Mapping Functions of the NACA Airfoils into the Unit Circle

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Title Mapping Functions of the N.A.C.A. Airfoils into the Unit Circle
Author(s) Imai, Isao; Kaji, Ikuo; Umeda, Kwai
Citation 北海道大學理學部紀要, 3(8), 265-304
Issue Date 1950-05-20
Doc URL http://hdl.handle.net/2115/34183
Type bulletin (article)
File Information 3_P265-304.pdf

Instructions for use
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
[Journal uf the .Faculty of Rciencc, I-Iukkaid6 University, Japan, Ser. II, Vul. III, Nu. 8, ]115().)
Mapping Functions of the N.A.C.A. Airfoils
into the Unit Circle
By
lsao IMAl*, Ikuo KAJI** and Kwai UMEDA***
(Received July 15, 111411)t
J\y meau>; of the method developed by one of us (1.I.), the mapping functi~ns bringing
thc X.A.C.A. airfoils into the unit circle have been determined numerically, as given il1
'fA!:1-'.cS II-XXXHI. Thc dependences of their Jlrst three Fourier-cuefficients un the shape
variables, i.e. call1bcr, camber position and thickness have been clarit-Ied, as given ill TABLES
I and XXXIV as well as Figs. 2 - 8.
1. Outline of the theory developed by
one of us (1.I.)Y)
In general, an arbitrary airfoil can be represented. parame
trically by
x = cos'l'l, ;1!=g(()),
{} = 0, " corresponding to the trailing and leading edges respective
ly. The region outside the profile in the z (= x + iy)-plane may be
mapped conformally into the region outside the unit circle in the
Z-plane by an analytic function:
+
1~f C·Z + z1 ) + C (·Lr"j) , (1)
where,
* Department uf Physics, University of T6ky6 and 1'ulllcrical Cumputation Bureau in T6kyo.
**
Department of Physics, Hokkaid6 University.
**"
Department of l'hysics, llukkaiclo Universit¥ and Nnll1crical COlnputation Bureau in T6kyo.
t Presented partly ilt the {~irst l1lceti"g of The Hokkaid6 Branch of The Physical SocielY
of Japan, December 8, 1946.
266 1. 1mai, 1. Kaji and K. Umeda
e" = A" + iB.,., (3)
C(Z) and hence C A", B" being small. Putting on the unit circle
n,
in the Z-plane, i.e. Z = f}O,
(4)
we get another parametrical representation of the profile
x = cos (J + A (0),
,= Ao + (1 + A_l +Al) cos (J - (B_1 - B sin 0
1)
+ :E (A.,. cos riO + Bn sin riO), (5)
H=.!
y = B «(J),
= (B_1 + B cos 0 + (A_1 ~ A sin 0
1) 1)
=
+ L, (B" cos nO - A" sin nO).
n=2
For any, l!'ourier series R(O):
R(O) == au + L=, ' (a" cos nO + b sin nO)
n
"~1
we can define the conjugate Fourier series R*(O):
== =
R* (OJ L, (a." sin nO - b. . cos n(J) ,
1£=1
= - - 1 I'>"lt: [R (~ + 0) - R (0) ] cot -~ d~ . (6)
2rr 2
0
Thus we get from Eq. (5)
x (8) = cos 0 + [y*(O) + Ao + ZA_ cosO - ZB_, sin OJ. (7)
1
This is the basic relation affording the possibility of the successive
a pproximat ion.
Assuming that the trailing and leading edges correspond to
o
=0 and rr+J respectively, we have geometrically the boundary
conditions, as seen in Fig. 1:
Mapping Functions of the N.A.C.A. Airfoils into the Unit Circle 267
•
Fig. 1 Boundary conditions of the litapping f,mction 0/ the air/o£l
j,,-ojile in the z-/,lrme into the unit m'C/e 111 the Z-plane.
x.(O) = 1, x'(O) = 0,
X (rr + a) = -I-s,t x' (rr + a) = 0, (8)
which give
2B = y*' (0), (9)
-1
(J _ y*' (0) + y*' (rr + fJ)
tan-- - (10)
2 2+s-y*(O) + y*(rr+ fJ)
1 .
