'-I Ln c H C-) C A S F/1' Oft m u0py NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1185 SYSTEMATIC INVESTIGATIONS OF THE INFLUENCE OF THE SHAPE OF THE PROFILE UPON THE POSITION OF THE TRANSITION POINT By K. Bussmann and A. Ulrich TRANSLATION "Systematische Untersuchungen ber den Einfluss der Profiliorm aul die Lage des Umschlagspunktes" Technische Berichte Band 10, Heft 9, und Vorabdrucke aus Jahrbuch 1943 der deutschen Luftfahrtforschung, IA 010, pp. 1-19 Washington October 1947 dat. ........ IEC Ap https://ntrs.nasa.gov/search.jsp?R=19930092185 2020-06-17T03:09:18+00:00Z
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'-I
Ln
c
H
C-)
C A S F/1' Oft m u0py
NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL MEMORANDUM
No. 1185
SYSTEMATIC INVESTIGATIONS OF THE INFLUENCE OF
THE SHAPE OF THE PROFILE UPON THE POSITION
OF THE TRANSITION POINT
By K. Bussmann and A. Ulrich
TRANSLATION
"Systematische Untersuchungen ber den Einfluss der Profiliorm aul die Lage des Umschlagspunktes"
Technische Berichte Band 10, Heft 9, und Vorabdrucke aus Jahrbuch 1943 der deutschen Luftfahrtforschung, IA 010, pp. 1-19
Washington
October 1947
dat.
........
IEC
Ap
https://ntrs.nasa.gov/search.jsp?R=19930092185 2020-06-17T03:09:18+00:00Z
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM NO. 1185
SYSTEMATIC INVESTIGATIONS OF THE INFLUENCE OF
THE SHAPE OF THE PROFILE UPON THE POSITION
OF THE TRANSITION POINT*
By K. Bus smann and A. Ulrich
The position of the beginningof transition laminar/ turbulent as a function of the thiclrness and the camber of the profile at various Reynolds numbers and lift coefficients was investigated fOr a series of Joukoweky profiles. The calculation of the boundary layer was carried out according to the Pohihausen method which may be continued by a simplified stability ca1cu1tion according to H. Schlichting (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 Joukoweky profile.
OUTLINES.
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 Untersuchungeniher den Elnfiuss der Pofilform auf die Lage des Umschlagspunktes.t! Zentrale f wissenschaftijches Berichtswesen der ..Luftfahrtforschung des Generalluftzeugmeisters (am) Berlin—Adlershof, Technische Berichte und Vorabdrucke aus Jahrbuch 1943 derdeutschen Luftfah±tforschung, Band 10 (1943), Heft 9, Sept. 15, 1943, ,IA 010, pp . 1-19.
2 .. . .. .... .. ACA TM No. .1185
IV. Results: . .. . ...
(a) Influence of the ca—value 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 or. tables for the separation and
instability points for all Joukowsky profiles (e) Mean.value of the laminar —flow distance of
suction and pressure side for all Jbukowsky profiles
V. Sim'mary . ... . . -
VI. References .
SYMBOLS .
x,y . rectangular coordinates in the plane
s profile contour length starting from the nose of the profile
- t wing chord
,tv • length of the profile contour from. nose to trailing edge (different for pressure and suction side)
U0 velocity of. incoming, flow
U(s) . potential velocity at the profile
boundary layer thickness according to Pohlhausen Pk
6* displacement thickness of the boundary layer
= --f.nondimensional boundary layer thickness
t dIJ' = z1 - ---
. . form parameter of the boundary—layer 0. profiles according to Bohihausen PZI
NACA TM No. 115
..4orm parameter according to. - ' . .••;. .PohflisenP6'
ffX), g(? . 'uniVersalfunctions'.of the boundary— \ I, \I' /j .. layer calculation .
5crjt.: pqsitio'n of the instability point,
measured along thecontour Of . the. note of th,e profile
5 p5 .' position 'of the—separation point according to PG method
I. STATEI€NT OF' T}tE PPOBLiFM
The position of the transition point laminar/turbulent in the frictional boundary layer is of decisive Importance' for the problem of the theretica1 calculation of the profile drag of on airfoil since the friction drag depends on it to high degree. The position of the transit-i on point on the airfoil is largely dependent on the pressure distribution along the contour 'of the profile and therefore, 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 tl,3,k) and J. Pretsch (2).
According to present conceptions the turbulence observed in tests develops from an unstable condition by a mechanism of excitation as yet little known; ,. therefore, the experimental transition point is alas to be expected a little further back than . the theoretical Instability point.
Knowledge of the., theoretical instability point is, nevertheiess, important for the researchon profiles, in particular for the drag problem, Recently a report
'An extract of this report was given in a lecture of the first—named au
thor at the Lilienthal meeting
for the discussion of boundary—layer problems in Gttingen on October 28 and 29, 1941.
• NACA TM No. 1185
was madeabout airfoil sections which, due to 'a position very far back of the instability d .transition point, have surprisingly small drag coefficients (laminar 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 CaV'alUCS 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 fork these investigations and the selection of a series of Joucowsky profiles was, therefore,natural. It was not adviable to take for instance the MACA series as a basis; the calculation of the poteiitilflow for such profiles according to the methods at present available does not achieve the accuracy which is required here.
II. EXTENT OFTflE INVETIGATI0N
A series of ordinary Joukowsky profiles of the relative thicknesses a/t = 0, 0.05, 0.10, 0 .15, 0.20, 0.25 and the relative cambers f/t = 0, 0.02, 0.0k, 0.08 were taken as a basis. (See fig. 1.) For,instance, J 415 stands for the Joukowsky profile of camber f/t = 0.04 and the thickness d/t = 0.15. The cregion which was examined IS 0a = 0 to 1 and the Re-urnber
'Ut" 8 range Re _ 10 to 10 • The complete calculations
were carried out only for the following profiles: 000, 005, 0 15, 025, 215, koo, 415, k2 5, 8007 815, and 825. The results for the remaining.profiles could be obtained by interoolatjon. Thus it was possible to obtain a result with tolerble loss of time in spite of the very extensive program ('four parameters); a certain amount of accuracy had to be neglected since the interpolation sometimes was carried out over three points.
