,A~A ~!itiiq
I 1 v i.  I
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM H10. 1185
SYSTEMATIC INVESTIGATIONS OF THE INFLUENCE OF
THME SHAPE OF TIE PROFILE UPON THE POSITION
OF THE TRANSITION POINT*
By K. Bussmann and A. Ulrich
The position of the beginning of transition
lamin ar/turbulent as a function of the thickness and the
camber of the profile at various Reynolds numbers and
lift coefficients was investigated for a series of
Jonkowsky profiles. The calculation of the boundary
layer was carried out according to the Pohlhausen
method which may be continued by a simplified stability
calculation according to H. Schlichtin, (4). A list
of tables is given which permits the reading off of
the position of the transition point on suction and
pressure side for each Jcuk:owcky profile.
OUTLI NE
I. Statement of the problem
II. Extent of the investigation
III. The calculation of the potential velocity and the
practical application of the boundary layer and
stability calculations:
(a) Potential flow
(b) Boundary layer and stability calculation
*"Systematische Untersuchungen uber den Einfluss
der Profilform auf die Lage des Umschlagspunktes."
Zentrale fri wissenschaftliches Berichtswesen der
Luftfahrtforschung des Generalluftzeugrneisters (ZIB)
BerlinAdlershof, Technische Berichte und Vorabdrucke
aus Jahrbuch 1943 der deutschen Luftfahrtforschung,
Band 10(1943), Heft 9, Sept. 15, 1943, IA 010, pp. 119.
NACA TM No. 1185
IV. Results:
(a) Influence of the Cavalue and of the
Reynolds number
(b) Influence of the camber of the profile f/t
c Influence of the thickness of the
profile d/t
(d) List of tables for the separation and
instability points for all Joukowsky profiles
(e) Mean value of the laminarflow distance of
suction and pressure side for all Joukowsky
profiles
V. Summary
VI. References
SYMBOLS
x,y rectangular coordinates in the plane
s profile contour length starting from
the nose of the profile
t wing chord
t? length of the profile contour from
nose to trailing edge (different for
pressure and suction side)
Uo velocity of incoming flow
Um(s) potential velocity at the profile
6p4 boundary layer thickness according to
Pohlhausen P4
6* displacement thickness of the boundary
layer
5P42 Uo
4 =  nondimensional boundary layer thickness
4 = T form parameter of the boundarylayer
o profiles according to Pohlhausen P4
NACA TM No. 1135
P6 form parameter according to
Pohlhausen P6
f~p4ji p universal functions of the boundary
f P 'g.P layer calculation
Scrit. position of the instability oirit,
measured along the contour of the
nose of the profile
sAP6 position of the s.paiatio.n oi.nt
according to P6 method
I. STATEMENT OF THE PFOBLEMF
The position of the tra:nition point laminar/turbulent
in the frictional boundary ia,. r is of decisive importance
for the problem nf the the~ eti,"al calculation of the
profile drag nf an airfoil s:'ncc the friction drag
depends on it to a high decrc. The position of the
transition point on the airfr,Il is largCely dependendent
on the pressure distribution along the contour of the
profile and, thercfcre, on the shape of the airfoil
section and on the lift coefficient. A way of theoretical
calculation of the start of transition (instability
point), that is, the point downstream from which the
boundary layer is unstable, was recently indicated
by H. Schlichting (1,3,4) and J. Pretsch (').
According to present conceptions the turbulence
observed in tests develops from an unstable condition
by a mechanism of excitation as yet little i:nown;
therefore, the experimental transition point is a.:a.ys
to be expected a little further back than the theoretical
instability point.
Knowledge of the theoretical instabil.ty point is,
nevertheless, important for the resear:oh on profiles,
in particular for the drag problem. Recently a report
1An extract of this report was given in a lecture
of the firstnamed author at the Lilienthal meeting
for the discussion of boundarylayer problems in
GOttingen on October '2 and 29, 1941.
NACA TM No. 1185
was made about airfoil sections which, due to a position
very far back of the instability and transition point,
have surprisingly small drag coefficients laminarr
profiles). Thus far no systematical investigations of
the influence of the shape of the profile upon the
position of the transition point have been made either
experimentally or theoretically. The following calcu
lation of the theoretical instability point is,
therefore, given for the first time in a sufficiently
large range of cavalues and Reynolds numbers to
achieve a greater systematization of airfoil sections.
In order to keep the extent of calculations within
tolerable limits only the two most important profile
parameters, thickness and camber were varied. A rather
convenient and accurate mode of calculation of the
potential flow for the profiles is important for these
investigations and the selection of a series of
Joukowsky profiles was, therefore, natural. It was
not advisable to take for instance the NACA series as
a basis; the calculation of the potential flow for
such profiles according to the methods at present
available does not achieve the accuracy ochich is
required here.
II. EXTENT OF THE INVESTIGATION
A series of ordinary Joukowsky profiles of the
relative thicknesses d/t = 0, 0.05, 0.10, 0.15, 0.20,
0.25 and the relative chambers f/t = 0, 0.02, 0.04,
0.08 were taken as a basis. (See fig. 1.) For instance,
J 415 stands for the Joukowsky profile of camber
f/t = C.04 and the thickness d/t = 0.15. The c region
which was examined is ca = 0 to 1 and the Renumber
Uot 4 8
range Re 10 to 10 The complete calculations
were carried out only for the following Profiles: 000,
005, 015, 025, 215, 400, 415, 4J, 800, 8.15, and 825.
The results for the remaining profiles could be obtained
by interpolation. Thus it was possible to obtain a
result with tolerable loss of time in spite of the very
extensive progJam (four parameters); a certain amount
of accuracy had to be neglected since the interpolation
sometimes was carrlel out .over three points.
