AERODYNAMIC CHARACTERISTICS OF SPHERICALLY BLUNTED …

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NASA TECHNICAL NOTE NASA TN D-3088

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, _ N 66--12 138¢_ (ACCESSION NUMBER} (THRU|

(NASA GR OR TMX OR AD NUMBER) (CATI=_ORY)

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AERODYNAMIC CHARACTERISTICS

OF SPHERICALLY BLUNTED CONES

AT MACH NUMBERS FROM 0.5 TO 5.0

by Robert V. Owens

George C. Marshall Space Flight CenterHuntsville, Ala.

NATIONALAERONAUTICSANDSPACEADMINISTRATIONWASHINGTON,D. C. - DECEMBER1965

/ / /

NASA TN D-3088

AERODYNAMIC CHARACTERISTICS OF SPHERICALLY BLUNTED CONES

AT MACH NUMBERS FROM 0.5 TO 5.0

By Robert V. Owens

George C. Marshall Space Flight CenterHuntsville, Ala.

NATIONAL AERONAUTICSAND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical InformationSpringfield, Virginia 22151 - Price $3.00

TABLE OF CONTENTS

Page

INTRODUCTION .............................................. i

MODE LS AND APPARATUS ...................................... 2

ACCURACY OF DATA .......................................... 3

RESULTS AND DISCUSSION. ..................................... 4

Normal Force Coefficient Slope ............................... 4Center of Pressure ....................................... 5

F orebody Drag Coefficients .......................... ....... 6

CONC LUSIONS ............................................... 7

APPENDIX ................................................... 45

REFERENCES. . .............................................. 51

,°°

Ul

LIST OF ILLUSTRATIONS

z

Table Page

I Test Conditions ................................... 2

Figure Title

1 Nomenclature and Model Characteristics .................. 8

2 Cone Families Tested ............................... 9

3 Typical Schlierens .................................. l0

4 Model and Balance System Used in 7 x 7-InchTunnel Tests ..................................... 1 i

5 Model and Balance System Used in i4 x i4-InchTunnel Tests ..................................... i2

6 Diagrammatic Representation of Transonic Flow

Configurations for Cones ............................. 13

7 Variation of Normal Force Coefficient Slope

with Mach Number for Various Cones (d/D = 0) 14


with Math Number for Various Cones (d/D = 0.4) ............. 15


with Mach Number for Various Cones (d/D = 0.8) ............. 16

i0 Variation of Center of Pressure with Mach

Number for Various Cones (d/D= 0.0) ..................... 17

li Variation of Center of Pressure with Mach

Number for Various Cones (d/D= 0.4) ..................... 18

12 Variation of Center bf Pressure with Mach

Number for Various Cones (d/D = 0.8) ..................... i9

i3 Variation of Foredrag Coefficient with Mach

Number for Various Cones (d/D= 0r 0)..................... 20

iv

LIST OF ILLUSTRATIONS (Cont'd)

Figure Title Page

14 Variation of Foredrag Coefficient with Maeh

Number for Various Cones (d/D = 0.4) ............. 21

i5 Variation of Foredrag Coefficient with Mach

Number for Various Cones (d/D = 0.8) , . , .......... 22

i6 Variation of Normal Force Coefficient Slope with

Cone Half Angle and Bluntness at M = 0.68 ........... 23

17 Variation of Normal Force Coefficient Slope with

Cone Half Angle and Bluntness at M = 0.94 .......... 24


Cone Half Angle and Bluntness at M = i. 05 ........... 25








Cone Half Angle and Bluntness at M = 2.75 ........... 29

• 23 Variation of Normal Force Coefficient Slope with

Cone Half Angle and Bluntness at M = 4. 00 ........... 30

24 Variation of Center of Pressure with Cone Half

Angle and Bluntness at M = 0.68 ................ 31


Angle and Bluntness at M = 0.94 ................ 32


Angle and Bluntness at M = i. 05 ................. 33


Angle and Bluntness at M = i. 50 ............... 34

V


Figure Title Page


Angle and Bluntness at hi = i. 97 ......................... 35


Angle and Bluntness at M = 2.75 ......................... 36


Angle and Bluntness at hi = 4.00 ......................... 37

31 Variation of Foredrag Coefficient with Cone Half

Angle and Bluntness at M = 0.68 ......................... 38

32 Variation of Foredrag Coefficient with Cone HalfAngle and Bluntness at M = 0.94 ........................ 39


Angle and Bluntness at M = i. 05 ........................ 40


Angle and Bluntness at hi = l. 50 ......................... 41


Angle and Bluntness at hi = l. 97 ......................... 42


Angle and Bluntness at M = 2.75 ........................ 43

37 Variation of Foredrag Coefficient with Cone HalfAngle and Bluntness at M = 4.00 ......................... 44

A-i Relationship Between Geometric Parameters of

Spherically Blunted Cones ............................. 46

A-2 Relationship Between Geometric Parameters of

Spherically Blunted Cones ............................. 47

A-3 Hypersonic Newtonian Normal Force Coefficient Slope

for Cones of Varying Geometry .......................... 48

vi


Figure Title Page

A-4 Hypersonic Newtonian Centers of Pressure for Cones

of Varying Geometry ................................ 49

A-5 Hypersonic Newtonian Wave Drag for Cones of

Varying Geometry .................................. 50

vii

DEFINITION OF SYMBOLS

Symbol Definition

A Area based on diameter D, (in 2)

CDb Base drag coefficient (__)

CDf Fore drag coefficient (CDt- CDb)

CDt Total drag" coefficient "

CN Normal force coefficient slope l_ad )

Dt Total drag (lb)

D Base diameter (in.)

d Diameter of spherical nose segment at station of tangency (d/2r)

dBluntness ratio

D

e Cone half angle (deg.)

L Length of body

L/D Fineness ratio

M Mach number

M Minimum Mach number for shock attachment for pointed coness

N Normal force ( lb. )

P Free stream static pressure (psi)

Pb Base pressure (psi)

Pt Total pressure (psi)

q Dynamic pressure (psi)

viii

DEFINITION OF SYMBOLS (Cont'd)

Symbol Definition

Rn Reyn°lds number( Vfi_PL )

r Radius of spherical nose (in.)

