Note on the results of some profile drag calculations for a particular body of revolution at supersonic speeds

TECHNISCHE HC " 'OOL \l.IF,GTUIGBOU • Rnru-ilitraal 10 - DEUT

Report No. 81

16 NOV. 195^

THE COLLEGE OF AERONAUTICS

CRANFIELD

NOTE ON THE RESULTS OF SOME PROFILE

DRAG CALCULATIONS FOR A PARTICULAR

BODY OF REVOLUTION AT SUPERSONIC

SPEEDS

by

J. R. WEDDERSPOON, B.Sc, D.C.Ae., and A. D. YOUNG, M.A., F.R.Ae.S.

JUIiï. 1954.

T H E C O L L E G E O F A E R O N A U T I C S O R A N F I E L D

Note on t h e R e s u l t s of Sone E r o f i l e Drag GcJ.culations f o r a Poar'ticuloj:- Body of R e v o l u t i o n a t

S u p e r s o n i c Speeds b y

-J,R, ¥edderspoon, B.Sc,, D.C.Ae.,

cjid

A,D, Young, n,A., r,R,Ae,S. of the Depcxtnent of Aerodynanics

S U LI M A R Y

Details additional to those discussed in College of Aeronautics Report No, 73 "'^e given of the method developed for the calculation of the profile drag of bodies of revolution at supersonic speeds and zero incidence. The method has been applied to a particulcj? body of fineness ratio 7*5 (see Fig, l) for ilach numbers ranging f rem 1,5 to 5»0, Re5Tiolds nunbers rcjiging fron 10 to 10 and transition positions ranging fron the nose to the tail end of the body. The calculations assi;ine zero heat transfer, The results indicate that the overall difference in profile drag befr.veen fully laminar and fully turbulent flow d^ecreases rapidly vri.th mainstream Ilach number and rather nore rapidly thcji does the corresponding difference for a flat plate, rjid Ctt I.Iach numbers greater than about 2 the profile drs-g of the body with fully turbulent flow is less than that of a flat plate (Pig, 14)»

-2-Ho\7ever, the effects of rotation in the f lav introduced by the

nose shock and secondary effects on the development of the boxmdary

layer caused by the modification of the external pressure

distribu-tion due to the boundary layer have not been tcJcen into account

in the main body of calculations and it appears that they roay

appreciably alter the calculated,, profile dre.g values for the high

end of the ï.ïach number range considered bringing the value for

the body vri.th fully turbulent flov/ nearer the flat plate value, A

further comprehensive series of calculations covering

B.

range of

fineness ratios is pl?jmed in v>±dch allowance for these factors

v.ill be made,

lll^HN NOTATION

b radius of base of body

c„ local frictional stress coefficient in tenus of

o undisturbed stream values of velocity and density

c„ local frictional stress coefficient in terns of density

n and velocity just aft of nose shock ( 2 T /p u )

C-, overall skin friction coefficient (based on surface area

of body)

C-. f o m drag coefficient (based on surface area of body)

P

C_^ profile drag coefficient (C^ + C^^ )

o ,, p

C, pressure coefficient (p-p )/p i

^o -

° °"'

C pressure coefficient for aft body

( P - P U ) / P K

1

HD

- ~'

D naximum dicmeter of body

P ( X ) , G ( X )

functions defined in equation (9) (see also Ref, I)

f, g function defined in equation i,2) (see also Ref, 8)

h(M. ,R ) function given in Ref. 1 (Pig, 2)

H ratio of displacement thickness to momentum thickness

of boundary layer (6 /O )

K hypersonic similarity parameter (M D / L )

L linear dimension used in K, e,g, length of head (Lp.)

or aft body (L^) "

M

lHach.

nmber

p pressure

p ^ pressure just in front of aft body

r radius of cross section of body

E Reynolds number

s distance measured along meridian profile from nose/l

S sizrface area of body/l

t see P i g , l(d)

u velocity component parallel to surface of body

X axial distance measured from nose/l

y distance measxared norme.l to meridian profile/l

x^

non-dimensional axial distance for head, i,e, x l / L ,

Xp. axial distance for aft b o d y neasiored from beginning

of aft bociy/Lg

Y ratio of specific h e a t s , assxjned equal to 1,4

e effective increase of cross sectional radius due to

boiindary layer

5 boundary layer thickness

6 mcnentxjm thickness of boxmdary la.yer (see equation 3)

6 displacement thickness of boundary layer I see R e f , 1 )

/\ generalised Pohlhausen parameter (see equation 4)