NACA TM No. 1185 :5
III. THE CALCULATION OP T}tE POTENTIAL VELOCITY
AND THE PRACTICAL APPLICATION :OF THE BOuNDARY
LAYER. AND STABILITY cALCULATIONS
(a) The' ' Potential- Flow .1..
The calculation of. the potential veldcity with its.. first and second derivatives along the profile contour forms the basis fora boundary layer and stability calculation. The potential flow about a Jbukowsky 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 profi•1 contour and for the velocity distribution follows:
• ;•.z=xIy
Coordinates in the complex plane
mapping function:2
cir1ë K-4mean camber line of the profile circle K —3cambered profile
a radius of the unit circle in the z—piane. R radius of the oircl.e to be mapped in the z—plane
t wing chord
tT length of the profile contour from nose to trailing LOfC (c.ferent for' suction nd pressure side) . •.
X0 y0 coordinates of the center of the circle to be mapped in the zpiane (circle K)
on
NACA TM No, 1185:
o, yl center coodina.té of the mapping circle of the mean camber line of th profile (circle Ki1.)
p varying angular' dordina'te of the conformal transformation
- zero lift direction. (See fig. 2'.)
a angle of attack of the airfoil referred to the theoretical' . chord
ag geometrical angle of attack referred, to the • :bitangent (See fig. 2.)
Profile nose: p =r.+
Trailing edge: p =-
xo • ••., ' • -' = thickness parameter;k =1
See table I. Yl 4= camber parameter
1 :
= arc cos '- -' = are sin - - • 1
\Jl+Ei
The profile parameters E • and • can be found
in table 1.
Profile contour:
_E
E (IkF+ 4 Cos
1
'T "H (1)
+ sin )(l
-] J
I'JACA TM •No. 118 7
N k2(1 + 22+ 6 12 + 2kl + Ek2cE1 Co + Ek5i
t 1+k,+ =2+ k + E
Nose radius P/t:
For symmetrical profiles the equation'
p 262
1+26 +1162 (2)
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 flose radius of the Joukowsky profiles is only little larger than for the NACA profile faniiy according to NACA report 460 for which PI =
Velocity distribution: - C
lim
UO = 2 in ((P —a). +sin (a + ) J P1(p) (3)
Stagnation points: Back p -
Front p=7r++2a..
N (!)-
- i.)2+)fk2(4.+ 2 2
8 NACA TM No. 1185
Velocity at the trailing.: edge:
urn
UM
a + ) r - (5)
Are length:
(dC11 2 +( CP
(6)
s as a function of cp 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:
idU u 0 d(p j ;17
(q —a)
+ 72ifl (p — a) + sin (a
++
sin cp)
N1 = (N 1)2 A
B = N1 N' - N Pi - 1) N 1 + 2kl + A cos
N' + + 4 k Cos T)
(7)
NACA TM No. .1185
The first derivative of the potential velocity with dU1
respect to the are length - was calculated
numerically from equations (6) and and from that
(graphically the second derivative dUm. ds2
Relation between Ca and a:
Ca 8ir -sin (a +
R kf + , 6 1^2 -_.
--- compare tables 1 and 2,
3 + 2 6 1. +
(b) Boundary—Layer and. Stability Calculation
After calculation of, the potential velocity with its first and second derivatives along the profile contour thee is a boundary—layer and stability calcu-lation to be made for each profile. The boundary—layer calculation according to- Pbhlhausen (5) was based upon the differential equai:'oi for the boundary layer thick-ness in the shape indicatc;d by Howarth (6).
Cl.Zh ____
T(s7E) :--ti---.+ z 24 (?)u' (8).
21n the meantime a simpler form of the Fohihausen equation was indicated by H. Holstein and T. Bohlen (10) where the momentum thickness appears as independent variable. For this method the sbcond derivative U" is unnecessary; the integration procedure is thus simplified considerably,
10 ' ']AC A TM No. 1185
The .follo*in : siiñbo1s stand for
t 2 d 2-1T
UfT, U , UTL ff_ _' , 0
o od
pil = boundary layer thickness ac &ordin&. to Poh1hauén pL1,
±
O2J -
' 'U •t
TJ? fom parameter according to Pohihausen
^.(A: p) and universai nmctions
2, ^(2 ^i\ 22 (X) =
27 r,\ o oO
2?
717
-0 '
(3o
tnitai conditions: 0 ,
At the stagnation point ' 0
0
X •7.O2
that is.,
- Xo - ____
40 z;_tYr_0
NACA TM No. 1185
11
Besides, ..
- 5.391 (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 z 0 is
known ,.but that' the initial inclination at the. stag- nation poInt z 0 1 also can be'determined. The latter
value is obtained by exact performance of the limiting lImiting
process urn . in (8) (H2wazth(6)). Withz 0 ' known
U-30 .. the .Integral curve' passing through the initial value z40 is easily found which otherwise Is not immediately p'ossible because of the s.ngularity of the Pohihausen equatiOn at the stagnation point.
For the profiles of the thickness d/t = 0, thatis, for the fist plate and the circular-arc profiles, the case where the flow does not enter abruptly (a = o) 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 layer Is here zero, that Th, at the leading.-edge there is: •.
(11)
I Profile Ca not abrupt flow entrance
0. 0
200 ' ,25
00 .. .
800 . 1
12 NACA TM No. 1185
The velocity near the leading edge of circular-arc profiles takes the same course as V:
Urn U00 + C VS + • • .
that is, U1 becomes with infinite like l/; the velocity has a perpendicular tangentwhich always • occurs when the •pontou of.•.the profile shows a sudden • change in curvature as it doeshere .(v. Koppenfels (8)).
near the leading edge for a circular-arc profile behaves like for the flat plate, that is; zk goes iii a linear relation to ,, s toward 0 .. 3
Taking these facts into considëratiö±i 'there results at the leading edge:
70' •Zt=!9. 12 0 . kô.
It has proved advantageous to calculate the liie elements z 4 directly from the equation (8) by means
of a,plotting of the curves. and (See
fig. 3.) This method is superior to the calculation of the line elements by means of the often used nomograms of Mangler (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 boundary—layer calculation yields for each profile for a givn..c value the nondlmensional boundary layer thicknessand the form parameter ?
as a function of the length of the arc s 'along the contour. The distribution of velocity u(y) in the laminar boundary layer is then obtained from:
3For the flat plate. z =3.O3 •s/t (according to
Pohihausen (5)).