NACA TM No. 1185 5
III. THE CALCULATION OP THE POTENTIAL VELOCITY
AND THE PRACTICAL APPLICATION OF THE BOUNDARY
LAYER AND STABILITY CALCULATIONS
(a) The Potential Flow
The calculation of the potential velocity with its
first and second derivatives along the profile contour
forms the basis for a boundary layer and stability
calculation. The potential flow about a Joukowsky
profile is obtained by conformal. mapping of the flow
about a circular cylinder. (See fig. 2.)
A short list of the most important symbols and
formulas for the profile contour and for the velocity
distribution follows:
z = x + Jyi
z = Coord.i.nates in the complJ]x plane
mapping function:
z
circle F:4)mean caemb,.r line of the profile
circle K  cair.be,'ed ;..: .[le
a radius of the ini.t circle in the zplane
R radius of the circle tc be mapped in the zplane
t wing chord.
t' length of the profile contour from nose to
trailing edge (d.ffr.rent for suction and
pressure sade)
Xo, Yo coordinates of the cent,er of the circle to
be mapped in rhe zilane (circle If)
NACA TM No, 1185
o, y1 center coordinates of the mapping circle of the
mean camber line of the profile (circle K4)
p varying angular coordinate of the conformal
transformation
P zero lift direction (See fig. 2.)
a angle of attack of the airfoil referred to
the theoretical chord
ag geometrical an.le of attack referred to the
bitangent (See fig. 2.)
Profile nose: q = r + 8
Trailing edge: 9 = 4
xo
a e = thickness parameter; k = 1 +
See table I.
yl
= = camber parameter
p= arc cos = arc sin 
1+ E1+E64
The profile parameters El e4' and B can be found
in table 1.
Profile contour:
x k + 1
t a 1 + E2 cos C e + + +
+ 3(( sn )( j+(1)
NACA TM ITo. 1185 7
N =k2 + 2e) 2 + ?k 1 + e cos p + k sin
t1 + + 62 (1(a))
a 2+ k + E
Nose radius P/t!
For symmetrical profiles the equation
t 261
1 + 2e + 4 2 (2)
1 1
is valid exactly. This formula may with a good approxi
mation also be applied to cambered profiles. The
numerical values in table 1 show that the nose radius
of the Joukowsky profiles is only little larger than
for the NACA profile family according to NACA report 460
for which P/t = 1.1(d/t).
Velocity distribution:
Um
U = 2 Isin ( a) + sin (a + 8) P1(c) (3)
Stagnation points: Back p =B
Front q = 7 + B + 2a
c) N (11 4)
PN(N 1)2 + 4k2 + l + 142 sin 2)2
NACA TM No. 1185
Velocity at the trailing edge:
lim cos (a + P)
k  U E
k 1+ 2
(5)
Arc length:
d.s /dX\2 d)2
d~p~ug^ +
(6)
s as a function of c is to be ascertained from (6) by
graphical integration or can be seen directly in an
enlarged presentation of the profile contour (t = lm).
Velocity gradient:
1 dUa1
0 d 2
o dcp
cos (Cp . a)
+ N7B sin ((p
A = 2k( +1+ A2
N1 = (N )2 + A2
B= N1 Nt N
 a) + sin (a ( I
sin p9)
N 1) N + 2k + 2 A cos
Nt = 2k 1 + e 2 sin p. + e k cos P)
(7)
NACA TM No. 1185
The first derivative of the rnotnctial velocity vith
dti11
respect to the arc length y was calculated
numerically from equations (6) and (7), and from that
d Um
graphically the second derivative 
ds2
Relation between ca and a:
ca = 8rn sn (a + B)
R k 1 + Fl2
t= ; compare tables 1 and 2.
3 + 2 ]. + + E
(b) BoundaryLayer and Sta.bility Calculation
After calculation of the potential velocity uvth
its first and second derivatives along' the profile
contour there is a boundarylauer and stability calcu
lation to be made fur er.. h prCfile. The boundarylayer
calculation according t) Pohlhau.sen (5) was based upon
the differential i quatj.an for the boundar layr thick
ness in the shapr incicat.~'d by Howarth (6) .
dz! f( l
z4 (:/7T .u+ z 4 ()U' (8)
2In the meantime a simpler form of the Pohlhausen
equation was indicated by H. Holste:in and T. Bohlen (10)
where the momentum thickness appears as independent
variable. For this method the second derivative U"
is unnecessary; the integration procedure is thus
simplified considerably.
NAOA Ti No. 1185
The following symbols stand for
U_
U.T, U'
S
t m
U ds
o
.2 d2
S "1= '. 2
o ds
6 = boundary layer thickness according to Pohlhausen P.,
f.
att
_" v tUo
4 = z 1' = form .ar.tter acccr'ing to Pohlhausen
fp x and g(' = universal functions:
S72 + 2
( 75 1
f(%) =
57 _.
5JL 3
2.
g(x) =;
6 L ,6o
630 630
c 2
J.
72
20L2
._M^
Initial conditions:
At the st cnstion point
= 7.052
that is,
Z4o T0 U UI
(9)
NACA TM No. 1185 11
Besides,
U"
z' = 5.391 U,2 (10)
The isocline method was selected for the solution of
the differential equation. The particular advantage of
this method is that not only the initial value z4o is
knoin, but that the initial inclination at the stag
nation point z40'. also can be determined. The latter
value is obtained by exact performance of the limiting
dz
process lim Tg in (8) (Eowarth(6)). With z4O' known
U)0
the integral curve passing through the initial value Zho
is easily found which otherwise is not i=mnediatPly
possible because of the singularity of the Pohlhausen
equation at the stagnation point.