1+2A_l =.. .. [2+s - y*(O)+y*(rr+o)+y*'(O) sin fJ]"
1+coso , . ,
(11)
An = - 2A_l - y*(O), (12)
i. Practical procedure of the numerical determination
of the mapping function Eq. (5).
Practically it is sufficient to take 40 equistepped values of the
parameter
2rr
O. = s -- , s = 0, 1, ......... , 39, (13)
. 40
But, for the airfoils 43012 and 63012, we took 80 points in view of
their peculiar form at the leading edges.
•
t z can be neglected, if the leading edge is rounded into :r: = - 1. But in fact, i;' the 'case
of the N.A.C.A. airfoils with strong camber the leading edge swells out over the chord
edge. We have taken e for the V-digit airfoil family into \lccount.
268 I. Irria~, I. Kaji and K. Umeda
ZERO-TH ApPROXIMATION
We take ,~
(8.) = cos Os' (14)
x(O)
It is naturally more favorable to take in place of Eq. (14) the exact
values of x (8.) of some other airfoil belonging ,to the same series,
if they were already known.
FIRST ApPROXIMATION
1
1. We evaluate on the profile the ordinates 1/) (0.) corresponding
to the abscissae (0.).
x(O)
2. ~/(!) (0.) are calcuiated by means of the Stirling interpolation
formula, sometimes up to A 7 being taken into account.
3. Approximating the integration in Eq., (6) by summation, we get
_y*CIl(O ) = _!_ y'(I)(8 ) + ~ ~ y(Il(8 )' cot krr
• 20 • 40 f:l 40 '
s+1G
J
= ~ yI(I)(8 ) + ~ [y(I)(O ) _'y(Il(O _ .)' ~ cot k-;:
20 • f:1 40 40 '
s+1G • 1G
(15)
which is performed in the usu~l way of the harmonic analysis. t
!N
[check] L, y*(O.) = O. (16)
•
8~O
4. The Stirling interpolation formula gives y*lCIj(O.).
[check] z":" ; y*'(O.) = o. (17)
a=!)
5. B-:'t is found immediately from y~'(I)(O), by Eq. (9).
6. By virtue of Eq. (10)" -;: + (~ or (J il'l found as the zero point
of the expression:
y*'(O) + y*'(O.) ~ cot (0/2) [2 - y*(O) + y*(O.)] , (18),
•
using its values at several points (810 - 02~) in the vicinity of ()20( =-;:).
l' We have prepared the tables of the functions (nj40)cot(krr/40) al1d (n(80)cot(k'rr/80) in the
range 1? = l-2500, k = 1-19 aml k' = 1-39.
Mapping Functions of the N.A.C.A. Airfoils into the Unit Cirde269
7. The Bessel interpolation formula gives y*(7!' + lJ)'.
8. Eq. (11) gives 1+2A~? or A~?
9. Eq. (12) gives A~I).
10. Thus we get oP)(O.) by Eg. (7):
aP)(tJ.) = y*(I)(8.)+A~I)+(1+2A~n cos o. - 2B~? sin 0.; (7a)
39
[check] L, X(I)( 0.) = 40 A~I) •
8=0
SECONP AND. :FIIGHER ApPROXIMATIONS
11. Again we evaluate on the. profile the. ordinates y(II)(O.) cor
responding to the abscissae (0.) ,
X(I)
12. The second ~pproximation proceeds in· the same way as the
first approximation runs, except using y(lI)(8.) - y(I)(8.) in place
of yOI)(O.) itself to simplify the computation of Eq. (15).
The approximation should be continued successively irdhe line
x(O)_, y(I)_, X(I)_ y(II)~ x(lI) ........... .
graph calc. graph' c~lc.
until the obtained X(N)(O.) coincide selfconsistently with the pre
cedi,ng X(N-l)(8.).