T t Total temperature (° F)

CP/D Center of pressure location (from base)

Angle of attack ( deg. )

ix

L

' AERODYNAMICCHARACTERISTICSOFSPHERICALLYBLUNTEDCONES

" ATMACHNUMBERSFROM0.5TO5.0

SUMMARY

• . . ,/_/_F5 mic /An expemmental investigation has been made to determine the aerodyna

characteristics of several cones at Mach numbers from 0.5 to 5' 0 and to estab!ish ]

the effects of bluntness. The results of the investigation show that the abrupt changes /

in aerodynamic parameters in the transonic region are closely related to the devel- /

opment of the shock wave pattern. Those parameters most affected were the normal /

force coefficient slope and the forebody drag coefficient. The center of pressure, ]on the other hand, remained relatively stationary for cones with moderate half- . . Iangles, except at very near sonic Mach number. The effects of bluntness were mini_

real at Mach numbers smaller than those required for shock attachment on pointed /

cones. At higher Mach numbers, bluntness effects were sizeable for the smaller \cone half-angles, but negligible for the larger ones. In every case, increases in \bluntness as well as cone angle increased the downstream movement of, the, center \

of pressure. _)/.c. //y'/ _..)

INTRODUCTION

The inadequacy of theory in predicting the aerodynamic characteristics

of blunted cones accurately creates a need for a thorough experimental investi-

gation. Such a study has been especially desirable in the transonic Mach number

range where no practical theory applies.

This experimental program was initiated to satisfy this need and to provide

designers of blunted-cone vehicles with data from a wide range of parametric varia-tions. The normal force coefficient slopes, center of pressure locations, and fore-

body drag coefficients were determined for twelve cones over a Mach number rangefrom 0.5 to 5.0. The models used in the investigation were grouped into four

families of different cone angles, each having three different nose bluntness ratios.

Eight models with two additional bluntness ratios were tested up to M= 2.0 [i].From this wide range of geometric parameters, the effects of bluntness and cone

angle could be deduced in addition to those of Mach number. The tests were con-ducted at the NASA, Marshall Space Flight Center, i4 x 14-inch trisonic and the

7 x 7-inch supersonic wind tunnels.

MODELSANDAPPARATUS

The investigation was conducted in three series of tests. Transonic and super-sonic tests were run in the trisonic wind tunnel over Mach number ranges from 0.47

to i. 93 and 2.74 to 4.96, respectively. To establish the continuity and validity of the

data, a supersonic test covering the Mach range from i. 58 to 4. 39 was conducted in the7 x 7-inch supersonic wind tunnel. The characteristic air properties during operation

of both facilities are given in Table 1.

TABLE I

TEST CONDITIONS

Wind Tunnel Test Facility M Pt Tt Rn/in"

(psia) (° F) x 10 -6

14 x i4 Transonic 0.5 22 i20 0.3520.7 0. 436

0.85 0.489

O. 95 O. 5101. i5 O. 535

1.25 _ O. 535I. 43 22 I V 0.5i3

i. 93 28 i20 0.590

i4 x i4 Supersonic 2.74 30 i20 0.4022.99 45 i20 0.55i

3.48 75 140 0.6574.00 i05 140 0.717

4.45 i05 180 0.55i

4.96 I05 200 0.402

7 x 7 Supersonic i. 58 14.5 90 0.350i.99 0.306

2.44 0.246

2.99 I i 0. i873.26 0. i62

3.60 I _ 0. i373.89 r I- ol174.39 i4.5 90 0.094

2

Twelve cone models representing four families of different cone half-angles of

10, i3.32, 25, and 50 degrees were tested at all of the above Maeh number ranges.The three cones within each family had bluntness ratios of 0, 0.4, and 0.8;i. e., one

was pointed, while the other two were trtfmcated and tipped with spherical segments.

Eight models, representing two additional bluntness ratios (d/D = 0.2 and 0.6) foreach of the cone families, were tested in only the Mach number range from 0.43 to

1.93.[ 1 ] The geometric characteristics of the models and nomenclature are given in

Figure i, and the configurations tested are represented in the diagrams of Figure 2.Some representative Schlieren pictures of the test models are shown in Figure 3 forvarious Mach numbers.

All models were mounted on a sting which contained a system of interchangeable

strain gauge beams as parts of a three:component balance. Because of limitationson the model size, the balance system used in the 7 x 7-inch tunnel test was housedwithin a faired portion of the sting aft of the model base. The balance assembly of

the 14 x 14-inch tunnel, however, could be fitted conventionally to the forward end of

the sting projecting into the models. Composition X-ray pictures of both balance

systems are shown in Figures 4 and 5.

ACCURACYOFDATA

All force and base pressure measurements were taken over the angle of attackrange from approximately -4 ° to i0 ° , which was traversed automatically in both tunnelfacilities. Because of the sting effects on base pressures, the accuracy of the total

drag is somewhat uncertain. By subtracting the base drag, which was determined froma base pressure measurement, the invalid component is removed. The resulting fore

drag, which consists of wave drag and skin friction drag, should be accurate at thedesignated Reynolds numbers.

Data reduction of the strain gauge and base pressure readings were achieved

through digitizing the analog signals, punching values into data cards, and finally

processing the card data into coefficient form by computer.

The accuracy of the data, estimated by several means, takes into considera-

tion the following:

a. Data scatter

b. Repeatability of a given test point

c. Agreement of data measured by several different strain gaugesd. Level of accuracy of previous tests

e. Continuity between the data from the two different test facilities.

