This article concerns the experimental determination of optimum nose-cones (minimum drag) of a body of revolution at supersonic and hypersonic speeds by means of ONERA magnetic suspension. The study concerns two groups of models, specifically: a group whose nose-cone has a profile in the shape of X(n); the AGARD B group whose nose-cone is plotted in accordance with a given law. The results obtained for the first group are comparable to those calculated with the approximations of Cole and Newton and the experiments carried out by Kubota

Dubois, G.

English

NASA Technical Reports Server

NASA TECHNICAL M&~O~ NASA TM-77325 
NASA-TM-7732519840008069 
THE DRAG OF MAGNETICALLY SUSPENDED WIND-TUNNEL MODELS 
WITH NOSE-CONES OF VARIOUS SHAPES 
George Dubois 
Translation of. "Trainees de rnaquettes de soufflerie comportant des 
ogives de formes variees en suspension magnetique, " La Recherche 
Aeronautique, number 87 - 64, March-April, 1962, pp. 47-54. 
OCT 1 :2 i?33 
LANGLEY RESEARCH CENTE:1 
LIBRARY, t~ASA 
HAMPTON, VIRG!NIA 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
WASHINGTON, D.C. 20546 AUGUST 1983 
111111111111111111111111111111111111111111111 
NF00308 
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;. 
';. 
STANDARD TITlE PAGE 
I. Fbpor' No. 
N~A TM-77325 
l. TUI. ond SuL.i.l. THE DRAG. OF MAGNETICALLY 5. R.por, Dol. 
SUSPENDED WIND-TUNNEL MODELS "WITH NOSE_~ __ A~u~qu=s~t~I~9~8~3~ ____ --1 
CONES OF VARIOUS SHAPES " •• p.,rOnD~~; Oigoniza,ion Cod • 
7. Author[a) " 
George Dubois, Chief ,Engineer 
Research Group; National Office'of 
Aerospace Studi'es and Re~pr!r("h 
9. Perform,n; O'II01izohon Nom. Q'ld Addr'-II 
" Leo Kanner Associates 
Redwood City, California 
"94063" 
.. P.r'or.Jn; O'IIOIIllo.ion R,por' No. 
10. Yo", UnU No. 
-n. -COn'roe! or (ironl No. 
NASw- -~S41 
13. Typo of R.por' ond Period Cour.1f 
-Translation 12. }ponso,;"; Agency Nem. end Add,.... • 
fiational Aeronautics and Space Adminis- lC. SpOllsorlng A;IIICy Cod. 
tration, Washington, D.C. 20546 
15. SlIppl.mtnlory Nolu 
Translation of "Tra'inees de maquettes de soufflerie compoitan 
des ogives dee formes variees en suspension magnetigue," 
La Recherc"he "Aeronautique, number 87 - 64, March-April, 1962, 
"Pp. 47-54. . -
16" A~Slto~' This article concerns the experimental d~termination' 
of optimum nose-cones (minimu~ d~ag) of a body of ~evolution 
at supe~sonic and hypersonic speeds by means of ONERA magnetic 
suspension." . 
The study c.oncerns two groups of moqels, specifically: 
a group whose-nose-~one hqs a profile'in the shape of Xn; 
the AG~D B group whose nose-cone is' plotted in accordance 
with a given law. " 
• The results obtained-" for the"" first group are comparable 
to those calculated "witH the "approximations of' Col~ and Ne,.wtor, 
and the experiments carri"ed out by Kubota." 
""This study "ca:r;-ried out as a resu'lt of" s'uggestions by 
Mr. Henri Le Boiteux~ Scientific Civics Oirector at ONERA was 
conducted "in a·small hypersonic wihd tunnel of OijERA, installE~ 
in the Laboratory of Physical Mech~nics of the Advanced Schoo] 
of Industria~ Physics an~ Chemistry (ESpeI) in collaboration 
with M. Bessac, engineer'of the ESpeI, "then a fo~rth year stud~nt. 