NACA TM No. 1185 :
= F 1 + ( 13) UM
with
•=a+
(1)4)
- tL ) . 1 1CL 'i2
)4 6 . . 2- PT
The results of the boundary-layer calculation for the profiles J 800 and . . J 025 have been plotted as examples .in figures Ii —and 5: the form parameter Xpt; and the .nondiiens1ona1 displacament thick- ..
/ut - . ness -- V' with . 6 for the dis p lace-ment thickness.
The following relation exists between the displace-ment thickñes and the boundary layer thickness according to Pbh:lh'ausen: • . .
- . .
The , displacement tiicknessof the fiat olate in longi-tudinal' flow (Xpj,. ' = o) is re presented gra'hicaliy
in figures ): .nd 5 for comaris on. Mae following equations are valid: -
(16) t
PLCLt • t y . -.
r
TLOJ TM No. 1135
• The profile J Boo (fig. !).. shows clearly that the displacement thickness .for accelerat,ed flow (suction side) is, smaller thanthe d'isplac?ment thickness of the flat plate whereas it is. larger for retarded. flow (pressure side). (Compare also fig. 16.)
From the boundary-layer calculation there result also the laminar separation pQint-s., According to the'four-term method of Pühausen s.earation occurs at -12, according to the si;-term method (see
below) 'at = -10 corresondin to = -9.6.
Flow hotoraphs 'have been taken ,n a •Lip.pisch smoke tunnel for apart Of 'the. calculated profile of - - .
sUot
models of 50-centimeter wing chord and atRe-numbers '5 of about .2 x.. 10, The r.oInts of separation have been ascertained from the flow graphs (figs. 6 to 11,' appendix). Figure 12 shows the exzperi.menthl and theoretical separation points f or- various profiles fOr comparison. Compare also table 3. The agreement is rather good.
AfterI-
has been ascertained a.' a function of i
the 1enth of the arc s there results the i::istabilit.y, point ' (s/t) .. from a stabi1ty calculation • ..•• (H. c ic1±ng ():)) based..on the six-term meth6d of Fohihausen. The P6-method is based on a one-parameter group (parameter-A O of boundary-layer profiles which can be represented by polynoriiais of the sixth degree. An invstigation of stability' was carried out for a number of these boüñd.ary-'layOr profiles in ()4); first,
• • the critical Re-number of the boundary iayr ''t.—;3)
crit. as a"function' :of . '? was obtained. The, critical
( U 1ö4 \ •• .
Re-number of the laminar J..ayer (' ) as a function
- • • : • • • ' crit.
Of . (fin'.' i) j., s then immediately known .1so
because of a universal relation betvieen and
indicated in (t4.).
NACA TA N.O. 1185 15
Once X(s/t) has been acertainad from the boundary-layer ca1clat1on .accdrding to Pohlhaisen 'S
IUm&\ method a critical ie--iumber 'L- ) may. be
coordinated to each point of the profile by means of figure 15. Moreover the Re-nurnber of the boundary
layer - can be calculated for each pint.ôf the i) U.t
profile at a certain :
• ___ m \!- (17) •••.. , =ç \lzLi bpL \J u
The location- of t.he instability point is then given by
I----- _.8.\ U /U ö"\
M
)Crit. 1)
IV. RESULTS
(a) influence of the c a-Value and the Re-nunberinfluence
Theresuits of the stability calculation, that is the position of the theoretical instability point (43
crit. for the sample profiles J 800 and J 025 are plotted in figures J LL and 15 against Ca with the Re-number
LTt asDarameter and furthermore against .-- with the
c-valueas psrarieter. The characteristic course of the curves, is the same for all 'profiles; the following statients crc aiIa: the instabili t y point trav€ls, with lnsreas2ng Ca at a constant Re-number, forward
16
'NPCATLI'N, 1185
on the: suction side, backward on the pressure side; the instability point tr-aveis forward on both suction. and pressure side with increaing Re-number at a' fixed ca_value. j This behavior is domorstra ,ted very clearly in figures 6 and 17 which represent the = velocity distributions for the top .profiles T800 and J 025 for the various Ca-values with instability and separation points. One can see in particular ,that the
- instability points of the suction side for Re-numbers Not
from io 5 to loT lie near the velocity maximum;
mostly the position of the ''instability point for Re, = 1O6 agrees well with the location of the velocity maximum. The pressure side of J . 800 in the case where the flow does not enter abruptly ( C a = I) 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 thit no relative velocity maximum exists • Measurements concerning the dependency of the transition point on the c a-value were taken. by A. Silvorsteir. and J. V. Becker (9). Thes'e tests showed (as a result) the same dependency of the transition point UpOil 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 he described as follows the instability point travels with increasing cembr, atconst:ant thickness; for all Ca_ValueS and Renumbers backward on the suctIon side, forward on the pressure side. This influence of the camber can be understood from the feet that - the stagnation point and therefore the region of the accelerated stabilizing flow trave], with increasing camber, 'backward on the suction side wh&reas because of the flow around the noe .e Qf the profile a region of considerably retarded destabilizing flow origlElatas. immediately behind the nose on the pressure side.. Figure iB ropresents as an example the results for profiles of the th1cKnes d/t 0415, with variable camber It for c = 0.25'' and again the Re-number as parameter. The urves for all thick-nesses and all c a-values have the same characteristics.
NL.CA TM No. 1185 17
• •. (c) Influence of, the 'Profile 'Thic1tness,
The dependency of the instability point on the thickness cannot be described in such general terms as the influence of the camber since this influence depends in 'the fol1owng'way on the c value: A
• •. •-rain. Ca not abrut flow entrance. , tha t. the
Ca value that corresponds, to the not abrupt entering.
of the flo (a 0) for the circular arc profile with the given camber, is coordinated to each. valuépf the camber:
fl t. The curys. ( s/t ) Cr$t versus d/t.
at 'a constant fjt show on principle tw6 'diffeient'. types (fig. 19)'
"'I. With increasing thickness, the curves
versus d/t start rroin a finite value and have a flat minimum: ' .•
On the suction side for Ca Ca for not abrupt flow changes. .•
. .^ . On the pressur e side for Ca — Ca for not abrupt. • , flow changes. . .
iii. The curves (s/t)t versus a/t rise starting from 0 with increasing thickness; hence, the transttion point riioves backward as follows:
On" the suction side for Ca > Ca for not ahiupt - flow' changes.