Based on these indications, the estimated accuracy of the faired data is

CP _ ±0 02, CN=±0.04, CDf= ±0.01.D "

RESULTSANDDISCUSSION

The results of the test are presented in a manner to enable the study of thevariation of normal force coefficient slope, center of pressure, and forebody dragcoefficient with Mach number for various cone configurations and also to show the

effects of cone half-angle (_) and bluntness (d/D). The basic data variations with

Mach number are given in Figures 7 through 15 for various cone half-angles and

are grouped according to bluntness. To provide data for the lower end of the Machnumber spectrum, the results of the transonic test published earlier [ 1] are in-cluded. Results from various other sources [2,3] are also incorporated when

possible. The interpolated values for various intermediate cone half-angles, repre-

sented by dashed lines, are also indicated in the above figures. By presenting cross-plots of the aerodynamic characteristics versus cone half-angles at various Mach

numbers {Figures t6 through 37), the effects of cone angle and bluntness can bestudied directly. Here again the data from other sources, notably from Reference

3, provided complementing key points.

NormalForceCoefficient Slope

With a few exceptions, the linearity of normal force coefficient with angle of

attack was maintained to approximately ±4 degrees. The normal force coefficient

slopes at zero angle of attack are presented in Figures 7 through 9 for three degreesof bluntness. For zero bluntness cones, the values of normal force coefficient

slope remain relatively constant once the local flow becomes supersonic, and the

levels agree well with hypersonic Newton±an theory, even for Mach numbers lessthan 5.0. For blunted cones, however, the values of normal force coefficients

slope for moderate cone half-angles decline at the beginning of the supersonic Math

number range, but tend to remain constant or increase slightly at the end of the

test Mach number range. An increase in C N values with still higher Mach numberswould be necessary to obtain agreement withhypersonic Newton±an values.

For Mach numbers smaller than M (minimum Mach number for shock

attachment for pointed cone) , no significanSt changes in CN_ were observed as thebluntness was increased from 0 to 0.4. For cones up to E = 30° , bluntness ratios

4

Of 0.8, however, produced very pronounced CNa peaks which occurred approximately: at those Mach numbers identified with shock a_tachment on pointed cones.

The cross-plots of. C._ versus cone half-angles at various Mach numbers

are shown m Figures 16 through 23. In general, bluntness effects were very small

for subsonic Mach numbers, but became sizeable at M > Ms for all cone half-angles,except _ = 50° .

Centerof Pressure

Since both the normal force coefficient and pitching moment coefficient were

generally linear up to +4° angle of attack, the variation of the center of pressure with

angle of attack was nearly negligible in that range. The values of the cone centers ofpressure at zero angles of attack are presented in Figures 10 through 12 for bluntness

ratios 0, 0.4, and 0.8. The least change of center of pressure with Mach number wasexhibited by the c = 25° cone family. For all others, a downstream or upstream move-

ment of the center of pressure was observed in the sonic Mach number region with asubsequent stabilization beginning at the transition of local flow from transonic to

supersonic speeds.

Figures 24 through 30 show center of pressure locations versus cone half-

angles at various Mach numbers. Up through free stream transonic Mach numbers,the effect of bluntness was greatest for small cone half-angles and decreased gradually

with increasing cone angle. While a bluntness increase from 0 to 0.8 at Mach 0.68caused the center of pressure of the _ = i0 ° cone family to move approximately

0.55 diameters downstream, the same bluntness increase on the E = 25° cones resulted

in a downstream CP/D movement of only 0.25 diameters.

For higher Mach numbers, the bluntness effect is largest for _ _ 10° ; thenit reaches a minimum for cone half angles of about 40 ° , and tends to increase again for

e = 50 ° . The agreement at M = 4. 0 of the center of pressure values with the relation-ship,

XCp L- [1-2/3 sec 2 e] ; oz_=0,D D

is good for all pointed cones, except those of _ = 50 ° . This equation assumes con-

stant pressure along a surface element, which is the case for cones in supersonic

flow. Reference 4 substantiates that the local flow regime for cones of _ = 50° isstill transonic up to a freestream Maeh number of 4.0.

For cones with e > 35 ° , the centers of pressure are located downstream of the

base under all conditions. Because the center of pressure is defined as the ratio

of pitching moment to the normal force, unsymmetrical axial force distribution at

5

angles of attack generates a stabilizing couple of sufficient magnitude to shift thecenter of pressure off the body.

ForebodyDragCoefficents

The zero angle of attack drag coefficients presented in this report contain

only forebody wave drag and skin friction coefficients.

Figures 13 through 15 show the variation of the forebody drag coefficientwith Mach number for three degrees of bluntness. The initial drag rise is nearlyidentical for all bluntness ratios up to the shock attachment Mach numbers for

: a given cone half-angle. For cones with bluntness ratios up to 0.4, the maximum

drag coefficients for a given cone occur at M , while peak values for largersbluntness ratios occur at higher Mach numbers. For comparison, the hypersonic

Newtonian drag values are included which were determined from

CDo= (t+sin 2 _) + 2 sin 2 ¢ - .

The first term in this equation represents the drag resulting from the sphericalsegment; the second term is the drag due to the truncated remainder of the cone.

In general, the effects of bluntness on the forebody drag coefficient of a

given cone are negligible for M < 1V[s , but are large at M > Ms and small conehalf-angles (see Figures 31 through 37). For cone half-angles of _ = 40 ° and

larger, the bluntness effects are almost negligible at all Mach numbers.

Abrupt variations of the data with Mach number occur for nearly all cones

regardless of bluntness. These phenomena are closely related to the mechanismof the shock wave development in the transonic region. This process, illustrated

in the diagrams of Figure 6 for pointed cones, is qualitatively described as follows:

As the free stream Mach number increases from subsonic values, a sonic

line forms at the shoulder of the cone, together with a main shock (nearly

normal) at the beginning of the wake. This marks the beginning of thetransonic range, which encompasses all free stream Mach numbers fromthe instant the'highest local Mach number reaches unity. [5] As Mach

number increases, the main shock grows and tends to move downstream,while at the same time a second shock wave approaches the cone vertexin the stream direction. Finally, this bow wave, whose shape and stand-

off distance depend on the geometry of the cone and the Mach numberattaches while retaining some of its curvature. The conditions behind the

shock are subsonic; this means that the downstream flow field (especially

6

at the shoulder) influences the entire shock. However, with further increase in

Mach number, a value is reached for which conditions behind the shock cease to

be subsonic. The shock angle decreases while the shock becomes straight.

