U

5" a
20 g


4) 3.

U e


III^i 3; 1^ r
ao FA

a / u 0


: ~ L O

I Ii
a a

s s

5 5
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I^S" 5\ lio
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