On the pressure side for Ca < Ca for not abrupt flow. changes.
ThresuJ:ts for the smrnetrical profiles at C a 'F 0.5 are ' represented as an exar.p1e in figure 20. For the .syrnrietrical profiles Ca for not abrupt flow changes 0,
•.'... ,that i3, the dependency : of the :instability point on the •
,., thickness d/t 'for all ' c 8 > 0 is of type II on the •
auction side, of type I.on tlae, pressure side.,
The flat r.n1Iaun1 ii curves of type I does, a. a some • cases, not exist at high.-n thers.(Re'"1O7.o 108),
'versus •-a/t " rises' f'Om"thé finite and (s/t)crjt value d/t '0 ".''.
' • ' '
• ": ,,
18 NACA TL1No, 1185
(d) List of Tables' for the' Sepaxation and Instability
Point in all Joukowst .cy Profiles
The total result of 'the boundary-layer and. stability calculations is représentéd by a graph of the. curves (.s/t)Ap 6 const. and. (s/t. ) ft . = con st., respec-
tively, in a system of axes thickness 'd/t - (See figs. 21 to 3Q. : ) A Drofile corres'Donds to each .oint of the p1ane In particular, tne 7rnmetrical profiles are coordinated to the toints of the d/t-axis, the circu1r areprofiles to.the.pointso'f the f/t-axis, and the flat plate corresponds to the zero point... Lift coefficient arid Re-number are considered as parameters,. One has therewith a catalogue of Joukowsky prbfiles that, make , it possib1e read off, for every
profile in the region d/t 0.25;. Of/t 0/03,
the position of the se paration oints.for 0 c (figs. 21 and 22) andthe. position of the irstailitj
point for: 0 Ca 1 and 105 Re 10. ...Figures 23 to 30 represent the curves (5/tent = const. for he
Ut
Ut 3 Reynolds numbers from Re 101' to 10 at
the . ca ValUes ' Ca 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 ' O.15 at Re 106 and are taken from these re presentations. ' (See page' 19. )
The most remarkable matter 'in.his gra phical repre-seritation is the location of the curve (s/t)Ap 6 9,. and
(S/t)crit .0, respectively, at the various . ca-values.
The position of this ZCrQ curve in the catalogue, for the instability points will be discussed; the same is valid, . for the searatjon points .(s/t)cnit. can 9]r.
appear for the flat plate and the circular-arc profiles on the suction side for 'Ca > Ca for not abrupt flow
changes, on the oressure sIde f or 0a < a for not
19 NACA TM No. 1185
0 i Id Lf\0H
(J) 00r-. .4
000 000
• 4) r1 .0 C.)
CD. •
4) 000
r U) çU)
0 'dU\ S...ce\OD
•r-I r HHO n . 000
• 000 0
4.) •l 'Drfl-I\ o HHO
0 H ^)S I I
• -S... 0 f)
II --.-•-4.)
5-..-
U) .4C) .i c— H'.0
• cj.-t O r-IH cs.j0 U)
S SI. 000
.5
0 2- 000
$1 4) • .'-
4) - 0 OOH CH
-S 4.)
•I S 000
•0 U) .1
0V •rl .rl
U\0'0 C'JC'JH
I 5.
O ti:) 000 - •
o cjc'iH I ,-... .5 . I
5--S. cc)
.0I C.)t\
20 NiCA TM No u85 -
The curve (s/-) = 0 always abrupt flow changes . crit. coincides with the fit-axis; it forms a part of the f/t-'axis which. is determined by the actual c a- value.
Therefore rio point (s/t) = 0 exists on the suction crit.
side for c 0 since c >0 a a for not abrupt flow changes for all circular-arc profiles. For the pressure side, on the other hand, (s/t) = 0 on the whole f/.t-axis,
cr1 t. There follows in the seine ay for Ca 0.25 that (s/t). = 0 for 0 = f/t < 0.02 on the suction side and for f/t > 0.02 on the pressure side. Pressure and suction side, therefor€, alw.ays comp ler1ent each other. The point which corresponds to the circular-arc profile with c = c a a for not abrupt flow changes (for instance J L00 at Ca =0.5, compare figs. 2 to 0), that is, the end point of th distance (s/t) 0 en t. is a singular point in. the following sense The point itself" assumes a certaii value (s/t (different
crit, for pressure and suctioi side), but infinite number of curves (s/t) =const. which are crowding
crit. together •asymp to tic ally . toward (s/t)crjt 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 inve s tigated In order to make more accurate statements nossibie. . However, only profiles with thicknesses dft > 005 which can:be analyzed quantitatively, are of practical interest.
For Ca = 0 1 (s/tY 1 is the same on suction and pressure side for the symmetrical profiles. Therefore the curves (s/t) = const. for thiction and pressure
cnit.. side would adjoin at C a = 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 C a ^- 0 also the curves (s/t) crit. = const.
- NACA TM No. 1185 21
have continuatións which correspond to the curves (s/t) const, for the rressure and c.rit. - suction side, -respectively, at the appertaining c a-value with inverted sign.
(e) Mean Value of the LamInar-Flow Distance on
Suction and Presuie Side for all Joukowsicy Profiles
In vIew of the development of laminar profiles the mean value of the laminar-flow distnce on suction and pressure side is int.erestln. Figures 31 and 32 show the curves mean value (sit ') const. in
CP1 L. the d/t, f;'b-Plane for various lIft coef fie Ients and
- 6 th o-nenies 101 and 10 7 [Cr
0.5 jt. Is + a Generally -the kcrit. suctor crit, preseure fol-I.owin- conclusll.ons are vali.d The profiles with the smallest mean value (1t) for a certain c -value erlu. a lie near the circular-arc profile to which this value Is coordinated as c
a for not abrupt flow changes. This profile will be for ca = 0 the flat plate, for C a 0.25 the profile J 200
10 for Ca = 06 the
profile JO0 and finally for C a 1 the profile j 800. lit.
