The calculation of the wave drag of a family of low-drag axi-symmetric nose shapes of fineness ration 4.5 at zero incidence at supersonic speeds

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE CALCULATION OF THE WAVE DRAG OF

A FAMILY OF LOW-DRAG AXI-SYMMETRIC

NOSE SHAPES OF FINENESS RATIO 4.5 AT

ZERO INCIDENCE AT SUPERSONIC SPEEDS

by

MDTE N0.^'4^»a]sü-QQt 10 - DELFT

MAI, 1954.

- - " - - ^ 1 ^ FEB. 1955

T H E C O L L E G E O F :. E R O W A U T I C S

C R A I-I F I E L D

The calculation of the -v^ave drag of a family of low-drag axi-symmetric nose shapes of fineness ratio 4.5 a-t zero

incidence at supersonic speeds,

-by-G.B, liarson, B.Sc.

S ÏÏ Hid A R Y

The pressure drag coefficients of a p£?xticular family of convex logcjrithmic xorojectile nose shapes in v/hich the nose angle is aji important parameter have been calculated over a range of supersonic Ilach numbers using a rapid approximate method due

5 to ZierikieT/icz,

The optimum nose angle for ininiiiium \/avc drag of these profiles for each Ilach number has been obtained. It is shoim that above ii = 1 . 5 ^ approximately, the optimum shape is similar to the hypersonic optimum profile pjid has the seme or less \7ave drag than this profile, Ho^v'ever for values of ï.i/P, \7here F is the fineness ratio, belo\7 0,5» both the hypersonic and the logarithmic optimum profiles have a higher drag thrai the so-called cubic profile (,Ref. 9 ) ,

-2-Gontents TECHNISCHE H O G E S C H O O L VLlEGTüïGBOoWKüNDE Kaaaalstraat 10 - DELfT Page Summary 1, Contents 2 2, List of Symbols 3 3, Introduction 5 4, Brief discussion on the derivation of the

family of loi7 drag head shapes 4 5, Discussion of v/ind ttmnel tests on the

logarithmic head shapes 6 6, The calculation of the drag of the

logaritlEiic profiles by the use of

the modified ogive of curvature method 6 7, Discussion of the results and comparison

v/ith the dxe-g of other nose shapes 8

8, Conclusions 10 9, Aclcnoi/ledgsments 10 10, References, 11

Appendix

Brief description of certain T/ell kno".7n

axi-symmetric head shapes, 12 Tobies

I , Equations t o the logarithmic p r o f i l e s I5 I I , Comparison of C-p^ values obtained using

the modified ogive of cuxva.ture method . v/ith those given by the method of

c h a r a c t e r i s t i c s and Van Dyke's second

order t h e o r y . 1_5

FicTures

1, Drag c o e f f i c i e n t s f o r lograritlxnic p r o f i l e s of 12° nose semi-rjigle at 11=2,0, v/ith v a r i a b l e end slope

2, Vcxiation of C-^ Tri.th nose semi-fJigle for

logarithmic profiles, " y. j = 0,077 'U ₁

3, Variation of C^ v/ith liach number for minimum drag logarithmic profile and inscribed cone

4, Comparison of logcjrithaic and other profiles 5, Drag of optim-uiu logexithnic and other profiles,

§2, List of Symbols C^, drag coefficient = -^^^^^f— = i C d/J' y' >-p U 7tR -^o o •„• o C pressure coefficient = p •^

P -

T-L length of head

li free stream Liach number p static pressure

p free stream static uressure R radius of head at shoulder

S(x) cross sectional area at distance x from nose

•p

t thiclcness ratio = ^

u free stream velocity X distance from nose

y radius of body at distance x f ran nose

dx

6 nose semi-angle,

S3» Introduction

The rapid approximate methods described in Ref, 1 for calculating the pressure distribution on convex a>:i-symmetric projectile head shapes at supersonic speeds have been used by

2

Bolton-Shaw to determine lo\7 drag head shapes. He applied the derivative fonjula and the log p-ö methods to determine the type of slope distribution required for minii-num drag

profiles. It vas noted from the pressiure distributions along these optimum profiles, given by the ogive of curva^ture method, that the wave drag decreased. -vTith increase in nose angle. It