3. Numerical results.
We have performed the above procedure for the following 32
N.A.C.A. airfoils,<z)
A. SYMMET~ICAL AI~fOIL GROUP (Standard 0012)
iii Thickne.ss Series (0000)
0003; 0006, 0009, 0012, 0015, 0018, 0021, 0025.
B. ASYMMETRICAL AIRFOIL GROUP
I. 4-DIGrr AIRFOIL FAMILY (Standard 2412)
Camber Series (0412)
0012, 2412, 4412, 6412.
ii. Camber Position Series (2012)
2212, 2312, 2412, 2512, 2612.
iii. Thickness Series (2100)
2406, 2409, 2412, 2415, 2418, 2421.
270 1. Imai, I. Kaji and K. Umeda
II. 5-DIGl'r AIRFOIL FAMII~Y (Standard 23012)
1. Camber- Series (03012)
0012, 23012, 43012, 63012.
11. Camber Position Series (20012)
21012, 22012, 23012, 24012, 25012.
iii. Thickness Serie.s (23000)
23006, 23009. 23012. 23015, 23018, 23021.
For convenience in evaluating th,e velocity q(O) at the airfoil
surface, we have calculated fUrther two following quantities:
(1 + 2A_l) sin 0. - 2B_l (I-cos 0.)
qc(o.) =
(X'2 + y'2)!
(1+2A_l) (1- cos 0.)+2B_ sin O.
q.(O.) = t
(X'2 + y'2)!
q(O,) = qc(O.) cos a + q.(O.) sin a,
where a is the angle of attack.
The obtained values of x,y, x', y', y*, y*', qc and q, for every
airfoil are given as functions of O. in TABLES II-XXXIIl,t together
with'the values of Ao; A_I> B_1 and lJ. The latters are collected as
functions of the shape variables of the airfoils in TABLE XXXIV as
well as in Figs. 2-8. In view of it, we can summarize their de
pendences on the camber, the camber position and the thickness
of the airfoils, as given in the following TABLE I.
TABI,E I. Dependences of Ao, A_I, B_1 and iJ on the camber, the
camber position and the thickness of the N.A.C.A. airfoils.
-B_1
camber nea(ri!n yc rienadseinpge)n dent I nea(rilnyc rienadseinpge)n dcnt lsitnroeanrgllyy dienpcerenadseinntg I increasing
I
camber nearly independent nearly independent linearly dependent
increasing
position (decreasing) (decreasing) . weakly increasing
I
thickness linearly dependent I linearly depen.' dent linearly dependent -I decreasing
strongly increasing very strongly increasing weakly increasing
t In all tables the numerical values of x, y, x', y', y*, y*1. Au, A-h B_1 and' e are being
multiplied hy 10000. <or the double-sign ± ill TABLES II-IX, the upper and the lower
are to be taken for () = 1-19 and , 21-39 respectively,
Mapping Functions oftheN.A.C.A. Airfoils into the Unit Circle 271
It can be enunciated thatAo and A_1depend practically on the
thickness only, while B_1 is affected primarily by the camb-er and
secondarily by the camber position as well as by the thickness, i.e.
B-J plays essentially important role for the aerodynamic character~
istics of airfoils.
In conclusion, we wish to acknowledge the financial supports.
of The Ministry of Education as well as The Numerical Computa
tion Bureau in Tokyo.
Refe~ences
(1) 1. IMAI: Journ. Soc. Aer01I. Sci. Jafan 9 (1942), 865.
1. lMAI and K. SAT6: Journ. Aeron. Res. Inst. TOkyo Imp. Univ. 247 (1945), 91.
(2)' E. N. JACOBS, ICE. WARD and R.M. PINKERTON: N.A.C.A. Tech. Rep. No. 460 (1933) .
. E.N. JACOBS, R.M. PINKERTON and H. GREENBERG: N.A.C.A. Tech. Rep. No. 610 (1937).
litstitute of Theoretical Physics,
Faculty of Science, Hokkaido University.