For the profiles of the thickness d/t = 0, that is,
for the flat plate and the circulararc profiles, the case
where the flow does not enter abruptly (a = 0) is
exceptional since there exists no true stagnation
point: the velocity at the leading edge has a finite
value different from 0. The initial value of the
thickness of the boundary lajer is here zero, that is,
at the leading edge there is:
.0 = 0 (11)
Profile ca not abrupt
flow entrance
0 0
200 .25
400 .5
800oo 1
NACA TM No. 1185
The velocity near the leading edge of circulararc
profiles takes the same course as Vs:
U = Uo + C / + .
that is, Ut becomes. ith s 0 infinite like 1I/v'
the velocity has a perpendicular. tangent which always
occurs when the contour of the profile shows a sudden
change in curvature as it does here (v. Kcppenfels (8)).
z4 near the leading edge for a circulararc profile
behaves like z4 for the flat plate, that is, z4 goes
in a linear relation to s towers. 0. Taking these
facts into consideration there results at the leading
edge:
? = 0; z, 05 (12)
o 40 Uo
It has proved ,advanta< us to calculate the line
elements z4t directly from the equation (8) by means
of a plotting of the curves f p and g p), (See
fig. 3.) This method is superior to the calculation of
the line elements by means. of the often used nomograms
of anglerr (7) with respect to accuracy and its equal with
respect to loss of time. Generally it will be sufficient
to determine the line elements for each value of the
abscissa s/t at two ordinate values only.
The bounndarylayer. calculation yields for each
profile for a given cP value the nondltcnsional
boundary layer thickness z4 and the form prrameter hp4
as a function of the length of the .rc s along the
contour. The distribution of velocity u(y) in the
laminar boundary layer is then obtained from:
3For the flat plate z = 34.03 s/t (according to
Pohlhausen (5)).
;IACA TM No. 1185
^'T '"P49 (4)
with
F =2 2 ) + ) 1
S1 y4(1 4)
6 =6 PL J 2p Lpj
The results of the bounda.rylayer calculation for
the profiles J 000 and J 025 hve been clotted as
examoles in figures IL ,nd 5: the forn parm.eter Xp
and the nondimensional disolacament tljick
ness " , with 6' standing for the displace
t
ment thickness.
The following relation exists betv:een the displace
ment thickness andthe bounrdryi layer thiclk'ness according
to Pohlhsusen:
5: = a'r. lx" ( 15)
The displacement thickness cf th e flat late in longi
tudinal flow P, = 0 s renpreente'td gra3 ically
in figures h and 5 for comrariJson. Tre following equations
are valid:
: = ot = 1.75 (16)
"t PI) t V FILL
.:Cc:. TI! No. 1105
The profile J 800 (fig,,;.) shows clearly that
the displacement thickness for accelerated flow (suction
side) is smaller than the displacement thickness of the
flat plate whereas it is larger for retarded flow
(pressure side). (Compare also fig,..16.)
From the boundarylayer calculation there result
also the laminar separation points. According to
the fourterm method of Pohlhausen separation occurs
at p4 = 12, according to the sixtermr method (see
*below) at Ap6 = 10 corresponding to p4 = 9.65.
Flow photographs have been taken in a Lippisch
smoke tunnel for a part of the calculated profiles of
Uot
models of 50centimeter wing chord and at Renumbers
of about 2 x 105. The points of separation have been
ascertained from the flow grsphs (figs. 6 to 11, appendix).
Figure 12 shows the experimental and theoretical separation
points for various profiles for comparison. Corpare .
also table 3. The agreement is rather good.
After hp has been ascertained as a function of
the length of the arc s there results the instabilityy
point (s/t) t from a stability calculation
crit.
(H. Schlichting (1).)) base on the sixterm method of
Pohlhausen. The P6methodi is based on a oneparameter
group: (parameter Ap6) of boundarylayer profiles
which can be represented by polynorials of the sixth degree.
An investigation of stability was carried out for a
number of these bbundarylayer profiles in (4); first,
the critical Renumber of the bo.undiary layer cr
crit.
as a function of Ap was obtained. The critical
s\
Renumber of the laminar layer \ as a function
crit.
of ?p4 (fig. 13) is then immediately known also
because of a universal relation between Xp6 and AP4
indicated in (4).
NACA TM No. 1185 15
Once pi (s/t) has been ascertni'il. firo the
boun.arylayer calculation according to FPhlhauce.L's
neT.hcd a crit'.cal Renurocr  may be
\. u .brlt.
coordinated to each point of the profile ty means of
fi urs 1.. rureover the Re.ulm!br of chu.. boundary
layer ca u calculat'id for each point of th
1 Io t
profile at ceyt9ln :
T, TI T5 t 0
 = j (17)
The loc..tion cf the insta.bility point i~ thn i.v n by
To* 'U *
m =m
 (1iS)
: .U /ccrit.
I'. 3 r.rLr3
(a) Infl'.ence of the caValue ani the Renumber
ihe r.j .ts of the stabilit. calculati.,, that is, tho
position of t th theoretical ins tatilit.) poitr,
'crit.
foi the samiple profiles J 00 J C025 ,re: plotted
in figures 1LL anr. 15 against ca with the Re.n.umbr
as oar',niter and furthF'rmrr: against  witI the
0
c .valsc as prr.mcter. lThe c tharacteriatic coarse of the
cuIrves is thc sa:ae for all profil,ss; the following
statements :.rc. vj.d1: the instabllity point travels,
with increaai.rng ca a.t a constant Renuber, forward
NACA TM No. 1185
on the suction side, bac!_ward on the pressure side;
the instability point travels forward on both suction
and pressure side with increasing Renumber at a
fixed c value. This behavior is demonstrated very
clearly in figures l6 and 17 which represent the
velocity distributions.for the two profiles J 800 and
J 025 for the various cavalues with instability and
separation points. One can see in particular that the
instability points of the suction side for Renumbers
Ut
from 105 to 107 lie near the velocity maximum;
mostly the position of thi instability point for Re = 10
agree s well with the location of the velocity maximum.