Under the latter circumstances, the flow about the cone is genuinely super-

sonic, and therefore, the shock is independent of the downstream flow field.

Further increases in Mach number thus affect the data very little.

The span of transonic Mach numbers for pointed cones varies according to the

cone geometry; thus, for _ "-- i0 ° the transonic regime extends from about M = 0.9

to M= 1.15, while for c =50 ° it may range from M= 0.7to M=4.0 or higher.

Consequently, considerable data changes are seen in these ranges.

The flow field development around blunted cones is considerably more complex.

On small half-angle cones, "the expansion around the rounded tip causes local supersonic

flow at free stream M < 1. 0. This supersonic flow is then altered by shocking back to

subsonic flow. At high Mach numbers with the 50 ° cone at a bluntness of 0.8, subsonic

flow undoubtedly exists all the way to the base because of the strong normal shock.

Thus, shock and flow development around blunted cones is not nearly so straightforward.

CONCLUSIONS

From the results of the investigation to determine the aerodynamic character-istics of blunted cones, the following general conclusions are formed.

i. The effect of bluntness on the forebody drag coefficient is small for all

cones at Mach numbers less than those required for shock attachment

on pointed cones. This is generally true for the normal force coefficient

slope, except at small cone half-angles and extreme bluntness.

2. For Mach numbers higher than Ms, the forebody drag coefficient andthe normal force coefficient slope are greatly inflaenced by bluntness.

3. For cones with half-angles less than 30 ° , the center-of-pressure loca-

tions are more sensitive to bluntness than to Mach number changes.For cones with e > 35 ° , the centers of pressure are located down-

stream of the base under all conditions and are influenced by changesin Mach number as well as bluntness.

4. With the exception of normal force coefficient slopes for the blunted

cones, the aerodynamic parameters for supersonic Mach numbers are,

in most cases, closely approximated by Newtonian theory for hypersonicflow.

7

_- L-----_ D Base diameter

___ r Nose radius

_-_ L Length of body

d Diameter at tangentstation

e Cone half angle

TUNNEL DIAMETER OFMODELS

14 x 14-inch 3.00 in.

7 x 7-inch 1.50 in.

CONFIG. £ d/D r/_ a/A L/D

A-I i0° 0 0 0 2.836A-2 i0o 0.2 0.102 0.04 2.352A-3 i0o 0.4 0.203 0.16 1.869A-4 i0o 0.6 0. 305 0.36 i.386A-5 i0o 0.8 0.406 0.64 0.903B-I 13.32 ° 0 0 0 2.112B-2 13.32 ° 0.2 0.103 0.'04 1.768B-3 13.32o 0.4 0.206 0.16 1.425B-4 13.32 ° 0.6 0.308 0.36 1.082B-5 13.32 ° 0.8 0.411 0.64 0.759C-I 25° 0 0 0 1.077C-2 25° 0.2 0.II0 0.04 0.922C-3 25° 0.4 0.221 0.16 0.771C-4 25° 0.6 0,331 0.36 0.620C-5 25° 0.8 0.441 0.64 0.469D-I 50° 0 0 0 0.420D-2 50 ° 0.2 0.156 0.04 0.372D-3 50° 0.4 0.311 0.16 0.525D-4 50° 0.6 0.467 0.36 0.277D-5 50° 0.8 0.622 0.64 0.230

FIGURE 1. NOMENCLATURE AND MODEL CHARACTERISTICS

8

o d d o o" _o _o _o "

I I I I I

I I I

Z0

__- _ _.- m _ m m

N

w

iiiiiiiiiiiiiill

M=2.44

..................ili:i_iiiiiiiiiiiiiiii

%

D-5 -- D-5_M=5.89 M=4.45::M=1,58

.................................................

FIGURE 3. TYPICAL SCHLIERENS

l0

o

FIGURE 4. MODEL & BALANCE SYSTEM USED IN 7 X 7-1NCH TUNNEL TESTS

II

iiii!̧ ilii!ii!i!!iii!i!ili!i!!i!!!!!!iii!iiiiiiiiiiiiiiiii........

F IGURE 5. MODEL AND BALANCE SYSTEM USED IN 14 X 14-1NCH TUNNEL TESTS

12

/_ _"_\/

LINE OF SONIC /,,,, IJ//VELOCITIES SHOCK / /

M<I "_"_4"'/"""i / M<I i/// / /SHOCK

-----..._. ,," M>I..._ ,,,,, M>_ f.,,./M<I

SHOCK

M>, S-/ M:>, / J_SHOCK

FIGURE 6. DIAGRAMMATIC REPRESENTATION OF TRANSONIC FLOWCONFIGURATIONS FOR CONES

13

SYMBOL

3.2 _ ........ HIGHEST FREE STREAM MACH __ , ;-.,-_L______ .....

_'a NUMBERFOR LOCAL TRANSONICFLOW j__-I-'_D

,,:I _ - ..... ENVELOPE OF MINIMUM MACH _ , _ _n- NUMBERS FOR ATTACHEDSHOCK ,.__cI_.._t_. -d ......2.8 -. INTERPOLATEDVALUES -- J _ --

REF. 3'=D' HYPERSONIC NEWTONIAN THEORY

Z _ _ '

c.) OPEN t4x t4 IN. NASA MSFC TUNNEL

2.4 -- FLAGGED 7x7 IN. NASA MSFC TUNNEL ,.-_ _ :

r,,r) i

F--2.0z -/C------Ey_;---- _ ------_W J • _.--Z_T-Z_-'I ' L_'.._ _ _ ......o- \" ....... LJ

_,,_ -,-n_0°

F..1.6 ,3.3o._-_s_ _.x '._ _ _---_----EJ---_

n," / ". '.. --

,:I 0.8 _ _ _ \ " '

" ._ I"><... ...........,......_ _----_

0.4 ---8.... , __(_. _.------q_ "_ ,a _,

00 04 O.B 1.2 1.6 2.0 2.4 2.8 3.2 3.6 .4.0 4.4 4.e

MACH NUMBER

FIGURE 7. VARIATION OF NORMAL FORCE COEFFICIENT SLOPE

WITH MACH NUMBER FOR VARIOUS CONES (d/D = 0)

5.2 ............... [ ..: ....