There seens to be an excer)tionai ease at Re 0 10 U
and c = 0.5 (fig. 31) which can be explained as fol1ows The circular-,arc profile for which at the considered c a-value the flow enters "not abruptly" (for instance J 1oo at c. = 0.5) is a sinpular point Lathe yt-, d/t-c1iagram Approaching thisproflie
-axis from two different sid-e one obtains two different limit values (s/L-1) crit.1 since once only tiiC suction side and once only the-pressure side cotr1butos to the mean value. Only for the sthgular point itself suction and •oressure side contribute so that this profile has a higher (/t) than. the
crit.
22' ; NAC/ TM No. 1185
profiles on the f/t-axis:iear it. 1± one 'now considers the curves mean value (s/t)' ='const. for alues crit.
f. or higher than the two, limit values (range I). These curves enclose the singular point and end at two points on the -f/t-axis. The remaining smaller mean
values (/t) generally, cover only a small region cr.it,
near the singular point '(iange II) where, with the present investigations as a 'basis* accurate statements are not possible.. Only for the.
Uot case - = 10 6 and c . = .0,5 the range II comprises'
a
all profiles of the series considered here s !Lnc6 on the pressure side the profile Loo at C a. =0.5 and
Re = 10 6 has no' transition Doint (s/t) = 1 and crit.
therefore the point c/t = 0.011.. ob1ains a high mean value (s/t) > 0.5. For this' case there are closed
crit, curves (/tY conet. and there exists a -profile(J 115)
with the smallest mean value (/t) •= 0.155 at
Re= cri . ,
Moreover, the following results are obtained from figures 31 and 2: All Joukowsky profiles have small mean values (s/t) ; for instance, the mean values,
cIi.
for practically important profiles with the camber -'
f/t = 0.02 and the thickness d/t = 0.10 to 0.20 at
Re-numbers of 106 to io7 are between 0,08 and 0.2. Those mean values are only to a small degree dependent on the lift coefficient; for instan9e, the mean values for the profile J 215 "'at Re = 10 0 and at lift coef-ficients Ca = 0 to 1 are between 0.16 and 0.175.
V • SUMMARY
A series of Joukowsky profiles with thick- tr nesses d/ 4- = 0.25 and cambars f/t = . 0 to 0.08 was investigated with respec. t to the position of the instability point for various lift coefficients and Re-numbers. The following resuitwas obtained: With
NPLCA TM No. 1185 ,: 2
increasin' Re-number ., InstabilIty point moves forward on suction and pressure sides with increasina: ca-value-it moves forward on the s"ction 'side, backward on the pressure side. The position of the' instability point as a function of t];iickne.,ss,'and camber of the profile is represeited n the '.'hape of a irphica1 list of tables 'which permits the- readin off of the position of the instability' point on suction , and pressure side as well as of the mean value of the laminar-f low distance on suet ion and prs sure side for' ech profile of the series.
Translated by Mary L, Mailer"' National Advisory Coirujttee for teronautics
C
24
NACJ TI No.. 1105
VI. REFERENCES
1. Schlichtiflg, H.: Uber die Berechrnng der kritischen Reynoidssthefl Zahi einer Reibungsschicht in beschleu.nigter ' urid verzögerter Str6mung. Jahrbuch 19 11.0. der deutschen LuftfahrtforsChUfl, P. I 97.
2. Pretsch, 3,: Die Stabi1!tt der LaminarstromUflg bei Druckgefälle und Druckanstieg. Jahrbuch 1941 der deutscheri LuftfahrtforsohUflg, p. I 58.
3. Schlichting, H.,: laminar/turbulent für elne ebene Platte bei kielnen ArITstellwinkelfl. Nicht verffentlichter Bericht.
. Sehlichting, H., and Ulrich, A.: Zur Berechnung des Urns chlagspunktes laminar/turbulent. Preisausschreihen 1940 der Lilienthal Gesellschaft für Luftfahrtforschuflg. Jahrbuch 1942 der deutschen Luftfahrtforschung, p. I 8.
5. Pohlhausen,K.: Zür n'ãherungsweisen Integration de Differentialgleichung der laminaren Reibungs-schicht.Z.angew. Math. U. Mech. Bd. 1, p. 252, 1921.
6. Howarth, L.: On the Calculation of Steadv. Flow in the Boundary Layer Near the Surface of a Cylinder in a Stream. ARC Rep. 163 2 (1935).
7. Mangler, W.: Einige Nomogramme zur Beredhnungder laminaren Reibungsschicht an elnem Tragflugel-prof ii. Jahrbuch 1940 der deutschen LuftfahTt - forschung p. I 16.
8. Koppenfels, If. V.: Two-Dimensional Potential Flow Past a Smooth Wall with Partly Constant Curvature. NACA TM 996 1 19-1.l. -
9. Silverstein, A., and Becker, J. V.: Determinations of Boundary-Layer Transitions on Three Symmetrical Airfoils in the NACA Full-Scale Wind Tunnel. NACA-Rep. No. 637 (1938).
NACL TM No. 1185 25
10. Holstein, H., and Bohlen ., T.: Ein vereinfachtes Verfahren zur Berechnung laminarer Reibungs-schichten, de dern Naherungsansatz von K. Pohihausen genugen (noch nicht verffent1icht).
NACA TM No. 1185
as
b-I I-b
0
a.
Cl)
Cl) c. E-.
c-i0 C.)
C,)
Cl)
0 C)
I-I a. 0 a.
N-- r-.r'.-ico . -.
-S.. •.._(j • • • . a.
r-1,0 03 - '0(0 C-..zf i-ISO 0) t- .-4SOc0 0J43 U\
0.-I NdSO14'..-4 L(\..ZN 0 i-I N ..O
N%,-4 Lr\- .-I UN- 00000 00000
0.-I aJ4 .0 000000 (\JS.O 00000
0 0 0. 0
ID -4 0) .
- I -soco.rso o,-Ic\J..'O
o'. i'o'oco cyLrS Lr.Lr.
c'.j.-i.-i 0c-co oo. -aJirs
rsO. r.j 00000
000.-INLt' 000000 . . . . . . 0 . 00000 . . . . .