The pressure side of J 800 in the case where the flow
does not enter abruptly (ca = 1) is an exception
among the above mentioned examples, since the flow from
the leading edge to the center of the profile is con
siderably increased so that no relative velocity
maxir;um exists. Measurements concerning the dependency
of the transition point on the cavalue were taken
by A. Silverstenr and J. V. Becker (9). These tests
showed (as a result) the same dependency of the
transition point upon the lift coefficient as the
present theoretical investigations.
(b) Influence of the Camber of the Profile
The influence of the camber upon the position of
the instability point can be described as follows:
the instability pointt travels with increasing camber,
at constant thickness, for all c values and Renumbers
backward on the suction side, forward on the pressure
side. This influence of the camber can be understood
from the fact that the stagnation point and therefore
the region of the accelerated stabilizing flow travels,
with increasing camber, backward on the suction side
whereas because of the flow around the nose of the
profile a region of considerably retarded destabilizing
flow originates iirarctdi.tely behind the nose on the
pressure side. Figure 18 represents as an example the
results for profiles of the thickness d/t = 0.15,
with variable carmber f/t for c = 0.25 and again
the Renumber as parameter. The Rurves for all thick
nesses and all cavalues have the same characteristics.
NACA TM No. 1185
(c) Influence of the Profile Thickness
The dependency of the instability point on the
thickness cannot be desorib.d in such general terms
as the influence of the caliber since this influence
depends in the following way on Lhe ca value: Ax
certain "ca not abrupt flow entrance", that 1s, the
ca value that corresponds to the not abrupt entering
of the flow (a = 0) for the circular arc profile
with the given camber, is coordinated to each value of
the camber f/t. The curves (s/b)crit. versus d/t
at a constant f/t show on principle two different
types (fig. 19):
I. With increasing Itlickness, the curves (s/t)crit.
versus d/t start from a 'inite value and have a
flat minimum:
On the suction side for ca ca for not abrupt
flow changes.
On the pressure side for ca C a for not abrupt
flow changes.
II. The curves (s/t)crit. versus d/t rise starting
from 0 with increasing thichnes:; hence, the transition
point moves backward as follows:
On the suction side for Ca > Ca for not abrupt
flow changes.
On the pressure side for ca < ca for not abrupt
flow changes.
The results for the symmetrical profiles at ca = 0.25
are represented as an example in figure 20. 2or the
symmetrical profiles ca for not abrut flow changes = 0,
that is, the dependency of the instability point on the
thickness d/t for all ca > 0 is of type II on Lhe
suction side, of type I on the pressure side.
The flat minimum in curves of tyDe I dos, in some
cases, not exist at high Renumbers (lie = 107 to 108),
and (s/t)crit. versus d/t rises from the finite
value d/t = 0.
NACA TIT No. 1185
(d) List of Tables for the Separation and Instability
Point in all Joukowsky Profiles
The total result of the. boundarylayer and
stability calculations is represented by a graph of the
curves (s/t)A 6 = cost. and (s/t)crit. = const., resoec
tively, in a system of axes thickness d/t camber f/t.
(See figs. 21 to 30.) A profile, corresponds to each point
of the plane. In particular, the symmetrical profiles
are coordinated to the points of the d/taxis, the
circular arc profiles to the points of the f/taxis,
and the flat plate corresponds to the zero point.
Lift coefficient and Renumber are considered as
parameters. One.has therewith a catalogue of Joukowsky
Profiles that make it possible read off, for every
profile in the region 0 /t = 0.25; Of/t = 0/00,
the position of the separation points for 0 = c 1
(figs. 21 and 22) and the position of the instaBility
point for 0 ca 1 and 105 = Re 10. Figures 23 to 30
represent the curves (s/tcrt = const. for .the
Uot .0
Reynolds numbers from Re = U 05 to 108 at
the cavalues .c. = 0, 0.25, 0.5, and 1 for suction
and pressure side. For instance the values indicated
in the following table for profiles of the camber f/t =0.02
and the thickness d/t = 0,10 to 0.15 at Re = 106 and 107
are taken from these representations. (See page 19.)
The most remarkable matter in this graphical repre
sentation is the location of the curve (s/t)Ap 6 = 0, and
(s/t)crit. = 0, respectively, at the various 'cavalues.
The position of this zero curve in the catalogue for the
instability points willbe discussed; the same is valid
for the separation points. (s/t)crlt = 0 can only
appear for the flat plate and the circulararc profiles
on the suction side for ca > ca for not abrupt flow
changes, on the pressure side for ca < ca for not
ITACA Ti
0 'U
r
S0
cn
c ;
iO
r i ~~^
i No. 1105
0
II
d
il
0.
II
; 0
I1
0
di
u
CO)
*
r,
P
to
0,~
4)'
)
U) W
C)
4)
ii
0
*L')
0 0 ,
* **
000
ou\
* a *
* *
000
0 0 0
r1 HO
000
00 c
C H l'0
0 o r
000
000
; C ; rl
* a0
000
0 0 0
4' ... ,
c J l',
000,
C .J "/i",
*S*
C C L
i
q
0
rii;L.L~Ls~L
20 ACA T2M io, 185
c The curve (s/t) = 0 always
abrupt flow changes crit.
coincides with the f/taxis; it forms a part of
the f/taxis which is determined by the actual cavalue.