(:3 SYMBOL - -

_: INTERPOLATED VALUES

E " t--_ 2.8 ..... =_9> HYPERSONIC NEWTONIAN THEORY , i <_ -D

"" OPEN 14x 14 IN, NASA MSFC TUNNEL _-.__._ __FLAGGED 7x7 IN, NASA MSFC TUNNEL t lrZ FILLED REF. 2

02.4 ..........

u;

2.0 ...................... " l .

E 1.6 .... __._ _ - -

0 - 13.30_ f I _"

,,, 1.2 15°"-// / /

o / . ;> l ..... _.,,. 20° _ EE}E_- .......o -_ -_ _ / " _,, 25o_.._._.__. / / c]- 0.8 _oo_ ......... /

o _%z 5oo_- _ _.___-,--J_'-0.4 "- "-.(t._.. _. _.;, Z_ _-YTF- ,,

kJ

00 04 0.8 1.2 1.6 2.0 24 2.8 5.2 5;6 4.0 4.4 4,8

MACH NUMBER

o_ FIGURE 8. VARIATION OF NORMAL FORCE COEFFICIENT SLOPE

WITH MACH NUMBER FOR VARIOUS CONES(d/D = 0.4)

3.2

_. SYM BOL _l Ti--_. -!<_ INTERPOLATED VALUES

--_ 2.8 .... =_> HYPERSONIC NEWTONIAN THEORY ; ___<_OPEN ,4xl4,1¢ NASA MSFC TUNNEL _"E/._ |

; FLAGGED 7 x7 IN, NASA MSFC TUNNELZ

(.)

2.4B

E 1.6 e:lo °-- /

," 20°--/! c;_'7"_" "\ _ "_-" _ '==--

J 0.8 _ 30o....... _.,' _ ......

n_ -_ 40°-- .... _ "/ _'_

0Z 50 °-.- -'(_ "_" ----" /-._...___.__ _.___.._ ..____]__.____-_-__ --_t-<- _]0.4 ---4_..__Tcw_.-...'_--CL_

'" ]

o i0 Q4 0.8 1.2 1.6 2.0 2.4 28 5.2 3.6 .4.0 4.4 4,8

MACH NUMBER

FIGURE 9. VARIATION OF NORMAL FORCE COEFFICIENT SLOPEWITH MACH NUMBER FOR VARIOUS CONES (d/D = 0.8)

I.o ] ' I I : T [I I ! _ i

i

I

o 0.8 ......,- , - ....._ '" -

0.6 t"-'" _ --- ,

] ' __-_ ........ ,____

_ 0.4 I ' I _ -_i I I ,

• i i

0.2u i

i "J _i'.W I \ _

a. -o.z --7.......... _, ----0 , SYMBOL ,

....... HIGHESTFREE STREAM MACH !IX:' I NUMBERFORLOCALTRANSONICFLOWI.zJ

zl'- -0.4 ..... [.... - ...... ENVELOPEOF MINIMUMMACH " i' L .......-'_ _ !

'!iW NUMBERSFORATTACHEDSHOCK j

0 : REF.INTERPOLATED3 VALUES --ii -<___-_i-0_6 _ _- i.... I _ HYPERSONICNEWTONIANTHEORY

{ I OPEN 14x14 IN. NASAMSFCTUNNEL iFLAGGED 7x7 IN. NASAMSFC TUNNEL i

........ ; " i :

0 04 0.8 1.2 1_6 2.0 24 2_ 3,2 3.6 4.0 4.4 4.8

MACH NUM_

"_ FIGURE i0. VARIATION OF CENTER OF PRESSURE WITH MACH

NUMBER FOR VARIOUS CONES (d/D = 0)

,.o j [,

__;,0oI_--_ _,_._,_..._,.._ --_ -a 0.8 .,

U

uJ 0.6 f15o ---..... L._.__

i i,

0.4 ' -- .

____20 ° _.

u 0.2 - 25°_ D"3CZD-" "-'- _""---t_---__

:3 0 .........(_. _or) .._.__Id ..i_ _ ....

D. .40 ° / "_ /'

,, -0.2 ......

o _._1_ -__ _------ G-- ----___

I- -0.4 /z / I I ............,. "- L----"I ---(O __, / •SYMBOL

-0.6 _- 5o0-- _ HYPERSONIC NEWTONIAN THEORY I _ _ D

j oP. ,..,.,__._._s.c_°.._ s -1 _ FLAGGED 7 x7 IN. NASA MSFC TUNNEL _ I

FILLED REF. 2i I ! 1 l,. 1 I I I ]

0 Q4 0.8 1.2 1.6 2.0 24 2B 32 3.6 40 4.4 4.8

MACH NUMBER

FIGURE ii. VARIATION OF CENTER OF PRESSURE WITH MACH

NUMBER FOR VARIOUS CONES (d/D = 0.4)

1.0 _

O. 0.8 ,__SYMBOL iNTERPOLATED VALUES ......... • ..._.__._ - 7

:_> HYPERSONIC NEWTONIAN THEORY'- : -_-_,(i.._/._._

• OPEN 14x 14 IN,NASA MSFC TUNNEL __ __

_"___I"_ 0"6 '0"4 -- ' " '0" {_"--- {_"'_ _! ! FLAGGiD 7 , 7 IN, _ASA MSFC TUNNFL' • i ' r' ' ' ......... : ' '_ i _.__,_____,,,__

_______.%_____(___ _ i_ _ o---_ _-_ --_,_ 0.2 _._! ____:_----_---_---_ _ _'

15° J

Oj -'--_ IW _" _ .........¢ 0 __ zo- I. ..... _____ ]0

v} __ 2so I _---C°---U_:J'<3-- -u'_ --{3-_7----- [] -Id ....-- _ --'_ "-=C>I_' 30°

' I ..... ' --o i .._.-- !no 40 ° ,

zw_ -o.4 _I ..................,I j._._-_ _----F-....__c__-_:__-F=-..........:_............................; c_ _>

-o.6 --_ .... ,"I .......