.--Iaj 000000
03 0)
.-I.--l.-l.-4,-I -4.-li-b -4.-I,-1. . S • • •
-4.-4.-1
0 a.
ID
S-s LrN 02 -'.O(0 u \ a-
0.-4cJSO'.S U\U%r1 t4 otr-
(0 PC\Q'.0 tC. oiOo
t--O .01
o00000 .
000000 S ...
000000r't-cs
00000.-I S 14.1-'
C') -
Ozj O '0 ON '0 ON UO\L(S 'O(0'O
4310LrsQu 0CM U\O\U' 0CM U\0\LfS
0000.-1 S • • • • 000014 • S. • S 0000.-I 0000.-4 S S • • • • • • •
0 hbas C)- (;-'.O .r'-* 'c
co C0'.O.4 0 0 N-.O a,-I-
CM E-,)'-Cj ?4 (\4 O'sO t-
i-I 0'.O Lrcx) 0 (\ $' 00 0,\(7\
051 t(\N4 LIQ 0_1--.-I IJ'O\ 0-.-1 Lt"Q i-b U'QN'\'0 0 o 00 .-s i-Is-I C1 00.-I .-I.-t (000.-Is--Is-I O00.-4s-I('J.
I s- -Is-Is - Is-I 14 s-4s-ls-4s-I.4 141414.-Is-b1I M 14 1I.-4.-l.-I
MIA
(\p(\t(\ '05050505050 ,)JNC\J,)JC'JC5J . 0000000 NNNCIJCIJN
000000 000000 C'JrJC'JCJC\IPJ _z,._ _ (0(0(0(00)0) '-0 '050 '.0 \O'.O
-:1- 00_0000 •o..S. 000000 . . . . . . s-Is-Is-IrIs-Is-b •o.S.. 000000
14sC5rIt4%0 .-4,c,s-4Lr.0 14 HNs4N\0 s-IPr\i-414\0 -ICO Nc0 (0 r(\0Zf ,e' . co le.a5 00.-Is-IN 00iIs-ICM 00s-Is4CM 00.-Is-I
(-0 o• o••• 0
0
4-b 0 I!"0 If'\O L(' 00.-Is-INN
0 t('0 If0Lf" 00.-Is-INN
0 U'0 It'0 Lf' 00s--Is-IC\JN
0 Lr'.0 I45OIf\ 00.-IN('J o 000000 NN(JNCMN -j- (0aco0ac0
a.
0.
SI
.I- 0 -Ic-.
a.
s--b
0 04-I 540 0
00 4) -4.r4 4) 0
ID 0 0.-I ILI 05
as
0
0 0 v-I
05 4)
wo 0
-0 ID
-4 0. 43
0
ID 05
U)
0
4) .00 0 003
•0 5403
00 :
s-bE-I S 0 s--I
14 43 VJ 0
.1-)•1 0 Ipr,
4) _•5_._ ID s-b-..
vf4.)
0 V 0) 14 0 0.0 0 0) 0)
.1 V 043
ID .glas 4J s-.4
i-b C) s0
0 £ 0 V .042 .4
.0 03 p-I 0$ ID 43 bO as 4) .-1 0
..-I OSI 04-. s-I'd 0) 4) 0 .014 .4
c.. co 01 P.S4C) ID a o c. 0 0 00 0
4.) 14 0.-I
01. a N 4) V
0 0-1-' a s-A a 4)0
NACA TM No. 1185
27
TABLE 2
THE ORETICAL ANGLE OF ATTACK a (DEGREE)
Profile
Ca
000 .005 010 015 020 025
O 0 0 0 0 0 0 .25 2. 2.2 2.]. 2.05 2.0 1.9.5 ^ : 2 ^:l
^.0 .O
9.2 4 2 7.6
C 200 205 210 215 220 225
O -2.3 -2.3 -2.3 -2.3 -2.3 -2.4 .25
. 50 2.3
-.1 2.].
-.2 1.9
-.25 1.8
-.3 1.7 5.1
-.1. 1 6.9 6. .6.2 5.9
Ca 400 405 410 415 420 14.25
0 -4.,6 -4,6 -4,6 -4,6 -4. 6 -4.6 .25 -2.3 -2.4 -2.
-2 L.-2.55 -2.6 -.2.
• 5 1
0 4.6
- .2 4.2 3.9
- .5 3.5
ft * 7 3.3
-. 3.1
Ca 800 805 810 815. 820 825
O -9.2 -9.2 -9.2 -9.2 -9.2 -9.2 .25 -6.9 -7.0 -7.1 -7.2 -7.25 -7. 05 -4.6 -4.9 -5.0 -5.2 -5 . 5 -5.
1 0 -.5. -.8 -1.1 -1.4 -1.
28
NACA TM No. 1185
TABLE 3 LAMINAR SEPARATION POINTS; COMPARISON
OF TEST AND CALCULATION
S = Suction side, D = Pressure side
Ca 0 0.25 0.5 0.75 1
S 0.929. 0 .891 0.835 0 0 Theor. D 0 o
J 14.00 S .92 .88 .83 .75 0 . -. Exper. DO 0
Thei,r. S .8925 .86 .82750
•755 D 0 0 0
3800S .88 .814 .80 .76 .73 Exper. •D 0 0 0 0
Theor. p997 .6596 D .997
3005S .70 .28 .10 0015
Exper. 1)
Theor. SD
.14025
.14025 .1.3 45.o8 .1491
.252
.592 3 025
Exper. S .145 .145
.142
.1485.385 .514.
.36
.60:g2
8 D
S .686 .630 .570 .1476 The or. D .200 .283 .377 .14914
314.15S 095 .86 .73 .65 .6
Exper. D .19 .29 .31 .33 .14
•77 .685 .6148 •59Li. Theor. D .OLi.27 .0562 . 0 9314 .19214
3 815S .86 .78 .71 .67 .63
Exper. D .o6 .075 .10 .12 .i6
Zero I
t
NACA TM No. 1185 29
000 200 we
005 205 405 805 _
- - =—-
010 210 410 810
020 2?0
E- - c1IiiTL
• E- cI 22S c1i 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.