}i coefficient of viscosity

0Ü exponent in temperature viscosity relation

cr E r a n d t l nu::iiber

vr

angle bet\-/een tangent to meridian profile and axis

X. ,''--'^ angles required for equivalent ogive of curvature

method (Ref, 7 ) ^ see equation 14»

suffix o refers to undisturbed stream values

suffix 1 refers to local values just outside the boundary layer

suffix n refers to values at nose just aft of nose shock

suffix

\i

refers to values at the surface

suffix H relates to the forebody or head

suffix B rela.tes to the boat-tail aft b o d y

suffix L relates to lominar flow

suffix T relates to turbulent

flow

suffix e relates to the effective shape allowing for displacement

due to the boundary layer,

-4-1, Introdaction

In College of Aeronautics Report No, 73 "the analysis was developed for calculating the prolile drag of aerofoils and bodies of revolution at supersonic speeds. This present paper

is concerned v/ith the details of the application of tliat analysis to bodies of revolution of the type illustrated in Pig. 1 having circular arc ogival heads, cylindrical centre sections and para-bolic boat tailed aft-bodies, and the results obtained for one body of fineness ratio 7«5 whose geometry is given in more detail

in the Appendix, These calculations cover free stream ivlach

r -I

numbers of 1 , 5 , 2 , 5 and 5 . 0 , Rejmolds nuiibers of 10 , 10 , end

Q 2

10 and with transition positions at 0, •=• I cjid I from the nose where I is the overall body length, A later paper vrii.ll

give the results of comprehensive calculations covering a range of fineness ratios,

The terra profile drag is here used (as in Report No, 73) to denote the drag that arises from the viscosity of the medium, and is thus the sum of the skin friction drag and the change in the vfave drag due to the presence of the boundary layer. This latter contribution to the profile drag is referred to as the

f o m drag. The analysis therefore involves the determination of the development of the boundra:y layer in the laminar and tui-bulent states, end the associated change in pressure distribution fron that given by inviscid flov»' theory due to the effective displace-ment of the body surface arising from the presence of the boundary layer,

The follox'vdng assumptions core made;-(i) The heat transfer is zero,

(ii) There is no separation of the bound.ary layer fron the body,

(iii) The value of the Erandtl number {<s) is taken to be 0,72, (iv) The value of y is taken to be 1,40,

(v) The value of the exponent (w) of the viscosity - tem-perature relation (|i = const, T ) is 8/9.

2, Details of the method of analysis 2,1, Pressxxre distribution

Before the development of the boundary layer can be follovfed the pressure distribution on the body must be dLetemined, in the first instance for inviscid flow,

Since it is ultimately intended to consider a number of bodies of different fineness ratio it is convenient to make

2 3

use of the hypersonic similarity law, * This law states tha.t subject to certain limitations of fineness ratio and Liach number similarly shaped bodies of revolution at zero incidence have the same non-dimensional pressure distributions at the same vsilue of the parameter K = M D/L, v/here M is the undisturbed strea;:i Ivïach nxmber, D is the majcirium body diameter andt L is the body

length v/hich might in this context be that of the head or boa.t tail alone ^i,e, L ^ o r L ) , Curves of the pressure coefficient against K for various positions dovmstrear.1 from the nose were therefore constructed,

To obtain the pressure distributions over the forebody the data from Ref, 3 v/ere used. The relevamt curves ere

reproduced in Pig. 2 where K_^ is the hypersonic paxaneter vri.th the head length L. as reference body length and C. =: (p-p )/p •

For the centre section the data of Ref. 3 for £'- cylinder folloiving an ogival hea.d \/ere plausibly extended and are shovm in Pig, 3 in the form used for these calculations,

For the boat tailed af t-body the results of the caJcula-tions by Praenkel vrerc used, supplemented for other vaJ-ues of Ko by calcula.tions using Van Dyke's second order theory,^ The

symbol IL, here d.enotes the hypersonic parameter -irith the tail length as reference body length. It should be noted ths.t for such calculations the conditions at the end of the centre section are assumed to be the sfime as the free stream conditions. Con-sequently, the non-dinensional pressures are given in the f o m of C = (p-Pi^)/?- f where JL is the pressure immediately ahead of the af t-body. The results of the calculations are shoivn in Pig. 4.

- 6 - TECHNISCHE HOaESCHOOL

V U E G I Ü i G B O U W K U N D c Kanaalïtfoat 10 - DELFT

It wa.s initially hoped that vri.th a lïach number range of 1,5 to 5oO the pressxare distribution for any overall fineness ratio between about 7.5 and 11 could, be foxmd vri.th a.cceptable accxiracy for the pxarpose of profile drag estimation f rem Pig, 2, 3 and 4. Hov/ever, the effects of rotaticn in the flov/ introduced a.t the nose shock have not been included in the calculations of Ref, 3 on v;hich Pig, 2 and 3 a^e based. Subsequent to the cpmpletion of much of the v/ork described in this paper the attention of the authors was dravm to Ref, 6 in v/hich allo-ivance for rotation effects is made and it is clear tha^t these effects can be ajjpreciable for values of IC, > 1,0 and therefore for the body considered at Hach nxaiibers f^reater than 2,5. For the

purpose of profile drag estimation their significance vri.ll be clearly less than for the estimation of vrave drag, nevertheless it is intendLed to investigate this point and subsequent calcula-tions that arc planned xvill be made v/ith allowance for rotation effects on the pressxire distribution,