272 I. bnai, 1. Kajiand K. Urneda
TABI,E II. N.A.C.A. . 0003 airfoil.
x -10000 - 9750 ;-9500 9000 - 8500 i-8000;-7000 1-660(fT=-50ocF'
0003 _Y_ ~_o ~±_~94.71~!30.7 ± 1J2:7 ± 210.0:1: _ 23411.:l::.~~'L~i.::L286~!} ± ':2!l.7.1.
1
____ ~_ _ ±~0~~1 ± 2ggg.21± 2~4.1 ±2~~.2 ± 4~~.21±6~~~2,± 80~~~41± 90~~.31 ~o~oo .
= a o.
Ao -35, 2A_l 258, 2B_1 0, =
__I I . I I'
. (} x y x' y' y* y*' qc qs
. .
0 10000 0 0 68 173 l~ 0 0 0
'1 39 9872 ± 11 ::;: 1633 72 1= 175 29 ± 0.9815 0.0773
2 38 9489 ± 24 ::;: 3219 106 182 49 ± 0.9841 0.1559
3 37 8866 ± 45 ::;: 4703 154 189 ::;: 46 ± 0.9898 0.2376
4 36 8019 ± 72 ::;: 6044 190 1= 195 .::;: 14 ± 0.9971 0.3240
5 35 6975 ± 104 ::;: 7228 21;;l 1I 193 ± 25 ± 1.0030 0.4155
6 34 5758 ± 138 ::;: 8230 1 225 I 186 ± 69 ± 1.0080 0.5136
7 33 4401 ± 174 ::;: 9016. 225 - 171 ± 124 ± 1.0134 0.6210
8 32 2937 ± 208 ::;: 9583 212 -- 148 ± 173 ,± 1.0178 0.7394
9 31 1403 ± 240 ::;: 9911 190 .117 ± .221 ± 1.0221 0.8730
10 30 - 163 ± 267 ::;: 9984 156 - 78 ± 274 ± 1.0274 1.0274
11 29 - 1721 ± 288 ::;: 9814 101 31 .± 318 ± 1.0323 1.2087
12 28 - 3216 ± 298 ;::.9411 32 21 ± 345 ± 1.0367 1.4268
13 27 - 4666 ± 298 8785 '38 76 ± 355 ± 1.0404 1.6978
14 26 - 5983 ± 286 7944 107 132 ± 355 ± 1.0446 2.0502
15 25 7152· ± 264 ::;: 6913 - 189 187 ± 341 ± 1.0489 2.5322
-
16 24 - 8146 ± 227 ::;: 5732 264 238 ± 298 ± 1.0508 3.2341
17 23 -- 8945 ± 182 ;:: 4416 - 319 280 ± 241 ± 1.0517 4.3808
18 22 - 9528 ± 127 ::;: 2996 - 372 313 ± 174 ± 1.0502 6.6304
19 21 - 9883 ± 66 ::;: 1505 407 334 ± 100 :f: 1.0293 13.0789
-
201 -10000 0 0 426 343 0 0 48.2049
TABIJE III. N.A.C.A.0006 airfoil.
I
----;--X-.----:lCO:O"'00c;:0-,.---:9:;;:;75::c0'-.-i---:::9-;;-;50"'0--.-1--=90iJO 8506 1-8000' - =70~O-=-0- c1~-6000 - 5000
0006 _y_ __0 _ ± I89.4± 261.41±355.4I,U~0'01,±,.468.2 i:_534~~± 573.8 ± 594.2
I
x - 4000 - 2000 0 2000 4000 6000 8000 9000 10000
Y ± 600.2 ± 580.2 ± 528.2 ± 456.4:± 366.4;± 262.4 ± 144.8,± 80.6 0,
--, = =
--'-~~-=-WS,-2A_~~5i6, 2B_l 0, (} O. ... - ----
I I I I
fJ x y x' Y' I if" y"'<1 qc qs· .