17. K.y'Word. (Selected fly Author[,») 18. Oh'rlbll'ion Sloltmmt 
" " 
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THE DRAG OF MAGNETICALLY SUSPENDED WIND-TUNNEL MODELS 
WITH NOSE-CONES OF VARIOUS SHAPES 
G. Dubois 
Chief Engineer, Research Group 
National Office of Aerospace Studies and Research 
Quantities and Notations 
Angle 
a : angle of the tangent to the profile at the point (x,y). 
Lengths 
R and 0, radius and diameter of the model (mainframe) • 
L = length of the model; 
1 = height of the nosecone; 
h = height of the blunted AGARD B nose-cone; 
p = radius of the tip of the blunted AGARD B 
Surfaces 
S = reference area (mainframe). 
Pressures 
p. = generating pressure; 
~ 
pl. = stopping pressure; 
~ 
nose-cone. 
PO = infinite pressure upstream (experimental section); 
p = pressure at the point (x,y) of the profile (nose-cone); 
2 qo = kinetic pressure = YPOMO 
-2-
*Numbers in the margin indicate pagination in the Iore~gn text. 
· ' 
/47* 
1 
Forces· 
RXp = pressure drag = cjpqos: 
Rx = total drag = cxqos. 
Nondimensional Coefficient 
A = 1/0 = elongation of the nose-cone; 
p /R = relative radius of the tip of the nose-cone; 
n = exponent of the equation of the profile of the nose-cone 
n (y -x ); 
= mach number of the experimental section; 
= Reynolds number referred to the length of the model; 
= Reynolds number referred to the height of the nose-cone; 
p - PO = pressure coefficient: = 
qo 
C
xp = pressure drag coefficient; 
C
x 
= total drag coefficient; 
Cxf = friction drag coefficient. 
I. Introduction 
I,l. We know that the Newtonian approximation makes it possible to 
determine the nose-cone of minimum resistance; it is on the basis of 
this scheme that we tested two groups of models for variable Mach 
numbers so as to determine this nose-cone. 
I, 2. Newtonian Flows 
The Newton scheme leading to the following expression of the 
pressure coefficient (fig. 1): 
'2 
K = P - Po = 2 sin2 8 = ~ y '2 
-p-- qo + y 
----- . --_. ----
2 
( 1) 
if Y = f(x) is the equation of the profile considered, is reduced for 
2 '2 
slender bodies to K = 2tg = 2y (2) 
, p 
(sine a-a- tga = y ). 
Taking into account the fact that the presence of the profile 
perturbs the flow in contact with the body and obtaining the balance /48 
of the quantities of motion, Cole /1//2//3/ finds that the correction 
.. 
term' yy must be added to expression (2): 
'2 .. K = 2 Y + yy p 
.On the basis of this theory Cole obtains a particularly interesting 
group of nose-cones of revolution of the shape y_xn • Introducing the 
value of Kp (3) in the expression of the pressure drag coefficient, 
C Rzp 8 I' K ' dx zp = --;:;- = D2 "yy • qo'" 0 (4) 
2 
(S = mainframe = n~ ), a simple calculation gives for a nose-cone which 
meridian is: 
. Y (I)R 
D/'.!= T 
II~ 311 - t C""=4A:!~ 
A= 1 being the elongation of the nose-cone; 
D 
-n -- 1: cone; 
n = ~: paraboloid of revolution. 
yA 
~ X 
---~-u------~~+-----+---,---~~ 
l 
Fig. 1: Elongation of the nose-cone: = 1/0. 
Key: (1) shockwave 
(5) 
(Ii) 
3 
2 The drag coefficient C is minimum for n = -3. The only accep-
. xp 
table value, because the necessary condition is: 
I 2 ~ n ~ 1· . 
The curves C /C cone as a function of n is specified on fig. 11 
xp xp 
(Cole) (see appendix 1) • 
If in the expression (4) Kp is replaced by value (1) (Newton ap-
proximation), we obtain: 
, - '16 J' yi' (",! 
corp ~ DI 0 1 + y': dr. 