-4-having a nose semi-angle greater than 24 • Since the ogive of curvature method v/as kno\7n to become less accurate as the nose angle was increased this result v/as suspect. A series of v/ind

3.4

tunnel tests ' was therefore performed in order to check the range of applicability of this method, and to find independently the nose angle for minimum drag over a range of Mach numbers, The results of these tests demonstrated in fact that the original ogive of curvature method underestimated the drag coefficient for profiles having large nose angles and that the nose angle for

minimum drag was considerably less than 24 • These investigations led hov/ever to the development of a modified form of the ogive of

5

c\irvature method which predicts \7ith acceptable accuracy the pressure distribution and the drag of head shapes having nose

semi-angles up to 30 •

This report describes the results of the application of this modified ogive of curvature method to calculate the drag coefficients of the series of head shapes developed by Bolton Shaw, The results are compared v/ith those for other well known minimum drag shapes, and the optimum shapes for different ïiach numbers are deduced,

§4, Brief discussion on the derivation of the family of low-drag head shapes

By the use of the derivative formula and the log p-O 1* 2

method , Bolton Shav/ derived integral expressions for the wave drag of a projectile of arbitrary shape. Application of Euler's condition for a minimum value of the drag integral in each case gave a differential equation from which minimum drag profiles could be obtained,

Because of the considerable difficulty in solving the differential equation in the case of the derivative formula, only one suitable profile v/as obtained. The log p-0 lav/ gave a

s In Ref, 3, these methods have been revised v/here necessary, and their range of applicability discussed,

profile v/hich represented minimum drag shapes for all fineness ratios collapsed into one curve by suitably scaling the axial and lateral ordinates. As this profile had infinite slope at the nose, it v/as impossible to apply the methods of Ref, 1 to find the pressvire distribution on this profile. However, except at the nose the type of slope distribution required for an optimuj'a shape projectile was indicated. It was found that this was very similar to that given by

1^ = a+

- ^ -

.

(1)

L ^ °

Equation (2) v/as used to describe the shape of the family of low drag profiles, referred to in this report as logarithmic profiles,

At the base of the profile, x = L,

j ^ = a + b log^ (j + 1 )

where P is the fineness ratio L / D , and D is the body dia-meter at X = L, a-, "b, and c may then be determined from the boundary conditions

^^ V JO dx at X = 0

(ii), jy'J^ = g at X = L.

At a fineness ratio of 4«5 ^^d. a constojit liach number of 2,0, Bolton Shaw determined by means of the original ogive of curvature method the drag of head shapes v/ith nose angles of 16 , 18,4 £>i^d 21 , for various values of the slope at the base, iy':.» For each nose angle, a minimum drag was found for y'L equal to 0,077 approximately. The drag decreased \a.th increase in the nose semi-angle 6 , and it ajppeared that a minimum drag, if present,

s

would occur at a value of 6 greater than 24 , s

- 6 - TECHNISCHE HOGESCHOOL

VLiiCI UiGBOU W K U N D E Kanaalstraat 10 - DELFT ^5 • Discussion of v/ind txinnel tests on the logarithmic head shapes

Since the ogive of curvature method v/as knovm to beccsiie less accurate v/ith increase in nose angle, wind tunnel tests v/ere performed ivith the logarithmic head shapes in order to assess its applicability and to detemnine the nose ejigle for optimum pi-ofiles over a range of Liach numbers,

The results for a profile v/ith a nose angle of 21 at M = 1,8 are described in Ref, 4> aiid those for 21 , 24 , and 30 profiles at M = 2.45 a^d M = 3»''9 in Ref, 3» These results showed that the original ogive of cinvaturu method considerably under-estimated the pressures over the rear portion of all these bodies, and that the discrepancy increased v/ith nose angles. In addition, it Was found that the nose semi-angle for minimum drag VTas about 15 over the range of lïach numbers tested,

It follov/s that the conclusion of Ref, 2^ namely, that the drag of a head shape decreases as the nose angle is increased, giving a minimum drag at a nose semi-angle greater than 24 , is not correct. The error of course derives from the use of the original ogive of curvature method outside its range of applicability,