I -
I
0 10000 0 0 349 0 0 0
134 ,::;:
1 39 9865 ± 22 ::;: 1709 146 353 I::;: 64 ± 0.9589 0.0761
23 3387 89846168 ±± 9492 :+:;: 43835695 321198 --- 336885 .::;: 10919 ±±: 00..99769524 00..21355219
4 36 7946 ± 148 ::;: 6202 394 394 ::;: 21 ± 0.9946 0.3232
5 35 6878 ± 214 ::;: 7373 426 -- 390 63 ± 1.0069 0.4171
67 3334 45266430 ±± 238513 ::::;;:: 89312642 I 445455 - 334744 ±± 214466 ±± 11..00215670 00..56218757
8 32 2786 ± 422 ::;: 9647 426 - 296 ± 354 ± 1.0357 0.7525
-
9 31 1245 ± 486 ::;: 9922 382 232 ± 464 ± 1.0459 0.8934
-
10 30 319 ± 540 ::;: 9957 293 151 ± 559 ± 1.0557 .1.0557
11 29 - 1871 ± 577 ::;: 9754 188 1= 58 ± 632 ± 1.0646 1.2466
12 28 - 3371 ± 598 ::;: 9313 64 46 ± 689 ± 1.0739 1.4781
13 27 - 4785 ± 596 ::;: S650 - 80 157 ± 720 ± 1.0832 1.7675 .
-
14 26 6079 ± 572 ::;:7800 239 270 ± 708 ± 1.0902 2.1897
-
15 25 - 7225 ± 522 ::;: 6755 382 378 ± 682 ± 1.0991 2.6536
Itl 24 - 8194 ± 452 ::;: 5596 519 481 ± 586 ± 1.1000 3.3853
17 23 8977 .± 360 ::;: 4320 633 561 ± 454 ±1.0934 4.5545
18 22 - 9545 ± 254 ::;: 2902 - 728 i 625 ± 347 ± 1.0859 6.8564
19 21 - 9886 ± 132 ::;: 1450 814 668 ± 195 ± 0.9889 12.5647
-
20 -10000 0 0 858 I 684 0 0 24.6543'
NIapping Functions of the·N.A.G.A. Airfoils into the Unit Gircle 273
TABI,E IIII. . N;A.G.A. 0009 airfoil.
x -10000 \-975-0'-'-[-,9500 -9,,000 [-8500 1-8000 1-7000 [-60,00 1-5000
1
0009 y ,0 ± 28",.0 ± 392.2 ± 533.2 ± 630.0,± 702.4 ± 801.8 ± 860.61± 891.2
- x-- 4000--=-20oo~I--0--:-1 2000 [ 4000 I---Woo- 1-8000-[9000- 10000 -
__- ,-_y ± 900.2 ± 8'70.4 ::&;;''794.2' ± '084.6:J:: 549.6. ± 393.4± 217.2 ± 121.0, 0
Ao = - :25~, 2A-1 == 767, 2B_I = 0, i3 = O.
() x y 1 x' 1 y' I y* I y*' qt qs
---0 ---.1-00~00-. ----~-0 -----.-0 - 92 - 516 0 0 --~0 -
1 39 9851 ± 20 'F 1871 220 - 533 'F 187 " ± 0.8938 0.D703
2 38 9423 ± 73 'F 3516 414 - 567 'F 189 ± 0.9397 0.1487
3 37 8705 ± 146 =r= 4982 503 - 587, 'F 93 ± 0.9763 0.2344
-
4 36 7863 ± 230 'F 6347 583 597 'F 18 ± 0.9930 0.3227
-
5 35 6772 ± 328 'F 7514 659 591 ± 99 ± 1.0093 0:4181
6 34 5514 ± 434 'F 8465 678 - 564 ± 246 ± 1.0258 0.5227
7 33 4124 '± 540 'F 9205 672 - 514 ± 388 ± 1.0394 0.6369
8 32 2635 ± 643 'F 9700 640 - 442 ± 541 ± 1.0535 0.7654
9' 31 1090 ± 738 'F 9926 5GO - 343 ± 709 ± 1.0697 0.9136
10 30 - 47.1 ± 816 'F 9918 423 - 220 ± 850 ±'1.0847 1.0847
-
11 29 - 2013 ± 870 'F 9677 267 78 ± 958 ± 1.0986 1.2862
12, 28 - 3499 ± 898 'F 9202 73 79 ± 1039 ± 1.1129, 1.5317
13 27 -'4883 ± 892 'F 8524 -- 143 246 ± 1070 ± 1.1.254 1.8365
14 26 - 6167 ± 853 'F 7661 366 413 ± 1050 ± 1.1357 2.2290
15 25 - 72~2 ± 777 'F 6636 - 592 573 ± 977 ± 1.1427 2.7588
16 24 - 8244 ± 669 'F 5476 - 786 7.17 ± 853 ± 1.1441 3.1\211
17 23 - 9006 ± 532 'F 4197 - 958 839 ± 691 ± 1.1355 4.7297
18 22 - 9559 ± 370 'F 2841 -- 1095 932 ± 486 ± 1.0,928 6.9005
19 21 - 9895 ± 190 'F 1401 - 1187 991 ± 283 ± 0.91l1·0 11.6542
20 -10000 0 0 - 1222 1018 0 0 17.6181
I
TABLE V" N.A.CA. 0012 airfoil.