Therefore no point (s/t)cr = 0 exists on the suction
rit.
side for c 0 since c >'.
side for a = a for not abrupt flow changes
for all circulararc profiles. For the pressure side,
on the other hand, (s/t) .0 on the whole f/taxis,
crit.
There follows in the same pay for Ca = 0.25
that (s/t) = 0 for 0 = f/t < 0.02 on the suction
crit.
side and for f/t > 0.02 on the pressure side. Pressure
and suction side, therefore,. always complement each
other. T'he point which corresponds to the circulararc
profile with c = c
profile ith a a for not abrupt flow changes
(for instance J l400 at c, = 0.5, cor: ro figs. 23 to 50),
that is, the end point of the distance (s/t)c = 0
crit.
is a singular point in the following sens: 'The point
itself assumes a certain value (s/b) differentt
crit,
for pressure and suction side), but an infinite number
of curves (s/t) = const. which are crowding
crit.
together asyrrtotically toward (s/t)crit. 0 run
into it. It is true, these relations for the very thin
profiles give only qualitative results from the present
investigations. An additional series of thin profiles
would have to be investigated in order to makJe more
accurate statements possible. However,, only profiles
with thicknesses d/t > 0.05 which can be analyzed
quantitatively, are of practical interest.
For c =0, (s/t) ct is the same on suction and
a crit.
pressure side for the symmetrical profiles. Therefore
the curves (s/t) = const. for suction and pnescure
crit.
side would adjoin at ca *=0 in a joint representation
of the suction and pressure side where for the pressure
side the measure of the camber is directed downward.
For values ca / 0 also the curves (s/t) i = cost.
cri#.
NACl TIT. ITo. 1185
have co,tj.inuations vrr.ich correspond to the
curves (s/t) = zont. for the ores.urc. and
f .a v al u e
suction side, reia, ectiveJly, at the anpertainring cvalue
wti h i:ver.t e si n.
(t) lean Vlu rnf the La..~inarFlow jlc tance on
Suction rnd Pressure Side for c.11 Jci:owc:ky rrofil:s
In ie,.v of th; Le.,vLOmnen^t of J :.inear profiles
the mieanl ';lu'. of the lwninarf'ic:: Mis 'sarcE or. su'.ictiocn
anld n.r.s iuri side .E Inte:'s, .. :gur s 51 sn': 32
shi>: t.n: curves r.:an .'i*.U'. (s/t ) = coi3t. it n
crlt.
the C /t, f/L.plane fo; various lift .,coLrfic,ent.: and
ti" . :.' er =. 10i. r.. 7 = *''.5
u _crit.
ts i, + s n,;raly the
hrit. EL.^ *rit. t'.lr ; .
fsllovio n., c c l,.s i5n. a; v.ali.l: The nLo ri.!cLe ith the
1for a c. :_ a
small r, meen value (s/'t) for a certain 
lic near th: ..4.rculariarc pr'f'ilie o v'hicch hs .'alu
is co:,ll i,'.'.tei as c *
is 'or n t,.s a i .jr :ot .u ..u t 11i c: at
L. p'ofle .'jill be for ca = Che f t Lt,
for c, = O.." th: nrofil J 200, f r c = 0.45 the
oroJfil. J 1.00 a '., f l or c= 1 the pr ofilc J 300.
i' t
ThIr seems to be an exceptional case at i. o = 1C'
and ca = 0.7. (fig. 51) which can be =:.... a
foilovs: ;T'.a circulairrc .forofil: for 'h;L.:> at the
consider c; .val .e ci. f'i.: ent:rs "lnou. oru:tly"
(for instance J 400 at ca = 0.5) i ,r t iilar :o.it
in the /'/t, d/tdiagramm. 1 ro;cbl..a thi: rofi.
./r F. t L .
on thre f/tax;iw 2rom t.': ,ifer"n: .. .:id;: c.' ,bt. ins
tg'o rij'"orent _ir..t vilues (s/t) crit sinre ,rnc. nly
thae s.r:tion s.ie and .nce ,nly ,t.h. irc issurc side
contribui'.s to the r.carn "al:u. 1 ., Onlj; C'or the 'si;.J;'lar
point itl' lf l':c .on and pr 3ssou.ra side o,L. _ti.ut1 so
th:;.t this pr'ofile has a higher (1/t) an th.
cr t.
NACA TM No. 1185
profiles on the f/taxis near it. If one now considers
the curves mean value (s/t) = const. for values
crit.,
higher than the two limit values' (range I). These
curves enclose the singular point and end at two points
on the f/taxis. The remaining smaller mean
values (7/t) generally cover only a small region
crit.
near the singular point (range II.) where, with the
present investigations as a basis, more accurate
sttements are not possible. Only for the
Uot 6
case 10 and c = 0.5 the range II comprises
v a
all profiles of the series considered here since on the
pressure side the profile J )00 at c_ = 0.5 and
6 a 
Re = 10 has no transition point k ) and
 ~crit 1 
therefore the point f/t = 0.0 i obtains a high mean
value (s/t) > O.5. For this case there are closed
crit.
curves (s/t) = const. and there exists a profile (J 115)
with the smallest mean value ( /t)cri = 0.155 at
Re = 106
T'oiorover, the following results are obtained from
figures 31 and 52: All Joukowsky profiles have small
mean values (s/t) ; for instance, the mean values'
crit.
for practically.important profiles with the camber
f/t = 0.02 and the thickness d/t = 0.10 to 0.20 at
Renumbers of 106 to 107 are between 0.08 and 0.2.