Hov/ever, since the modified ogive of curvature has been shov/n to be sufficiently accurate for engineering purposes over a v/ide range of Ivïach numbei-s and head shapes having nose semi-angles up to at least 30 (see Table II), it has been possible to tackle again the problem of Ref, 2 on a more satisfactory basis. The next section describes this investiga.tion and the results,

S6, The calculation of the drag of the logaritlrcic profiles by the use of the modified ogive of curvature i:iethod

The calculation of the pressure distribution and wave drag of arbitrary head sha.pes at supersonic speeds by the use of the modified ogive of curvatiore method is described in Refs, 3 and 5» Ttie method depends on the fact tliat the ratio of the static to the stagnation pressure at a point P on the surface of an arbitrary head shape at a free stream Ilach nuiiiber II , is practically the saine as at P on the equivalent ogive of

curvatiire at P, for the saine value of 11 , The static pressure on the equivalent ogive of curvat\ire can be calc\ilated from an equation giving the decrease in pressure from the nose to the point P on the ogive surface as a function of the decrease in pressure from the leading edge of a tv/o dimensional aerofoil having the same profile as the ogive to a corresponding point P on the aerofoil| provided that the J/Iach number and pressure just

12

downstream of the nose and leading edge are the same. This equation may be va-itten

Vviiere \ depends only on M and the nose angle X- of the equivalent ogive of curvature, ^Values of X are given by Zienkiewics in Refs, 1 and 5)» The modified ogive of curvature method differs from the original method in that the stagnation pressure used in calculating the pressure on the local ogive of curvature at each station is taken as that behind the bow shock wave at the nose of the equivalent ogive of curvature at that

station. In the original method the stagnation pressure behind the bow shock wave at the nose of the actual body was used through-out,

This modified method gives results which agree well vyith the experimental pressvire distributions described in Refs. 3 and 4, and also with calculations made for a few particular ca^es by the Method of Characteristics and by Van Dyke's second order theory

(Table II), The values of the pressure coefficients G , obtained from ejcperiment, could be predicted generally v/ith a possible error of at most + 6 per cent, and the calculated drag coefficients, C^, within +_ 3 per cent. Since the results below have been calculated by the modified ogive of curva.ture method v/ith a possible error of at most + 2-g- per cent, comparisons betv/een values of 0^. for different profiles of the family should be possible to this order of accuracy,

In order to check the value of the end slope, ' y' ;'., as determined in Ref, 2 for the optimum profile, the v/ave drag v/as computed by means of the modified ogive of curvature method

_8_ TICHNISCHE HOGESCHOOL

VLIEGTÜÏGBOUWKUNDE Kanaalstraat 10 - DELFT

for a number of profiles designed for different values of | y' L » but each having a nose seni-angle of 12 , fineness ratio of 4,5^ and at a Mach number of 2,0, The results are shov/n in Fig, 1 , It can be seen that the drag coefficient varies very slov/ly over this range of y''.. It appears reasonable to assume, therefore, that the value of ;y'i^ = 0«077 used in Ref, 2, may be retained as an optimum value without serious error. With this value for the end slope, a series of logarithmic profiles of fineness ratio 4,5 were derived having nose semi-angles of 9«75 > 'IS , 15 j 21 24 and 30 • Their equations are given in Table I,

For each of these profiles the drag coefficient v/as calculated, using the modified ogive of curvature method, at Ilach numbers of 1,5, 1.8, 2.0, 2,45, 3.19 and 4,0. The results are plotted in Fig, 2, It can be seen that the minimum drag head

shape has a nose semi-angle of about 15 at II = 3> falling to a value of 12 approximately s-t II = 1 »5»

In Fig, 3 the minimum values of C_ at each Mach number are plotted against Ilach number. These results indicate that the minimum value of C^^ decreases v/ith increase in liach number. The

scatter of the ccmputed values about a smooth curve in Fig, 3 does not exceed +_ 2-g- per cent v/hich is v/ithin the estimated order of possible error of the calculations. The drag coefficients of the inscribed cone (nose semi-angle é 21 ') over the same range of liach numbers are also given in Fig, 3» It is found that the drag coefficient of the optimum logarithmic profile is 7^ per cent of that of the inscribed cone at II = 1,5, and 82 per cent at

M = 3 . 5 .