• 'l• j*irj I—
t. Zero lift-axis
Of
• _
TbeO!eC —
:^ ^d c h ^Of Id' ^ Profile Parameter: (thickness par.
ve = 1 (camber par.)
Figure 2.- Explanatory sketch to the Joukowsky transformation (schem.tic).
30 NACA TM No. 1185
Ix
150
lri!_,il, 0 -2 -' -6 -8 -ID -12
Stagnation point
A-0 r/AJl(AJ
SOOS IIAJ
0
02
Figure 3.- Auxiliary function f(X p4 ) . and g(X p4) for the boundary-layer
calculation.
MMMEMEMEEMM MEN uiu•U•SRIuU
• IN INIVIIINN No ME
P6 P6
Suction aide gp = jaainer separation point, according to P4
pressure aide pp5 .aainar separation points according to P6
w k---tt- -- fl7 I I I I I I 1 I fr4l1tRP6
brupt" entering is flow
Figure 4.- Profile J 800, boundary-layer calculatin: form parameter X p, and displacement thickness
RIP HIM
NACA TM No. 1185
31
A
, 1 mom * MEN ME INS U.
Suction side RP4 — Laminar separation points according to P4
r- Pressure side APg Laminar separation points according to P6
te 0
-_,
Mill
01 02 0 0.4 05 05 01
Figure 5.- Profile J 025, boundary-layer ca].cualtion: form parameter *-1IUt
and displacement thickness
NACA TM No. 1185 33
ao C,,
—4,6 0
—2,3 0,25 0 0,50) 2,3 0,75 4,6 1,00
Figure 6. Profile J 400A smoke tunnel photographs (Re = 2. 10°).
*) not abrupt entering.
ao C. d".,
—9,2 0 —6,9 0,25 —4,6 0,50 —2,3 0,75
0 1,00)
Figure 7. Profile J 800, smoke tunnel
photographs (Re = 2. 10). Laminar separation points see Fig. 12. *) not abrupt entering.
NACA TM No. 1185 35
6
0 () 2,2 0,25 4,4 0,54) 0,6 0,75
1,00
Figure 8. Profile J 005, smoke tunnel
photographs (Re 2.10).
0 0 1,9 0,25 3,8 0,10 5.7 0,75 7,65 1 1,0)
Figure 9. Profile J 025, smoke tunnel
photographs (Re = 2.10). Laminar separation points see Fig. 12.
NACA TM No. 1185
37
-
Coth,,r.
-4,6 -8,1 0
-2,5 -6,3 0,25
-4,2 0,50
1,45 -2,2 0,75
3,5 1 -0,2 1,00
Figure 10. Profile J 415, smoke tunnel
photographs (Re =- 2.10). Definitions of oc and OC g see Fig. 2.
-9,2 -11,3 0 -7,2 - 9,3 0,25 -5,7 - 7,25 0,50 -3,2 - 5,25 0,75
-1,1 - 3,25 1,00
Figure 11. Profile J 815, smoke tunnel
photographs (Re = 2.10. Laminar separation points see Fig. 12.
_ -01
0.1
NACA TM No. 1185
39
15) = Suction side - Theoretical (Di = pressure side —0--- Experimental
Ut, IL Ia I
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.
40
NACA TM No. 1185
------- IIIII1lIIIIP! ----u--------r.------•••• ••*
1UlUUIiIlUI ENE MOMEMEMIMM ENE MUMMINEM •••iinuwi•••• •u•uuuuuu•i•i uIIIIu!iIuuIu1 ------------
iiiIi!-IIIII iIIui..Iu..
Figure 13. Universal relation between the critical Reynolds number ( crit. and the form parameter X
NACA TM No. 1185
41
10.7
06
0.5
0.4
03
0.1
qe— P10'
I0 ___
106
Ca 10 a 025 05 075 10
Al08
07
0.6
05
04
0.3
0.2
01
I tJfrj. _____
800
N
"Not abrupt entering of the flow
------------- I 1-UL
= 0 025
10' 105 - 10 197 108
- Suction side pressure side
Figure 14.- Profile J 800: Result of the stability calculation, (f)crjt versus Uot and c.
U
42
NACA TM No. 1185
,?e
----------
foe
--- --------- --- jo
-: i---: -------. ---.- _ --
0.25 05 0.15 10
0.
10' 10'1 106 . i0 . 108
Suction side ----- pressure side
Figure 15.- Profile J 025: Result of the stability calculation, versus Uot
and Ca.
Of
0.4
0.3
0.2
0.I
10
- 14P 108
U _._ Laminar separation
Instability points
"Not abrupt" or I o. Lthe -flow
Ca
S--.tio. id.1 N
e-10
a,Y r • ca _____ _____
-0.2 0 :^P g7251
aide .S,PII!---r"-
14
a,
w
a:
NACA TM No. 1185 - - 43
points
tering
' Not abrupt" entering. O f the flow
- 0 0.2 44 0 08
Figure 16.- Profile J 800: Velocity distribution with instability and separation - Uot
points at various Re =
and Ca - values.
Li
II
41
44
030 02 04 46 48 (0
rn pointa
I
deV. id. J V.
1111 ___
1/It_ U
44 NACA TM No. 1185
Figure 17.- Profile J 05: Velocity distribution with instability and separation U
points at various Re = ° and c - values. U
Profile I ''' 10 ID 1Jfr I -
- w hui1 i Pressure side--- _ /W1VJJ5Re-
__ffJ7j5 \ VAlz
rn' lo t os 0.1
fog
\111^
LW) 0.04 0.06 408 rn;0.
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.
I rn=n•.=nicI
0
iO
-
---too
R 005
I0Iv v6
Suction side Pressure side
NACA TM No. 1185
45
Is'
Figure 19.- Characteristics curves about the Influence of the thicknsss 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 now changes; pressure side: Ca Ca for not abrupt flow changes; pressure side: Ca <Ca for not abrupt flow Changes
profile 000
I en•I co.zs =01
10'
10'
IIiIIIjjjIjjIIIIiiiiii 025 ____________too
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.