These mean values are only to a small degree dependent
on the lift coefficient; for instance, the mean values
for the profile J 215 at Re = 10 and at .lift coef
ficients c = 0 to 1 are between 0.16 and 0.175.
a
V. TYn .
A series of Joukowsky profiles with thick
nesses d/t = 0 to 0.25 and chambers f/t = 0 to 0.08
was investigated with respect to the position of the
instability point for various lift coefficients and
Renumbers. The following result was obtained: With
NACA TrL I'o. 1105 23
increasin: enumrtber1;, th: ijntab!!ity ;.oint r.ves
forward on sactic'a nrid o'res;.!re Eid ei wilhi i.nctressin
Cavalue it moves for ,;rd i..n the auction side, bric!:ward
on the pressure side. Tlie position of thV. i.il.t.'.'ility
point as a function oi thickness and csarber of blhe
profile is represented in the chape of a ;rapth'ical
list of tables which rerNit; the r:adin ff of the
position. of the instability point on suction and pressuoe
side as well as of the m,an value of cL' l:jitinnrflow
distance on suction iiand pressure side for e :.ch profilee
of the series.
Translated by 'ary L, ;' hler
INational Advisory
Coirmittee for aeronautics
NACA T_ Ao. 1105
VI. REFERENCES
1. Schlichting, H.: Uber die Berechnung der kritischen
Reynoldsschen Zahl einer Reibungsschicht in
beschleunigter und verz6gerter Str6mung.
Jahrbuch 1940 der deutschen Luftfahrtforschung,
P. I 97.
2. Pretsch, J.: Die Stabilitat der Laminarstromung bei
Druckgefdlle und Druckanstieg. Jahrbuch 1941
der deutschen Luftfahrtforschung, p. I 58.
3. Schlichting, H.: Berechnung des Umschlagspunktes
laminar/turbulent fur eine ebene Platte bei
kleinen Anstellwinkeln. Nicht ver6ffentlichter
Bericht.
4. Schlichting, H., and Ulrich, A.: Zur Berechnung des
Umschlagspunktes laminar/turbulent.
Preisausschreiben 1940 der Lilienthal
Gesellschaft fur Luftfahrtforschung. Jahrbuch
1942 der deutschen Luftfahrtforschung, p. I 8,
5. Pohlhausen, K.: Zur naherungsweisen Integration der
Differentialgleichung der laminaren Reibungs
schicht.Z.angew. Math. u. Mech. Bd. 1, p. 252,
1921.
6. Howarth, L.: On the Calculation of Steady Flow in
the Boundary Layer Ihear the Surface of a Cylinder
in a Stream. ARC Rep. 1632 (1935).
7. Mangler, W.: Einige ITomo ramme zur Berechnung der
laminaren Reibungsschicht an einem Tragflugel
profil. Jahrbuch 1940 der deutschen Luftfahrt
forschung p. I 16.
8. Koppenfels, W. v.: TwoDimensional Potential Flow
Past a Smooth Wall with Partly Constant Curvature.
NACA TM 996, 1941.
9. Silverstein, A., and Becker, J. V.: Determinations of
BoundaryLayer Transitions on Three Symmetrical
Airfoils in the NACA FullScale Wind Tunnel.
NACARep. No. 637 (1938).
JNAC TIM Io. 1185 25
10. Holstein, H., and Bohlen, T.: Fin vereinfachtes
Verfahren zur Berechnung lamlnarer Reibungs
schichten, die dem Naherungsansatz von
K. Pohlhausen genugen (noch nicht veroffentlicht).
NACA TM No. 1185
,,I Cr
N *A *G *
H \O Qt C O C
00000 00000 00000 00000
O 0e H
0 0 0 0
0
40
II
oHNO~~ OOOHN O000 H 0 H(L0
00000 000000 000000 000000
0 .... .... ..... .H .. H
03
S 0000oR 000000 000000 Oo0
00 do OOi< N NiM xO 0 r40 N
o ** .* *** ** .
00
l0) OOOOHI OP0 0000 CO C i H 0000
O jI H'c3r4 H H H r4'* 4 NH4 't H dq 4 3 H 10 H4 0H
b" o PC% Or\D (fio~o_ rCM 0c c a' o
000000 000 00 (M 0m CM CM (m
lt'idrd' cococo 0202 ( D\%O\OsO>
000000 000000 HH r r
*0*** *** *
000000
AN' oa H it o i o HN 'r\o H^ Maoi HoN
H H OOOOHN OON OOHN OOHHN
0 0 C 0 0
o 00oooooo 0nVmNM =a W0202 0
94
0
0
s H
0 0
0
0
43 0
o
0 0
0
. C
0 B
OD .0
*0
in10
W o
0
b
a aC
0
S. 0.
0
E m
N 0*
aa
0
a ) "
a eM
V H
u H
0 *
&0 1;
0
H
H 00
.4 01

1"Co
NACA TM No. 1185
TABLE 2
THEORETICAL ANGLE OF ATTACK a (DEGREE)
Profile
Ca
000 005 010 015 020 025
0 0 0 0 0 0 0
.25 2.5 2.2 2.1 2.05 2.0 1.9
.5 4.6 .: .2 4.1 .o 5.8
1 9.2 .4 2 .0 7.6
Ca 200 205 210 215 220 225
0 23 2.3 23 2.3 2.3 2
.25 .1 .2 .25 .3 
.5 2.3 2.1 1.9 1.8 1. 1
1 6.9 6.5 6.2 5.9 5.6 5.