§7, Discussion of the results and comparison with the drag of other nose shapes

The results of these calculations show that the minimum drag logarithmic profile has an optimum nose seni-angle v/hich varies from 12 at H = 1,5 to about 15 at M = 4, Betv/een M = 2,0 and M = 4,0, the minimum is not sharply defined and changes in nose angle of + 2 do not affect the drag by more than +_ 2 per cent. It will be noted from Pig, 1 that the drag

at least at a Mach number of 2.0, For the purpose of coniparison with other nose shapes the logarithmic profile having a nose semi-angle of 13*5 and an end slope of 0,077 T/ill therefore be taken as a suitable standard in the Mach number range betv/een 2,0 and 4.0,

Although a number of so-called optimum head shapes have been derived, only the Von Karman, the lighthill, the hypersonic

optimum and the cubic profiles will be compared v/ith the logarithmic profile. The results quoted, hov/ever, can be regarded as

typical for all low drag head shapes. These four profiles are shov/n in Pig, 4, together v/ith the logarithmic profiles of 12 , 13*5 and 15 nose semi-angle. It will be seen that the Von Karman, Lighthill and cubic profiles, v/hich fair smoothly into a cylindrical body at x = L, lie close together^ but considerably above the profiles having finite slope at x = L, Near the nose (X/L C 0 , 2 ) , the hypersonic optimum curve is almost identical with the logarithmic profile of 13*5 nose semi-angle. For values of X / L > 0,2, the shape of the hypersonic optimum profile lies slightly

above the logarithmic profiles. In Table II, values of C-. calculated by Zienkievïicz using the modified ogive of curvature method are compared v/ith the values given in references 7 and 9 for the Vor. Karman, linear, and cubic profiles. It vd.ll be seen that the errors due to the approximate method do not exceed about 2 per cent, so that differences in C^, greater than this may be regarded as significant,

In order to ca^pare directly, the drag results for nose shapes v/ith different fineness ratios, the hypersonic similarity law has been used. This states that the ratio of local static to free stream static pressure at corresponding points on geo-metrically similar ogival heads is a function only of the

hypersonic similarity parameter ï/j/P, where P is the fineness ratio. This lav/ has been fotmd to hold for all the shapes considered in this report at Mach numbers above about 1,8, It

se A brief discussion of each of these profiles is given in the appendix,

-10-enables us to represent al]. the available drag data on a single diagram of M G^^ plotted against Il/P, The results are shovm in figure 5» It v/ill be seen that within the accuracy of the calculations, the logarithmic profile v/ith a nose semi-angle of 13,5 has the same drag as the hypersonic optimum shape for

IV'P> 0,5 (e,g, M > 2,2 for our fineness ratio of 4,5). This is not surprising, since as noted above, these two profiles are very similar near the nose, and a closer resemblance could be obtained by use of a slightly smaller value of the end slope,

i y' j., for the logarithmic profile. It is knov/n (Fig, I) that this alteration in 'y'L v/ould hardly affect the di-ag,

Belov/ M = 2 a smaller nose angle rmist be used on the logarithmic profile in order to maintain its optimum shape, and in this region the logarithmic profile has less drag than the hypersonic optiiiumc For }Ii/¥ < 0,5, however, the cubic profile has slightly less drag than either the logarithmic or the hyper-sonic optimum shape,

§8, Conclusions

The calculations of the drag of logarithmic projectile nose shapes (ref, 2) have been revised and extended using the modified ogive of curvature method (refs, 3 and 5)» It is found

that the drag of these head shapes at Ilach numbers betv/een 1,5 and 4.0 is equal to or slightly lov/er than that of the hypersonic optimum profile. There is also little difference between the shapes of the tv/o profiles. The nose serai-angle for minimum drag varies from 12° at M = 1 , 5 to 15° at M = 4.0,

Although the drag of the logarithnic profile with a nose semi-angle of 12 is lower for M < 2,0 than that of the hyper-sonic optimum, it is slightly greater than that of the so-called cubic profile,

§9, Aclcnov/le dgement s

The author wishes to acknowledge the supervision and

Author Title, etc, Bolton Shaw, B,T7,, and

Zienkiev/icz, H.K,

Bolton Shaw, B.Yf,

Mar son, G,B,, Keates, R,E,, and

Socha, ¥, Zienkievra.cz, HoK,, Chinneck, A,, Berry, C,J,, and Peggs, P,J, Zienkiewicz, H,K» Von Karman, T, Mettam, H,S,, and Ireland, B,

The rapid, accurate prediction of pressure on non-lifting ogival heads

at supersonic speeds,

English Electric Rep, No, L,A,t,054, June, 1952, To be published in the A,R,C, Current Papers series,

Theoretical investigation of minimum drag projectile shapes for supersonic speeds,

English Electric, Rep, No, L,A,t, 040, October, 1952.