x -·10000 1-9750 - 9500 1-9000 1-8500 1-8000 1- 7000 1-6000 1-5000
0012 y. o ± 378.8 ± 523.0± 711.012=_84Q~I:±:_ 936~1± 1069.0± 114]:<2± 1188_2
~, x - 4000 [- 2000 O. 2000 4000 6000 8000 9000 10000
Y ::b 1200.4 ± 1160.6 ± 1058.81,± 912.6 ±. 732.8:1: 524.6,± 289.61.± 161.41 0
Ao= - 319,2A_ = 1 (J)34, 2B. . :;=· 0, .(J :~ O.
1 1
() ~. x I y, I x'· I y' I y* I ytll I . qc ' qs
1- ,.
0 10000 0 0 715 0 0 0
1 39 9853 ± 44 'F 1867 236211 -- 727' 'F T41 ± 0.9111 0.0717
2 38 9419 ± 106 'F 3619 477 - 756 'F 209 ± 0.9340 0.1479
3 37 8725 ± 196 'F 5175 665'1 - 787 'F 166 ± 0.9601 0.2305
4 36 7805 ± 313 'F 6514 805 I - 803 'F 28 ± 0.9882 0.3211
5 35 6690 ± 446 'F 7655 885 - 794 ± 148 ± 1.0126 0.4194
6 34 5411 ± 588 'F 8587 920 - 756 ± 340 ± 1.0336 0.5267
7 33 4004 ± 732 'F 9286 901 - 686 ± 546 ± 1.0538 . 0.6458
8 32 2506 ± 869 'F 9744 834 - 585 ± 750 ± 1.0730 0.7796
9 31 956 ± 992 'F 9948 726 -- 451 ± 950 ± 1.0926 0.9332
10 30 - 606 ± 1094 'F 9896 560 - 287 ± 1138 ± 1.1132 1.1132
11 29 - 2141 ± 1165 'F 9613 337 - 96 ± 1286 ± 1.1331 1.3266
12 28 - 3614 ± 1198 'F 9114 80 114 ± 1380 ± 1.1514 1.5848
13 27 - 4993 ± 1189 'F 8414 - 207 335 ± 1418 ± 1.1682 1.9063
14 26 - 6248 ± 1134 'F 7537 - 503 556 ± 1390 ± 1.1818 2'.3194
15 25 - 7353 ± 1032 'F 6508 - 786 768 ± 1294 ± 1.1902 2.8733 '
16 24 - 8287 ± 888 'F 5363 -- 1060 959 ± 1122 ± 1.1863 3.6512
17 23 - 9033 ± 702 'F 4120 - 1270 I 1118 4: 889 ± 1.1619 4.8398
18 22 - 9575 ± 492 'F 2757 - 1413 I 1238 ± 653 ± 1.1007 6.9497
19 21 - 9896 ± 258 'F 1345 - 1579 I 1321 ± 381 ± 0.8321 10.5733
20 -·10000 0 0 - 1668 I 1353 0 0 13.2310