Ca 400oo 405 410 415 420 425
0 .6 .6 4.6 4.6 4.6 4.6
.25 2.3 2.4 2.5 2.55 2.6 2.7
.5 o .2 .2 .5 .7 
1 4.6 4.2 5.9 3.5 5.5 3.1
Ca 8oo 805 810 815 820 825
0 9.2 9.2 9.2 9.2 9.2 9.2
.25 6.9 7.0 7.1 7.2 7.25 7o5
.5 4.6 4.9 5.0 5.2 5. 5
1 o .5 .8 1.1 11.
NACA TM No. 1185
TABLE 3
LAMINAR SEPARATION POINTS; COMPARISON
OF TEST AND CALCULATION
S = Suction aide, D = Pressure side
ca o 0.25 0.5 0.75 1
S 0.929 0.891 0.855 0 0
Theor. D 0 0
J 400 S .92 .88 .83 .75 0
Exper. D 0 0
Theory. .8925 .86 .8275 755
Theor D 0 0 0 0
J 800
S .88 .84 .80 .76 .73
Exper. D 0 0 0 0
Theor .997 .6596
Theor D 997
J 005
JS05 .70 .28 .10 .015
Exper. D
D
S .4025 3535 .P08 .252
Theo D 4025 .34 491 .592
J 025 Exer. 45 42 385 :6 .32
D x45 .485 :54 6o .68
The. .686 .60 570 .476
Theor D .200 .283 .377 .494
J 415
J415 s .95 .86 .75 .65 .6
Exper. D .19 .29 .31 .33 .:0
Theor. 7 .68 .648 .59.
Thr D .05 .0934 .1924
J 815 s .86 .78 .71 .67 .6
Exper. D .06 075 .10 .12 16
NACA TM No. 1185
000
oos
010
S015
020
f0
E^2
200
205
210
220
405
410
800
fo0
_^ = s
Figure 1. Joukowsky profiles: thickness d/t = 0 to 0.25; camber
f/t = 0 to 0.08. Profile number: for instance, J 415 stands for
the Joukowsky profile with f/t = 0.04 and d/t = 0.15.
Zero liftazis
S/
Zero
Profile parameter: E=' (thickness par.)
4=01 (camber par.)
Figure 2. Explanatory sketch to the Joukowsky transformation schematicc).
42S
NACA TM No. 1185
SStagnation point
0 2 4 6 8 w / 2 46 8 12
Figure 3. Auxiliary function f(X P4) and g( p4) for the boundarylayer
calculation.
according to P4
according to P6
entering
Figure 4. Profile J 800, boundarylayer calculation: form parameter X P4
and displacement thickness 
NACA TM No. 1185
Stagnation point
4 
2 
S  b 
1 RP=12 1\ \
 Suction side Pl4 Laminar separation points according to P4
P pressure side RPe Laminar separation points according to P6
40o
~45
L.1.r
Plain plate
with C, : 0
O_
y
7 I2 04 0.5 06 U7
Figure 5. Profile J 025, boundarylayer calcualtion: form parameter X P4
6a* Ut tst
and displacement thickness t .
AfMl
 t
I I
r J^",
i ri
\ ~ ~ ~ 0 r T v
 ... ... =i
^l
1.1
7 z
NACA TM No. 1185
A I ,, I
2, U7
4,6 ],Ou
Figure 6. Profile J 400 smoke tunnel
photographs (Re = 2. 10 ).
') not abrupt entering.
I]
U 1,01" J
Figure 7. Profile J 800, smoke tunnel
photographs (Re = 2. 105). Laminar
separation points see Fig. 12.
*) not abrupt entering.
NACA TM No. 1185
\ c.., r
II Ia ll
Figure 8. Profile J 005, smon:e tunnel
photographs (Re = 2.105).
0 0
1,9 0,25
3,8 0,50
5,7 0,75
7,65 1,00
Figure 9. Profile J 025, smoke tunnel
photographs (Re = 2.105). Laminar
separation points see Fig. 12.
NACA TM No. 1185
',5 6,3 0,25
3,5 ,2 1,07
J,5 nA
Figure 10. Profile J 415, smoke tunnel
photographs (Re = 2.105). Definitions
of oc and oc see Fig. 2.
9,2 11,3 0
7.2 9. 0.25
5,7 7,'25 1,50
3.2 ,' 0,75
1,1 I,2i 1,10
Figure 11. Profile J 815, smoke tunnel
photographs (Re = 2.105J. Laminar
separation points see Fig. ]2.
NACA TM No. 1185
IS = Suction side Theoretical
(0 = Pressure side Experimental
Figure 12. Laminar, separation points A P 6 versus ca, comparison of test
and calculation for the profiles J 800, J 025, and J 815.
NACA TM No. 1185
Umr *
Figure 13. Universal relation between the critical Reynolds number ( v ) crit..
and the form parameter P
P4"'
NACA TM No. 1185
IfIrt
06
0,5
O4
0.3
0.2
,at
47
0.6
0.5
0,3
0.2
0,1
105
106
10'
0,2ccL^5 Qi108
02S 0.5 075 1.0
105 106 107 108
 Suction side  pressure side
Figure 14. Profile J 800: Result of the
Uot
t and
v
stability calculation, ( )crit. versus
Ca
NACA TM No. 1185
h  
10
025 05 ,75 10
 Suction side  pressure side
Figure 15. Profile J 025: Result of the stability calculation, ( crit. versus
t crit.
Uot
and ca.
NACA TM No. 1185
I / U I 'LL 'Nt abrupt" entering
of the flow
I 42 44 0, 48
I / ^ pressure side I
Figure 16. Profile J 800: Velocity distribution with instability and separation
Uot
points at various Re = u and ca values.