MENOMOREEM SPIPOPIAPNINFOOld
jig a
Üü{0 III (113 Ui (List
46 NACA TM No. 1185
C00 .j Ca =025
iLl IL' I
MEMMEMMEN MMMMMEMMM
I
MMMMEMMMMM 1.OMMMMEWBOM
fit
)1I//Y,IA I I I I I 405 41 415 42 tz&ct
Co =05
'11/ I I _j_-----t--T 405 at 415 42 0.25
CQ = 1.0
a
am IS WINEW,
"ISEWO MEMOin
- 405 41 4(5 42 425 0 ((05 41 0.15 (22 as
Figure 21.- 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; suction - side.
I/f Ca=0
fit Ca025
1319
1111
OPENOW-1
EIMEWWROM wwWWWOMMI
l^E.Pi.P.Ap
popmd am
Ca O'S At Ca1,0
IUUPLrn1
7 a m i,
404
402
- .- - ---ro
405 41 415 42 Q25t
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.
fit Ca 0J
MI MEVIA-Mi WAVINAME
MWOM 11t
NACA TM No. 1,185 47 -
Pt Cc 0
AtCo 25
NOW
I-f/f
-+---_.!! a Of V QØ 2 425* 455 4! 555 42 025t
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 = i05
COAt
AI I I M-610 SSION RMI, MMISM a ft-- - MIN
NEME,HOW MMIN h M
EMIq -q- ME" WE aMM-SM,M,
I -.J2I - 415 1 d
9115 as os 02 425
Co 1.0
IN
LO
d d %5 af'5 4 42 42Sf
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 =
-1- I-OifL I I If 005 0' a1502 025 0
Ca05 . _flt
t-°5 45 L4 005 4! 515 02 0250
CD= 10
M4M
fk
1-00.06
008
0.02
d
48
NACA TM No. 1185
1k
Ca =0 Ca025
a
I
Mg
MEM-^
-U- a.. 505 0.! Sf5 02 425 0 0.05 at 015 02 025
Figure 25.- Position of the instability . point (s/tcrjt as a function of the thickness of the profile d/t and the camber of the profile f/t; suction side; Re =
10 6 .
At .Jit C4=425
006
0.04
0.02
d
o.IbI 025 .00
::
—
u!4I L 005 or ojs as ass 0 sos 51 ojs 02 025
I/f Ca=O'S Cf 008
006
004
002
a
- 005 01 0.55 52 025 0 505 0,! 0.15 02 000 54
Figure 26.- Position of the instability point (s/tcrjt as a function of the thickness of the profile d/t and the camber of the profile f/t .; pressure
side; Re = 10
iL!1 in him
rvi NW AVAVAVA —MAPAPAV
NIVEMSEAWEMMA MA-A
. EVA
0.15
0.04
.102
NACA TM No. 1185
49
CaO. f/f ca-025
i
OPION gio 0W^ w
I; a rom00900. _
lg. Cl 415 42 ff 425 0D2 I 1D C
Ca-05
'-__I \'SJ Wt 415 41 0.15 42 D25
fit Ca= 10
ib 4h. I M 7MonM-a W%Mbb OR,
q-h1howift.
- 0.55 Cl C'S 42 025 455 01 il5 02 415
Figure 27.- Position of the instability point (s/t)crjt as a function of the thickness of the profile d/t and the camber of the profile f/t; suction side; Re = 10
Ca -0 CC =025
OL " i,', - 01*5 0.05 SI 015 0.2 (225
Cij=05
Ca10
- W
MINIMP1,10-
PPP"
4" '4*5 51 0.15 02 4725 - - 005 01 015 02 025
Figure 28.- Position of the instability point (5/t)rit as a function of the thickness of the profile d/t and the camber of the profile f/t; pressure side; Re = i07
-VA- 005
1102
rn -
-
MAVWAV W
VIECIMAIMISUMV 'A0
"APPENAVENAF
—U—, IMM
I waaM MINE P1va
606
0,04
402
a Id
50
NACA TM No. 1185
At cGO
co=O, 25
0.06
0,04
052'
a 5 at 615 02 025
0,05 1 0,15 0.2 025 t 0.05at 015 , 02 0,25
Figure 29.- Position of the instability point (sIt) crit as a function of the
thickness of the profile d/t and the camber of ' the profile f/t; suction
side; r = 108
Jit fi, ' ca =a2f
006
0,04
062
_
___________ L 605 01 615 02 025 t 005 41 , 15 42 025
A Ca=O.5 '4CaO
0,08
406
002
405 41 415 42 425 t 0,05 41 205 62 0,25 1
Figure 30.- Position of the instability point (s/t)crjt as a function of the
thickness of the profile d/t and the camber of the profile f/t; pressure
side; Re = 108
W..'07100,
NEW_ 310 ww^%0
--N.
i
wo. ft- *aINqhh... Mal N— I MI, , EN hM.,
404 ft6 rz a 0,65 01 0.15 62 0.25
At Cd=O,5_ Ii: 10=1,0
008
008
0,04
0,02
d
No 1, N— M,16 lk ON46 MI
MIL_________ a
0 64l_
7 IMVWM'AIAA
ViA
MEANS"
J)l25 V /1 / I 0.0 .. 41S 0.2 525
.2,
0175
•
075
0725
505
0.01
0.02
0225
I75
LA 5055
6
0.55
024
0.02
a 0.05 51 575 0.2 025 00125 0.05 0.1 05 02 57
f/I Cc =0,5 '/1 ta-1 40723
NACA TM No. 1185
51
C00 JO Ca25
P -U mpg -0. a mb,
i1 I I I •
sac o.i 515 0.2. 025
ft.
Ca-OS
IWAUR!ii PJIt(K3EIi•
-U-T11111.
505 8.1 515 (22 511 (205 (27 U.l 'V (V'
Figure 31.- Mean position of the instability point (/t) crit. for pressure and
suction side at Re = io6 . ()crit.= 1/2 (crit. suet. side + 5crit. pressure side)
(o0 C57Q25
2100
004
002
SOC
0,82
a
0.___)
RIS
••t••\0/25
4125
a 0,148 005 01 075 0.2 525 ' - 0.85 0,0 015 52 0.25
Figure 32.- Mean position of the instability point (/t)crjt for pressure and suction side at Re = 107 • ()crjt. = 1/2 ( scrit . suct. side + 5crit. pressure side).
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