2,2, The laminar boundary layer

From Ref, 1 (equation 33) we have that the laminar

boxmdary layer noraentun thickness, 0, at any station s. can be ' obtained from the fon^iula

j''*1

Ipi-o^'i = - 7 1 I p i - i ^ " ' v ^ (0

• "1 V^'^^l 'J o

\Tdiere p. and u. are the velocity and density just outside the boxmdary la.yer, respectively, made non-dimensional in terras of

the corresponding quantities (p ,u ) just aft of the nose shock at the nose of the body. The Reynolds nxjmber R is similarly defined by

u I p ^^n

v/here I is the overall length of the body,

The distance g is measured alon,g a meridian profile of the

body and is made non-dimensional by dividing by Z, The functions f and g are given by

f = 9.072 I 1 +

0 , 3 6 5 ( Y - O

i'I^ (

" • \ ( 2 )

= 9 . 1 8 + 1 c436 11^ - I I 1 + • ^ ^ ' ^ 0-^" ï/I^ ! i g = 9 . 1 8 + n 4 3 6 ir - j I 1 + - ^ ^ - ^ o-^ 1,1^ ! ^ where M i s t h e Ilach number j u s t a f t of t h e nose shock a t t h e

n n o s e of t h e b o d y , The momentxm t h i c k n e s s 0 i s d e f i n e d b y .

i^^ / \ / \

9 = 1 ^1L_ f i . . Z . Cos •,?') ( 1 - i ^ l . d y . . ( 3 ) ( 1 - ^ :

V ^1 /

p . u . \ r / \ u , ƒ '.,' o

v/here y i s neasxired normal t o t h e body sxirface, 5 i s t h e boxmdary l a y e r t h i c l c n e s s , r i s t h e x'adius of c r o s s - s e c t i o n of

t h e b o d y , and Vf i s t h e a n g l e betv/een t h e t a n g e n t t o t h e m.eridian of t h e body and t h e ajcis. A l l d i s t a n c e s a r e nade n o n - d i m e n s i o n a l i n terms of t h e body l e n g t h I,

The r e l a t i o n betv/een s and t h e n o n - d i m e n s i o n a l a x i a l d i s t a n c e , x , i s g i v e n b y

-.X

f

2~ii

I' / dr \ i u o

Given the shape of the body and the pressxire distribution the quantities p. and u. as functions of s(or x) can be readily evaluated on the assxjmption of isentropic flo\7 outside the boxmdary layer aft of the nose shock. Hence from equation (I)

the distribution of 0 along the body can be determined, The skin friction distribution can then be obtained from (equation 34, Ref. I)

c 2 T ( A + 1 2 ) U , '^ w ^ '^ 1 ! : f t i

\ , , , . , , ( 4 )

•f ~ 2 - 3 R , f .0 n p u -^ n ^n n vfhere T i s t h e f r i c t i o n a l s t r e s s , a n d i ' ^ l 2 2

•A = R ' ^ . 0 f p,|i i . _{n d a ^^^'€i}

The v/all v a l u e of t h e v i s c o s i t y , IJ._, i s g i v e n , xri.th t h e assxmp-t i o n s made, by assxmp-t h e formula

-8-The contribution of the skin friction in the laminar boxmdaary layer to the overall skin friction coefficient is then

i o i'^T ! P u^

% - S i °f '""o ^ i • 2 ^^^ L i : n p u

• L •-' o ... "^o o 2

v/here S, Z is the surface area of the body, and suffix o relates to quantities in the unddsturbedt stream and sxiffix T to t h e t r a n s i t i o n p o i n t , Prom t h e d e r i v e d d i s t r i b u t i o n of 0 t h e d i s t r i b u t i o n of t h e d i s p l a c e m e n t t h i c k n e s s can be o b t a i n e d u s i n g t h e r e l a t i o n s 5* = H,0 / where . '^ • • • , , . . . , , ( 7 ) Ii = 2.55 ^1 + 0.277 ^ V • ^

2 , 3 , The txurbulent boxmdary l a y e r

From e q u a t i o n ( 3 6 ; of Ref, 1, and talcing t h e recoiTimended v a l u e s of n and Cp (6 amd 0,00878, r e s p e c t i v e l y ) f o r t h e r a n g e of Reynold-s nxinber c o n s i d e r e d v/e have