The minimum cxp corresponds to a certain function determined on 
the basis of the previous expression (7) • A.J. Eggers, J.R. Meyer, 
M. Resnikoff and David H. Dennis /4/ found that the theoretical profile 
is actually very close to the simple profile y-x 3/4. The calculations 
of Cxp have been carried out for n = 1, 3/4, 2/3 and 1/2 (see appendix 
2) and the curve Cxp/Cxp cone was plotted as a function of n, fig. 11 
(Newton) • 
II. Experimental Arrangements 
11.1 Means of Investigation 
The test on the model defined in number 22, were carried out in 
a hypersonic wind tunnel with gusts of the ONERA. 
The drag was measured with the ONERA magnetic suspension method 
described in articles /5/ and /6/. 
In this wind tunnel in which the Mach number varies for these 
tests from 3.75 to 6.3, the adjustable generating pressure from 0.5 
50 8 atm makes it possible to obtain a Reynolds number per cm between 
104 and 7 by 104 • 
4 
Since the models have small dimensions {D = 6 rom, L = 60 mm)the 
Reynolds number referred to the~ngth of the body is low; at maximwu 
Rl~ 4 by 105 • This corresponds if the model is on the scale 1/1~0 of 
an -aircraft flying at M = 5, to an altitude of the order of 50 kID. 
It may be noted that the static pressure PO in the experimental 
section is, according to the value of the previous quantities, between 
100 and 1500 pascals. 
All the measurements are recorded without any difficulty as a 
function of time (time of the gust ~40 seconds). 
II.2 Definition of the Models 
TWo groups of models were chosen: 
n 
a) groupy-x (figs. 2 and 3) • 
----1-~-
I. 
l=128 ... 
'(0 
- II 
I:::) 
Fig. 2: Model: cylinder plus nose-cone, y_xn : (case n = 2/3). Elong-
a tion of the nose-cone: A = 2.13. 
---- -------- --".1 .. ~ • 3 n:r-, 2 n=-3 1 n--2 
Fig. 3: Models in xn 
(nose-cone) • 
5 
These models consist of a stra:Lght cylinder and a nose-cone whose /49 
profile is given by: 
n = 1, cone; 
n = 3/4; 
n = 2/3; 
. D (':r)-y = - - , 
. 2 l . 
n = 1/2 (paraboloid of revolution). 
To have an element of comparison, we took again the models tested 
by Kubota /3/ who determined their pressure drag coefficient C . 
xp 
These models have an elongation nose-cone: 
A= liD = 2.13. 
Note: According to Kubota's notations, is replaced by: 
o = D = 1 = 0.235. 
21 2I 
b) AGARD B group, elongated to 10 D (the AGARD B model has an 
elongation of 8.5 D) (fig. 4). 
6 
x 
L 
i i 
----W----- ~ 
I i 
Fig. 4: Elongated AGARD B model (10 D). Nose-cone: length = 3 D 
(model number 1).8 a = 18°26', D = 6 mm; L = 60 mm, 1 = 18 mm 
Nose-cone: 
Elongation: A = 1 = 3 (model number 1). 
. D 
Equation of the profile (type A, by Mr. Maurice Roy) : 
:1: [" J (:1:)2' T' "(X\3] 
,r':=3 1- 9 IT +5', p)', 
Blunted models, number 2, 3, and 4 according to the scheme 
specified on fig. 4. 
h = height of the blunted nose-cone. (variable); 
h = 1 - x + p - r tg 8 
with p = r 
tg8 
The elongation A of the nose-cone is specified in fig. 5 as a 
function of the relative radius of the tip p/R. 
Note: The control of each nose-cone was carried out with the 
profile projector. The dimensions are respected to within plus or minus 
0.01 mm. 
II.3 Calibration of the Drag 
A very thin wire, fixed to the base of the model is connected with 
an auxiliary balance (fig. 6). The effort applied is automatically 
balanced by the current passing through the so-called drag coil. This 
current is measured (deviation of the spot) and a calibration curve is 
plotted for each model. 