NACA TM No. 1185
 Laminar separation points
So Instability points
N>
5" o25
0 2 __ ,6
0 42 Q4 0, a to
Figure 17. Profile J 025:
points at various
Velocity distribution with instability and separation
URe tand ca values.
Re =  and ca values.
V
Ca=0.25;d=415
It'f. I
Profile
', 215.. .1
015
U S' l _*___~1 215
15  ,uj
II 41
Suction side
Pressure side
Figure 18. Influence of the camber upon the position of the instability point for
profiles of the thickness d/t = 0.15 with ca = 0.25. A = laminar separation
point; M = maximum velocity; S = stagnation point.
urn(
u0 1l
N0
If.1
11117/
L SSuction side Vm
pressure side Va
\coO
lb
Re 
'Os
tO'
'Ga
L~d
t
Figure 19. Characteristics curves about the influence of the thickness of
profile upon the position of the instability point. Suction side:
Ca < Ca for not abrupt flow changes; suction side: ca > ca for not
abrupt flow changes; pressure side: Cae ca for not'abrupt flow changes;
pressure side: ca < ca for not abrupt flow changes
profile 000
=Y2: M, n 0O5
:E '= :* ^
A 025
~,rs & ~^^'
LA. ^.
F~xraA
Figure 20. Influence of the thickness of the profile upon the position of the
instability point for symmetrical profiles with ca = 0.25. A = laminar
separation point; M = maximum velocity; S = stagnation point.
.NACA TM No. 1185
NACA TM No. 1185
os at 415 42 '"i
rz'~ziz'
VL '_ # _
t ca =05
a 7 IS
r^ Y~ T~~t
_. _ ____ _, ,_"
a40S a as 12 U st
t co =1,0
\ s s s
qo4 1 4 If o2 w t
Figure 21. Position of the laminar separation point
of the thickness of the profile d/t and the camber
side.
(s/t)AP 6 as a function
of the profile f/t; suction
Figure 22. Position of the laminar separation point (s/t)AP 6 as a function
of the thickness of the profile d/t and the camber of the profile f/t; pressure
side.
fh
Co=0
006
cOi
sui
U
Off
Ca=025
o.n
4 fi vIS s t
Figure 23. Position of the instability point (s/t)crit. as a function of the
thickness of the profile d/t and the camber of the profile f/t; suction
side; Re = 105
00 Or IL, 42 41S t
k caO,5
ar / /
noe
w0 ;f.r"*, (i. ,V .US. .
^ ''.` '* v
^~J ,\ __ _L
f/I
a
C =0,25
__0 ZIA
Figure 24. Position of the instability point (s/t)crit. as a function of the
thickness of the profile d/t and the camber of the profile f/t; pressure
side; Re = 105
NACA TM No. 1185
cro=o
Gas a a S
w "i /. ',
7^/ ^
SI ___ ^ .
f _ '^  g "^ .
OM ai c U as r si
: 
:r_
s 4t
F I
2 1
Ud at." u 0(
a' 82i f
NACA TM No. 1185
fit co=1O
0,02
05 Q1 #5 e2 US 
0 \ d
0.05 010 02 025 t
Figure 25. Position of the instability point (s/t)crit.as a function of the
thickness of the profile d/t and the camber of the profile f/t; suction
side; Re = 106
002
ft' ca= 0,5 02
o0S of O5i 02 25 t
S'/t /Ca = O42
0.02
00 o. 0. OJ5 2 025 t
008 7S ca IK 0
^ _r
~f/ '\ **}! *ut . ^ ~~ ^ C _
Figure 26. Position of the instability point (s/t)crit. as a function of the
thickness of the profile d/t and the camber of the profile f/t; pressure
side; Re = 106
NACA TM No. 1185
Figure 27. Position of the instability point (s/t)crit.as a function of the
thickness of the profile d/t and the camber of the profile f/t; suction
side; Re = 107
Figure 28. Position of the instability point (s/t)rit.as a function of the
thickness of the profile d/t and the camber of the profile f/t; pressure
side; Re = 107
dl (lid i05 G 17.f
NACA TM No. 1185
ft Ca=OS5
0,o07s ^ ~ *si \
0.05 tr 0S 02 025 t
Sca= 0o.25
406
0.02
0.0.
02
o.S 5 0.5 42 25 t
ca 1,0
aas at 41s a2 025 t
Figure 29. Position of the instability point (s/t)crit.as a function of the
thickness of the profile d/t and the camber of the profile f/t; suction
side; RF = 108
a a =0
Q6 5 I t W .02 O25 t
0.*L 7
40 0577
ca =425
405 at 4s5 42 025
Ca 1.0
uOs ar RU 02 o t
Figure 30. Position of the instability point (s/t) crit as a function of the
thickness of the profile d/t and the camber of the profile f/t; pressure
side; Re = 108
NACA TM No. 1185
il Ca 
,' /
/ .t
CaO,5
*i f
Sca=025
LI .' .a 0
ca=iO
,i. ." I I r .'5 r
Figure 31. Mean position of the instability point (/t)crit for pressure and
suction side at Re = 106. (S)crit.= 1/2 (Scrit. suct. side + Scrit. pressure
side)
ca 0
~ o=
ii ..
Figure 32. Mean position of the instability point (s/t)crit.for pressure and
suction side at Re = 107. ()crit.= 1/2 (Scrit. suct. side + Scrit. pressure
side).
Cfa 25
CI
UNIVERSITY OF FLORIDA
S1262 08106 6611 111
3 1262 08106 661 4