II50 _ I U t .......... l

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 32 3.6 .4.0 4,4 .4.8

MACH NUMBER

i,-w.

FIGURE 12. VARIATION OF CENTER OF PRESSURE WITH MACH


1.8 I I I I ] i I I I I I. i I I. I['_ SYM BOL

ENVELOPE OF MINIMUM MACH _ --_

NUMBERS FOR ATTACHED SHOCK _ ; __-_f]. }!

1.6 -- _.__. INTERPOLATED VALUES ____.-- qL;, cl" LD

=C> HYPERSONIC NEWTONIAN THEORY

1.4 -- OPEN t4x 14 IN. NASA MSFC TUNNEL ___ II "

i : I ' L _ '' 1 i' I ' °i ' P _ ! (_,

_ 0.6 . _i [] []

40 ° -L....

_- 30o - _25

15 o _\_ _F Is.s°_..-TZ. o () o

00 0.4 0.8 . 1.2 1.6 2.0 2.4 2.8 3.2 5.6 4D 4.4 4.8

MACH NUMBER

FIGURE i3. VARIATION OF FOREDRAG COEFFICIENT WITH MACH

NUMBER FOR VARIOUS CONES (d/D = 0)

I.,8 t I II I ' t r r t--I

1.6 , © E N THEORY ...... -<-I. / , L/ , "

,,.,..1.4 I

o (}t@ - ---(_.._ __

1.2 ..4Y"

- / [U.I,Ll .,...-.._ _-----.-L_

o,o jr_ --I--C_ .._/n.. 0.8 .....

° )"' .._ _,.....,,9

/ _-_0.6 "El-.... '"-" "_ "_ _ .............

I , "_"" _Q_ .-..._..

¢=50 `',_" _ ------- _ ---[3-- _-t ? [] .... IE0.4 - 40° ..... _ _ .......................

20°\_[_../' _ '_ --- _:_------Zx- -- ,& z_, _ ----'\ _ ____-------c,, c C o (; C _'0.2 - r_.3"-_,.LJ-

00 04 0.8 1.2 1.6 20 2.4 2.8 3.2 3.6 ,4.0 4.4 "4,O

MACH NUMBER

FIGURE 14. VARIATION OF FOREDRAG COEFFICIENT WITH MACH


lo8 l _ I I I I 1

-'-L----_

INTERPOLATED VALUES "_..

1.6 _ HYPERSONIC NEWTONIANTHEORY_ -- _--_-.-_--_.--ol ..... D

SY vIBOL 14x14IN.NASA MSFC TUNNEL "'_'_"'_--_ I"

_-1.4 1¢_ s1

I-" 1.2 / G_ <_ _ ---_ ]-----_ _'-'-"z ._u

- dw.

o _, -- _

0 /"

0.8 f -;_ l / _ _ _ _ "_

_o , jl L_--0.6 .E"--_ J ------ _ _ -

6=50 "" / //50° --_ _]//

0.2 -- 25"-. _j

20: _ / _/o\/-"15 ...

- ,oo.t_'3°"_.cfJ0 i (31 ,

0 Q4 0.8 1.2 1.6 2.0 2.4 2.8 32 3.6 4.0 4.4 4, 8

MACH NUMBER

FIGURE 15. VARIATION OF FOREDRAG COEFFICIENT WITH MACH

NUMBER FOR VARIOUS CONES (d/D -- 0.8)

- r [ F--- ..... -T ! I, i i i i I

_---|--4 i _ i i

f ! I '' ! i

2.8 t ! .........-,-........_-

i___ SYMBOLi 0 O "I

" O.q, 'Q ' ,_ / A MID I

,_ 2.4 I -3L--- ' l- O 0.6 tO_ I--_ ' i _ 0.8

I- ! __.q__ OFEN PRESENT DATA15' i i FLAGGED REF. 3 I

• I

=" I '2.0 _ , -- r........._-........... 1I

G I iIii

1.6 F---,-- I -Z

'" 1.2 I IO , _. i

o ! I i

i

_j _ EI

JI1

0 I ,

O IO 20 .30 40 50

CONE HALF ANGLE, £ (DEG.)

FIGURE 16. VARIATION OF NORMAL FORCE COEFFICIENT SLOPE WITH

CONE HALF ANGLE AND BLUNTNESS AT M = 0.68

23

i

2.8 iSYMBOL

© 0 "(> 02.

a A 0.4 > d/D,,_ 2.4n. o 0.6--_ U 0.8.--- 1OPEN PRESENT DATA --

FLAGGED REF. 3Z ! ! _ i

(..) 2.0 \ ! ,

u2 _ , ,a. z, t t9 1 !_ 1.6 I

z _ !

_w I

u. I,,, 1.2 _\\ I I

w0 .

o !u. 0.8 I

o >z 0.4

J

ii I

0 F0 I0 20 30 40 50

CONE HALF ANGLE, E (DEG.)

FIGURE i7. VARIATION OF NORMAL FORCE COEFFICIENT SLOPE WITHCONE HALF ANGLE AND BLUNTNESS AT M = 0.94

24

FIGURE i8. VARIATION OF NORMAL FORCE COEFFICIENT SLOPE WITHCONE HALF ANGLE AND BLUNTNESS AT M = i. 05

25

I II!!

2.8 _ !i ' I

SYMBOL

0 0 "1

t-_ I A 0.4 k2. d/D,,¢e:: 2.4 0 0.6 - "

U 0,8.[

t OFEN PRESENT DATA _ IFLAGGED REF. 3

U _2.0

I \'b.\_ i ir,D I _\\ \ , ! ! !

,., ,.. . ! , ,4- i !o_ -------_ ...... ---+- ....-rII.

0 1.2 ...........

Uri-Ou.

J 0.8 _

Z

0.4 i_ 7

O f_ .

0 I0 20 30 40 50


FIGURE i9. VARIATION OF NORMAL FORCE COEFFICIENT SLOPE WITH

CONE HALF ANGLE AND BLUNTNESS AT M -- i. 30

26

l I i t I I i _

' tt ! ,

• li I_ il 1! i2.8 _ i ' ' /

i SYMBOL L

L- O O " --

ta ,,x 0.4 > d/D _,0 0.6

e::: 2.4 -- --_ _ 0,8.