7 
·J 
L A=-D 
3 
2,8 
2,6 
2,4-
2,2 
~ 
(+/' 
" 
2 
° 
K , 
V -3 
I ........ 
"" 
I 
'" 
~~-
P 
R 
~ 
0,2 0,4- 0,6 
Fig. 5: Elongation of the nose-cone as a function of the relative radius 
of the tip £.. • 
R 
1. Model number 1 (AGARD B) (x = 0, Q.= 0) • R 
2. Model number 2 (x = 1, e...= 0.117) • R 
3. Model number 3 (x = 3, (L= o • 34) • n 
.. 
4. Model number 4 (x = 6, e...= 0.624) • P.. 
Remark: The curves specified on fig. 6 show that for a same 
force the current varies according to the shape of the profile of the 
nose-cone. Actually since the models in xn have the same length, the 
more rounded -the nose-cone, -therefore the more magnetizable matter it 
contains, consequently the lower the current has to be for a given 
force. 
Thus for a force of 12 g we have: for the cone (n = 1) a deviation 
of the spot of 159 rom, and for the paraboloid of revolution (n = 1/2) 
for 148.5 rom. 
11.4 Specification of the Measurements 
8 
-generating pressure p.: + 0.5 %; 
, 1 
stopping pressure p.: + 0.5 %; 
1 -
static pressure (experimental section): + 1 %i 
Mach number M: + 1 %. 
Drag R : + 0.5 % for R = 7 gramsi x x 
+ 1 % for Rx = 3 grams. 
Drag coefficient: C = Rx x 
qS 
Negligible error on S (+ 0.1 %) : 
·1~~1 ~ 1 0 /. c., ....." /0.' for low drag (+ 2 percent) • 
200 -r----..,..----- --y-----
Spot I A C 
mm \-1- 0 I-j f 
150 -~-E"- -I i 
1 
2 
4. 
8 I 
mO~----~----~--~--~~-----+-----r--~-r----~ 
I 
50+-----~i--.~~----_+--~_+----~----~------r_--~ 
o ~----~------~------------~~----~~----~------~-----~ 
. 0 2 4 6 8 10 12 14 g 
Fig. 6: Calibration of the balance 1: n = 1 2: n = ~ 3: n 
1 4: n = ~ A. nose-cone; B. drag coil; C. experimental section; 
~ 
D: wire (nylon 20v ) E. grams. 
III. Experimental Results 
2 
= 3" 
III,l. For each model and for a given Mach number, C
x 
total was deter-
mined as a function of the Reynolds number ~ whose variation was 
9 
obtained from that of the, generati~g pressure p .• 
~ 
, " 
o,~+--------+--------+--------+------~ 
0 ... 
...... 
.......... O'2+----2 ... ~_~_------~--------+-------~------~~------~-------; 
---
............ -"""1 
---
---- -------~_ Cn 
, ------ ------- -
o'~OL5~----~------2-.1+0~5~----~----3-.~m~5~,----~------~-m+5~--~m~L~ 
Fig. 7. - 111 ~ 3,75 Pi = 0,3 a 2 atm. A n -= t 0 n'= 1/2 ~n = ~/~ + n = 3J~. 
III.2 Group of Hodels with Nose-Cones in the Shape of xn 
The results are specified in figs. 7, 8, 9 and 10. 
As may be observed, for low ~ the C
x 
is 1.5 to 2 times greater 
than when ~ has the largest value; the limiting laminar layer on the 
profile is thickened as RL decreases and in corrolation the total C
x 
increases, very consistently with theoretical predictions (fig. 7). 
To compare our results with those calculated from the approxi-
mations of Cole and Newton and the experiments of Kubota /3/, we plotted 
on fig. 11 the curve Cxp/Cxp cone and Cx/Cx cone as a function of n. 