An experimental investigation of the pressure distributions on five bodies of revolution at Mach nuiobers of 2,45 and 3.-19.

College of Aeronautics Rep, No, 79. Experiments at M = 1 ,8 on bodies of revolution having ogival heads.

A,R,G, 15,586j PJi, 1854, Jan. 1953) To be published in the Current Paper series,

Further developments of some approx-imate methods for predicting pressure distribution on non-lifting ogival heads at supersonic speeds,

To be published as an English Electric L,A,t, Report,

Resistance of slender bodies moving with supersonic velocities, v/ith

special reference to projectiles, Trans, A,S,M,E, Vol«54, No,23, December, 1932,

The Von Karman and other nose shapes, Pairey Aviation Co, Rep, No,15.

Perkins and Jorgensen,

Leslie, D«C,Ii,, and Perry, J,Do

Investigation of the drag of axially symmetric nose shapes of fineness ratio 3 for H = 1,24 to 3.67. N.A,C,A. Rep, RJI, A52H28,

The development of a nr^se contour

of minimum wave drag for Project 502,

-12-Noo Author Title, etc,

10, Lighthill, H,J,

11. Kopal, Z,

1 2 , Z i e n k i e v / i c z , H.K,

13. Young, A,D,

Supersonic flow past bodies of revolution,

A,R,C, R, and II. No. 2003.

Supersonic flov/ of air around cones, H,I,T, Tech, Rep, No, 1,

A method for calculating pressure distributions on circular arc ogives at zero incidence at supersonic

s p e e d s , u s i n g t h e Brandtl-Lfeyer flov/ r e l a t i o n s ,

A,R,G, C u r r e n t Paper No, 114,

The c a l c u l a t i o n of t h e p r o f i l e d r a g of a e r o f o i l s and b o d i e s of r e v o l u -t i o n ai-t s u p e r s o n i c s p e e d s ,

C o l l e g e of A e r o n a u t i c s Rep, No, 73»

APEEI'DIX

Brief description of certain v/ell Icnown low drag axi-symmetric head shapes

1, The Von Karman optimum nose shape

For low or moderate liach numbers, it can be shov/n {see Ref, 1 0 ) , that the wave drag of a slender body of revolution as the thickness tends to zero, is given by

Til C^l

1

where S(x) is the cross sectional area of the body a.t a distance X from the nose,

This expression v/as used by Von Karrnan to derive an optimum profile for a projectile nose of given fineness ratio v/hich fairs smoothly into a cylindrical afterbody, (This condition, that L^'J^ = 0, v/as decided by the fact that equation (l) is finite only if S'(x) is continuous,) The optimum profile, as derived by means of the calculus of

variations, is given by

and its slender body theory dra.g is

% = ^^'

where t is the thickness ratio, R / L ,

Unfortunately, this profile has a blunt nose, and does not satisfy the asstmiptions on v/Mch slender body theory is based, Its true drag cannot therefore be calculated by any theory at present available. The drags of slightly modified Von Karman bodies, v/ith either conical or cubic shapes near the vertex, have been ccmputed using Van Dyke's second order theory (Refs, 7 and 8 ) ,

These results are in reasonable agreement v/ith experimental measurements described in Ref, 8,

2, Lighthill's Linear Profile

For the reasons given above, Lighthill regarded the Von Karman profile as inadmissible as a projectile shape, and recommended the profile

y = t f J3-2I

sometimes IcnovvTi as the linear profile. This has a pointed nose, and fairs smoothly into a cylindrical afterbody. Its slender body theoiy drag is given by •

C^ =

lét^.