_'_ L I OPEN PRESENT DATA

I 1 FLAGGED REF. 3 -- -$

Z I FILLED REF. 2 IU }

2.o ' !

__ _ _I1.Wo 1.2 l lU

ldU

0

" t0.8 '.J

0Z

0.4 _7

00 I0 20 30 4,0 50



CONE HALF ANGLE AND BLUNTNESS AT M = 1.50

27

L

SYMBOL

p.. O 0 "• >a A 0,4 d/D

2.4 O 0.6£7 0.8.

OPEN PRESENT DATA

FLAGGED REF, 3Zu I

2.0 t

I-- 1.6 _ Iz

,=- \M.

o 1.2 '

U

o 0.8

Z

0.4?

0

0 I0 20 :50 4.0 50

CONE HALF ANGLE, £ (DEG.)


CONE HALF ANGLE AND BLUNTNESS AT M = i. 97

28

r ij i t

i I iI i ,2.8 1 1 I ! I ,

' SYMBOL ;

I L ',.-,i ...... I ' - o o h - -

__e::'_ .,' A¢, o.4i d/D0.6 t2.4 I .... -V

! _ o,8

Z .... . -- FLAGGED REF. :3 -(..)

I 1 FILLED REF, 2

o _

o ----.._,.......

,,_ 0.8

0 L_ .,

z

0.4 .....

.....

0 I0 20 30 4'0 50

CONE HALF AMGLE, E (DEG.)


coNE HALF ANGLE AND BLUNTNESS AT M = 2.75

29

l t ISYMBOL IF

o o ; ---_2.8A 0.4 > a/D I

O 0.6u o.8.,

OPEN PRESENT DATAa,,_ 2.4 FLAGGED REF. 3 --"

._ FILLED REF. 2

"" i s !

Zu 2.0

1.6 .......k-

w 1.2 _____;_____ ______.x

ot_ 0.8_i,,= !

Q: 70Z I

o.41 !

0

0 I0 20 30 40 50



CONE HALF ANGLES AND BLUNTNESS AT M = 4.00

30

FIGURE 24. VARIATION OF CENTER OF PRESSURE WITH CONE HALF

ANGLE AND BLUNTNESS AT M = 0.68

31

i.o ................ I

i SYMBOL Ii 1 , i

o.e o o ! Ia _\\ I i ¢_ o.2[ I

_¢ \\\ L , ...... A o.4 _ a/D _

\\_ o o._/\_,_l)' i _' o.sj i

\ I\ \ '\ ', t i i

g \ I\\\ '\1 \'\\\ I !,': t I l ,

o._ _,\\i ,ILl

° ["' ,\ "4"-,%,,," o . "_. _\_',_,\_\'._'-. t

i- -0.2z1,1,11to

\

_O_ 4

! ,

/

-0.6

I

0 I0 20 30 40 50




32

I.O

SYMBOL

0.8 o o " -t

(> 0.2 I

/_ 0.4 > d/D -- ,

_' O.6 OPEN PRESENT DATA -- '

FLAGGED REF. 3

,

' t

D,. t _"

0

_" -0.2hiI"ZU_I

-0.4

>

-0.6

7

0 I0 20 30 4.0 50



ANGLE AND BLUNTNESS AT M = i. 05

33

1.0 I '

SYMBOL0.8

° °i,., (:> 02

A cl/D0,'_

O, 0.60.8

_.o._ __ o_._ _s_,,o_,_I i\ \\_ FLAGG__DRFF.3 i

°_ _\\\_ I---I , '0.4 ' ! ''I \_ \\\ I ,

,."I,

-o.2 t

-0.4

-0.60 I0 20 30 4,0 50


FIGURE 27. VARIATION OF CENTER OF PRESSURE WITH CONE HALFANGLE AND BLUNTNESS AT M = i. 50

34

1.0 ' I ' !

, I

() I ! I

(_ SYMBOL

0.8 z o o ....C> o.2

0 _ 0.4 > d/D _

<> 06

u _ o.8., tFLAGGED REF.:3 I

g ! '!

ILl 0 ""

ZI--ILI-0.2 \. x-X ....

laJ0 >

-0.47

I

iL

-0.60 I0 20 30 40 50


FIGURE 28. VARIATION OF CENTER OF PRESSURE WITH CONE HALFANGLE AND BLUNTNESS AT M = i. 97

35

FIGURE 29. VARIATION OF CENTER OF PRESSURE WITH CONE HALFANGLE AND BLUNTNESS AT M = 2.75

36

o i iSYMBOL

°_ \ ° _>. -- , ,

\ o o;_ __ _ O.1• oPe. P,ESE,TOATA,

0.6 , F_GGEDREF.3 !FILLED REF. 2

! l

k

_ 0.2W

= \

,,. ,..._. 0

o <'-.., <

t- -0.2 >Z "_

0 x

-0.4 _7

-0.6 j....

0 I0 20 30 40 50



ANGLE AND BLUNTNESS AT M -- 4.00

37

-- t I- I ! ,: t J I

SYMBOL , I

1.4 I-------I ------F'_I"..... z_0 0.40 _ d/D I_i_" J'i'i U 0.8 iOPEN PRESENT DATA .t I

,.2 ! - FLAGGED REF-3 j

1 ' I! i I t ,

I--" 1.0 I i _ !t ' r'g ,

E i' I

_ j r F

e o._ 1 i , i

r

/ ,I ,

i0.2 _ I

......

0 I 0 20 50 40 50


FIGURE 31. VARIATION OF FOREDRAG COEFFICIENT WITH CONE HALF


38

.... 1 i16 l ! ' I

1.4 :

o OA _ diD ......0 o.6_7 o.8, I

1.2 OPEN PRESENT DATA ._FLAGGED REF. 3

t ' i ....." t )

1.0 _. !

_-,.., ..... ! _. ..

E 0.8_ ' t _o !

i

•"- 0.6 ......