10 
A C; 
~5+-~----r-----~-------+-------r------~ 
O,~+-------~~~~~--+---------4---------4---------~ 
O 3 +-----+-----+----------t----~~__=___t_-=--~ ,
~210~5~----~-----2-.~~O~5~----~-----3-.~10~5~--~m; 
... .. 
Fig. 8. - M ~ 4,8 ~ n = IOn = II'.! • n··~ 2/3 + n = 3/4 Pi = 0,9 a ~,I alm.· 
ACx 
0,3 i-~:---_+----+_---_+---__l 
O,2r-----~------_+------4_----~ 
Fig. 9. - ~I = 5,3 .l " = J C n = I 12 • n = 2/3 + n = 3/~. 
11 
These curves in Table number 1 indicate that the minimum C is /52 
x 
obtained for n 2/3, M varyi~g from 3.75 to 4.8 and n 3/4 for M 4.8. 
The tendency is rather towards 3/4. 
Table 1 
VaJeurs du rapport Cn/Czp c'n, _ Valeurs du rapport CrIC" c'.' 
Cl) A = 110 = 2,13 (2) ,. = 110 = 2,13 Experiences ONERA Soufflerie :S2.PC 
Approx. Approx. Ex·per. II = 3,75 ~[ = 4,8 M = 5,3 .M = 6,3 n KUBOTA 
,i;L = 3.10' '~L = 3.10' '~L = 3.10' '~L = 2.10' COLE NEWTON :M = 7,i 
3/4 O,i02 I· 0,84 I 0,905 0,94 [;] 10,851 ~ I I I I I 2/3 I 0,666 0,852 0,882 0,92 0,90 0,89 0,95 
1/2 oc 1,132 1,1.,)5 t 0,96 O,9i 0,98 
Key: (1) values of the ratio C Ie ; (2) values of the ratio 
xp_ xp cone 
ex/ex cone' = 1/0 =2.13 ONERA Experiments S2.PC wind-tunnel. 
~~+-------~~~~~----~-------+------~ 
O,3+-~------~---------+~~~~~~--------+---------~ 
o,20~----~~-------140~5---------~------2-.-W~5~---~m~L 
Fig. 10: 
( • n = t 
J + n = 1/2 
M ~ 6,3 E' I 0 II = 2/~ 
, A n = 3/ .. 
_.,"'-_._-----
12 
Cxp-,-.---.-----r--~_.----_r--__, 
CXP (C6nl) 
~2 ~~+-+---+-----~--~~--~----~ 
Cxconl 
1+-----€~~~----+---~~---+~~ 
0,5 
., 
n--2 
0,6 0,9 
Fig. 11: A= 2.13 Experiments of Kubota. O~I=7.7Rdcm=6,8,to' 
Elongation of the nose-cone A= 2.13. Experiments in the S 2 PC wind 
tunnels (ONERA) 0 ~i ~ 3.75/ . 
+ 111 ~ 4,11 \ RL = 3.10', RI = 6,~.101. 
<1 111 ~ 5,3 
• 1.1 ~ 6,3 RL = 2.10', RI = ~,3.10'.' 
III.3 Group of Blunted AGARD B Bottles 
The curves of fig. 12 show that the minimum C
x 
is obtained with 
model number 2 whose nose-cone is very slightly blunted ( P/R = 0.117) • 
The effect on C of the roundness of the tip indicated in fig. 13 
x 
for RL = 300,000. 
IV. Conclusions 
The magnetic suspension allows precise measurement of the drag of 
bodies of revolution. 
The few results obtained on models in the shape of xn and blunted 
AGARD B indicate that at high speeds (M = 3.75 to 6.3) and for Reynolds 
" 
:1 
13 
number between 105 and 4 by 105 , .slightly rounded nosecones are less 
resistant than pointed nose-cones •. Th~s the profiles y-x2/ 3 in super-
. d" d 3/4. h . d" d son~c con ~t~ons an y-x ~n yperson~c con ~t~ons seem to correspon 
to the optimum in this respect. 
The same holds true for the slightly blunted AGARD B profile 
(radius of the tip in the order of tenth of the radius of the model). 