In Ref, 9» Leslie shov/s that Lighthill's profile is the first of a series of low-drag profiles, and suggests tv/o improved shapesj the so-called cubic

shapes,-y = tf (5.10f.10(ff-4(fj^J

1 2

v/hose slender body theory drag i s given by C.^^ = 4 T t , and

the quintic shape, vAich i s specified by

2 i • . 5 •.51

S " (x) = ^ ^ 13 (1-2

- 10 H-2 ^ 1 + 15 i'1-2 y] I

^ i \^ l i •' '. J j ' - •'-'' I

1 2 and has a slender body theory drag coefficient C^p, = 4 T ^ t .

1 4

-Using the method of c h a r a c t e r i s t i c s , Leslie and Perry have made

accurate c a l c u l a t i o n s of the drag of the cubic p r o f i l e at Ilach

numbers below 2 , 0 , and have shov/n t h a t t h i s p r o f i l e has l e s s drag

than any other known pointed body v/Mch f a i r s smoothly i n t o a

c y l i n d r i c a l afterbody. Accurate drag resiiLts have not been

published for the q u i n t i c p r o f i l e ,

The Hypersonic optirmm Nose Shape

This i s the optimiffii shape for a non-faired body of

given fineness r a t i o , according to F e n a r i , I t s derivation i s

ISaood on Nev/ton's impact theory, and the p r o f i l e i s closely

approximated by the 3/4 pov/er c u r v e , - ^ = -r^ ( f f •

The r e s u l t s of Ref, 8 shov/ t h a t t h i s p r o f i l e has l e s s

v/ave drag than any other knov/n pointed body at liach numbers

above 1.8,

Ti^BLE I

EQUATIOIIS TO THE LOGiJlITmilC ."ROFILES

The e q u a t i o n t o t h e f a m i l y i s

Z L

X -u T / A X / L \

^ ^ - b log^Q ( 1 + ^ y .

The values of the constants a,b, and c are given in the table below,

P r o f i l e 1 2 3 4 5 6 Nose Semi-angle 9 . 7 5 ° 12° 15° 21° 24° 30° a .0398 . 0 4 0 4 .0555 ,0^20 ,0636 .0655 b .1455 .1033 .0562 .0372 .0329 .0273 .479 ,260 .115 .050 .0374 .0236 Ti\BLE II

Comparison of C_ values obtained using the modified ogive of c\jrva.ture method, vith those given by the method of character-istics and Van Dyke's second order theory.

P r o f i l e Von Karman L i n e a r Cubic Method ( a n d Ref, No,) Van Eyke (Ref, 7) C h a r a c t e r -i s t -i c s (Ref, 9) 1 1 P 6 4 , 1 7 2 , 8 4 4.17 4 . 1 7 M 2 , 0 1.5 1.5 2 . 0 Given Method 0,0226 0.0511 0.1062 0,0494 0,0460 i ^D ^ Modified Ogive of C u r v a t u r e 0,0230 0.0500 0.105

I

COLLEGE OF AERONAUTICS. NOTE N o . l O . FIGS I.2.& 3

h-J

SUDPE OF PROFILE AT * = l-O [^J,]

DRAG COEFFICIENTS FOR LOGARITHMIC PROFILES

OF 12° NOSE SEMI-ANGLE AT M«-20 WITH VARIABLE END SLOPE

FIG. I. 1 ( ) \

k,

(REF. 11.) - ANCLE ONE . ^ ^ ' V . ) 3RAC 3FILE ^

NOSE SEMI-ANGLE &&

VARIATION OF Cp WITH NOSE SEMI-ANGLE FOR LOGARITHMIC PROFILES

[b,l,= O - 0 7 7

FIG. 2 .

MACH NUMBER M .

VARIATION OF C^ WITH MACH NUMBER FOR MINIMUM DRAG LOGARITHMIC PROFILE

AND INSCRIBED CONE. , ^ ^ FIG. 3

LOGARITHMIC ( s , • 13 5 ° ) LOGARITHMIC f o , - 12^ - • LOGARITHMIC f i , , - 15°) HYPERSONIC OPTIMUM

COMPARISON OF LOGARITHMIC AND OTHER PROFILES

) • A' " / / " • 1 - 8 < 1/ « 2 - 2 5 i f /

f/

HYPERSONIC SIMILARITY PARAMETER

DRAG OF OPTIMUM LOGARITHMIC AND OTHER PROFILES FIG. 5 .