0.4 . !

f ' i[g '

0.2 / t!

• ] ,o I I

0 I0 20 30 40 50


FIGURE 32. VARIATION OF FOREDRAG COEFFICIENT WITH CONE HALFANGLE AND BLUNTNESS AT M = 0.94

39

I • , i i i

i { ' i J! _ { { ]

i i _ { !, SYMBOL Z

1.4 ! o o"l [0,4Z_ diD

--- _ .... 0 o.6 ' i{ _ o. ! { {

1.2 OPEN PRESENT DATA { I i{ FLAGGED REF. 3 {' ; i '

l,.o {

5 t

o'_ 0.8 {I i _

_ 0.6 i

A0.4 '_ /// .......

(

S y

0

0 I0 20 ,50 40 50

CONE HALF .ANGLE, E (DEG.)

FIGURE 33. VARIATION OF FOREDRAG COEFFICIENT WITH CONE HALFANGLE AND BLUNTNESS AT M = i. 05

4O

i

1.6 I i

i i !II

1.4 .0 --f--

i

O_ i£_ -'1.2 OPEN DATA -4

FLAGGED

E 1 /_ i_ 0.8 i

° / _Y

." 0.6 / ',,o

0.2 /_ _/

I

0 I0 20 30 4.0 50


FIGURE 34. VARIATION OF FOREDRAG COEFFICIENT WITH CONE HALFANGLE AND BLUNTNESS AT M = i. 50

41

1.6 I It

, 1

SYMBOL t=

"4 I o o1 -_--Z_ oA ). d/D _[

0 0.6 t! _ _ o.8., ! I

1.2 I OPEN PRESENT DATA _ I_

FLAGGED REF. 3 i S P

J|

,=.,

_. 0.8

e_/

0.2 I : 1z_

0

0 I0 20 ,:30 40 50


FIGURE 35. VARIATION OF FOREDRAG COEFFICIENT WITH CONE HALFANGLE AND BLUNTNESS AT M =- i. 97

42

1.6 I I '

k ]

1.4 SYMBOL _

o o_Z_ O/t d/D

o o.;] __z o.1.2 OPEN PRESENT DATA /p"

FLAGGED REF. 3

_ i.o_ ,/

g,

_ 0.8

_o.6 - ,/,9 _" /

0.4 ////_

/0

0 I0 20 30 4.0 50


FIGURE 36. VARIATION OF FOREDRAG COEFFICIENT WITH CONE HALFANGLE AND BLUNTNESS AT M = 2.75

]

43

1.6 - " i _ r, i i j

I SYMBOL i1.4 --t--_

0 0

0 0.6

i _ O.B1.2 OPEN PRESENT DATA

! FLAGGED REF. :50= t ;

I I I /,_'

_,-5

0.6 s:_o

//./

I

0.2 i_. ,,

m

0 I0 20 3,0 40 50

CONE HALFANGLE, E (DEG.)

FIGURE 37. VARIATION OF FOREDRAG COEFFICIENT WITH CONE HALF


44

APPENDIX

To allow quick estimation of the basic aerodynamic characteristics by the

Hypersonic Newtonian Impact Theory (Ref. 6) , the necessary relationships of cone

geometry parameters are presented in Figures A-i and A-2. The charts of theoretical

normal force coefficient slopes, center of pressure locations, and wave drag coefficients

are given in Figures A-3 through A-5

45

FIGURE A-i. RELATIONSHIP BETWEEN GEOMETRIC PARAMETERSOF SPHERICALLY BLUNTED CONES

46

2O

I0

00.2 0.4 0.6 0.8 1.0

BLUNTNESS RATIO, d/D

FIGURE A-2. RELATIONSHIP BETWEEN GEOMETRIC PARAMETERSOF SPHERICALLY BLUNTED CONES

47

FIGURE A-3. HYPERSONIC NEWTONIAN NORMAL FORCE COEFFICIENT

SLOPE FOR CONES OF VARYING GEOMETRY

48

• I

0 0.2 0.4 0.6 0.8 1.0

BLUNTNESS RATIO, d/D

FIGURE A-4. HYPERSONIC NEWTONIAN CENTERS OF PRESSURE FOR

CONES OF VARYING GEOMETRY

49

i

,.6 i '/1.4 /f /

E (DEG.) ./7-"/ / ',

/50o I /

_, IA!o,.o _/ 1 /

" 0.8'. /

,, _///_: 0.6Q /30 ° , J/'J/

o. __.s /" " I h--20° _-'--L _.-_t

_ 5o "_..

0.2 /-// "// -""'_,oo. _>__-ti-_,......io -"I ' ,.L-_E;r

__-_ I ;I

0 0.2 0.4 0.6 0.8 1.0

BLUNTNESS RATIO, cl/D

FIGURE A-5. HYPERSONIC NEWTONIAN WAVE DRAG FOR CONES OFVARYING GEOME TRY

5O

REFERENCES

i. Felix, A. Richard; ABMA Report DA-TN-42-58; "Force Tests of a Related

Family of Twenty Nose Cones for a Range of Mach Numbers from 0.47 t o

i. 93." July, 1958.

2. Marks, Alma S. ; US AOMC-ARGMA-OML Report 648F; "Normal Force,

Pitching Moment, and Center of Pressure of Several Spherically BluntedCones at MachNumbers of 1.50, 2.18, and 4.04." April, 1958.

3. Gendtner, W. J. ; Convair Report ZA-7-017; "Sharp and Blunted ConeForce Coefficients and Centers of Pressure from Wind Tunnel Tests at

Mach Numbers from 0.50 to 4. 06." June, 1955.

4. NACA Report i135; "Equations, Tables and Charts for Compressible Flow."

Ames Research Staff; 1953.

5. Liepmann, H. W. and Roshko, A. ; Elements of Gas Dynamics, John Wiley

and Sons, Inc. , New York; November, 1956.

6. Grimminger, G. , Williams, E. P., and Young, G. W. ; Journal of the Aero-

nautical Sciences, November, i950; "Lift on Inclined Bodies of Revolution

in Hypersonic Flow."

NASA-Langley, 1965 M125 51

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