The advances achieved in the technique of magnetic suspension will 
allow such measurements of ex at higher Mach numbers, in a larger range 
of Reynolds numbers with simultaneous remote measurement of the base 
pressure, an indispensable element for a complete discussion of the 
tests. 
14 
0,5~------,--------'-------'--------~------'-------' 
O,~+--------P~-----+------~r-------+-------~-------i 
~2~-----+------+-----~------~--t--~------~ 
0, "OL5':"""""--....J....---2-.,+O-::5:-----..L---3-.-,0~5~--~~ 
Fig. 12 - AGARD B : ],I = 5,3 
no 1 + p IR = 0, x = O. 
no :!. p/H= O,117,.r = I. 
no 3 <1 p/H = 0,3'0, x = :J .• 
na ~ 0 p IR = o,62~. :r = fl. 
.1 
I 
! 
I 
I 
i 
I 
-I 
P 
R 0,5 '--"'---+--....1---+----'----f--'~ . 
o 0,2 0,4- 0,6_ . 
Fig. 13: Effect on the roundness of the tip on ex. 
Rr, = 300,000 P relative radius of the tip. 
R 
15 
Appendix I 
Approximation of Cole 
Taking an expression (6) n = 1, 3/4, 2/3 and 1/2, we find (fig. 11): 
n = 2/3 
1 C"I> 2 
CzP = 3i.." 
' C,. dill. = 3 = 0,666 n = 1 (cone) CzP = 2').% 
,n = 3/" C - 45 C .. ~ " __ O,7 r'3 zp - --_ 4 ' -- v 128/.- CZI> eMe 
n = 1/2 Czp = co Cz" = ac , C
ZII .... I (paraboloid of revolution) • ,. 
Appendix II 
Approximation of Newton 
Starting from the expressions (5) and (7) and introducing, the 
elongation of the nose-cone 
A = 1 
D 
the following expressions are found for the pressure drag coefficient 
Cxp : ,n = 1 (cone) C .. " = ft).22+ 1 
• ',' 3 [ 2 2 ] 
n = 3/"- Cz .. = f"l I - l'l -+- P Log (1'2 + I) 
taking 
• 8), 
n = 2/3 
n = 1/2 
r=-3 
C -....:!.... t - - Log (9)..2 + 1) '[ 1 ] O-~i.2 9).' 
(paraboloid of revolution) 
Cr .. = 8~2 Log (16i.! + I). 
For A = 2.13 (tested models) we find the following values for the 
ration C /C 0.84 for n = 3/4, 0.852 for n = 2/3 and 1.132 for xp xp cone 
n = 1/2 (curve fig. 11). 
Manuscript received 6 November 1961 
16 
Bibliography 
/1/ J. P. Guiraud: Some theoretical aspects of the study of hypersonic 
flows. Technical discussion of the ONERA of 13 December 1957, not 
published. 
/2/ J. D. Cole: Newtonian flow theory for slender bodies. J.A.S. 
vol. 24, number 6 (June 1957) •. 
/3/ Toshi Kubota: Investigation of flow around simple bodies in hyper~ 
sonic flow. GALCIT, memorandum number 40 (June 25, 1957). 
/4/ A. J. Eggers, J. R. Meyer, H. Resnikoff and David H. Dennis: 
Bodies of Revolution having minimum drag at high supersonic 
airspeeds. N.A.C.A. Report 1306 (1957). 
/5/ M. Tournier, P. Laurenceau and G. Dubois: ONERA magnetic sus-
pension. Rarefied gas dynamics. Proceedings of the first inter-
national symposium held at Nice. Pergamon Press 1960. 
/6/ Georges Dubois and Charles Rogue: On a method for measuring the 
base pressure. Measurement and visualization on a magnetically 
suspended cylindroconical model at MO ~7.6. La Recherche Aero-
nautique, number 79 (1960). 
17 
End of Document 


The drag of magnetically suspended wind-tunnel models with nose-cones of various shapes

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