0,010536 R^ ! G ( x ) r ^ ^ / ^ e x p | ; | F ( X ) g , d ^ : + l r ^ e ) j

O ' • px

' i 6 T-,/ \ ds . ' ;exp 5 5 n x ) -Q^ » ^ i

• x^ ( 8 ) and t h e sl:in f r i c t i o n c o e f f i c i e n t i s g i v e n b y ( e q u a t i o n 37» Ref, I )

c^ = 0,01756 R^ p.,u^ G(x) 0 ' ' / ^ , i

"" / where P(x) = ^ j ^ , ^ ^ ( H + 2 ) - l i ^ | \ vS)

/•u \ - l / 5 i G(x) = ? - ^ I . h(M, ,R )

and the functions h(lï. ,R ) and H are given in Fig, 2 and 3 of Ref, 1,

From the distribution of 0 the distribution of the

displacement thickness 5 is readily obtained fi-cm the relation

6* = H,e.

The contribution of the skin friction in the turbulent boxmdary

layer to the overaill skin friction coefficient is

A " ] 2

' O p ,U

^P„ - I S : . °f •^o*'^^. 2 •

^^^'

T t ' n S p u.

2,3, The f o m drag

I t i s shovm i n Ref, 1 t h a t the effect of the boundary

la.yer on the external flow can be talcen as equivalent to t h a t of

displacing the sxirface outv/ards through a distance e nor^nal to

i t s e l f T/here e i s relented to the displacement thiclcness o by

the equation

2

6* = e + ~ - . (11)

o

"1e therefore require t o develop a method of acceptable accxiracy

for c a l c u l a t i n g the change i n the e x t e r n a l pressxire d i s t r i b u t i o n

due to t h i s ciisplacement of the sxirface v/hich nay be assxiraed t o

be small except perhaps over the t a i l ^

The method adopted i s as follox"/s. Consider the head

f i r s t J i f vre v/ere to apply t h e ' e q u i v a l e n t ogive of cxxrvature '

method of Bolton-Shav/ and Ziehkiev/icz t o the displa.ced shape,

T/hich vre v i l l denote by sxiffix e, then a t each point v;e would

r e q u i r e to determine the equivalent hypersonic pajrar.ieter, K.^ ,

and the equivalent distance dovmstream from the nose x^ ,L n^y

means of the formulae (see Ref, 7)

Kjj = 2M t a n - ^ . . ( 1 2 )

£

/and , , ,

X The distance e is here non-dinensional in terms of the

overall body length Z.

1 0 -and

S i n !.'^ e

Xj, = i l ^ / . (13)

The a n g l e s '''•• and ''y' a r e d e f i n e d i n t e n n s of t h e o r d i n a t e r , and i t s d e r i v a t i v e s w i t h r e s p e c t t o x b y t h e fomula.e . ! 1 + r ' + r , r " / 7 = Cos-^ .' 2§ 2 S _ ^ ( '"e •; (1 + r ' 2 ) o ' \ and .(14) -1 , . -1 / , de 1 - • = t a n r ' + T ~ •' o i o dx / ^' = t a n r ' = t a n ( r ' + I o We t h e n w r i t e ÓK^ = Kg - Kjj e and A x j j = x^^ - x^^ , e

and assxffiiing t h a t t h e c o r r e s p o n d i n g pressxire change i s s m a l l v/e have

/ dC \ f dO \

A c - i —22. ' r- K. + ' T^° < A X (15) •-'^p^ - \ Ö K j j / ' ' ^ - ^ ^ V d Xjj/ • -- ^H * ^^^^ vjhere t h e q u a n t i t i e s \ ^ ^ .' and f -r-^— • a r e d e t e r m i n e d fz'om

t h e cxirves of F i g , 2 ,

The forLi d r a g c o n t r i b u t i o n of t h e head i s t h e n g i v e n b y

p ' „ vS 1.1 ,' -^o •^ H ' o j o

For the f o m drag contribution of the af t-body a similar though somev/hat more cca^plicated method is used. It was assximed that the displaced profile is parabolic in form like the actual profile finishing with a base radius of cross-section equal to 5/9 the maximum radius of cross-section. Since the displacement e will grow more rapidly over the fonvard portion of the tail than r decreases d(r +s)/dx is zero at some point

X . doxmstream of the beginning of the tail (see Fig, Id), The displaced tail must therefore be as.^.uined to bogin at this point, writing Lg for the effective length of the displaced tail, D for its maximu::x diameter and t for the length of the actual tail minus x . (all lengths being made non-dimensional in terms of Z the overall length of the body) then we have from the assximed parabolic formula for its profile (see the Appendix) that

2 ^ •^^^ ^e

he = 9(D -2b ) ^^7)

v/here b is the radius of cross-section of the displaced profile in the plane of the base of the actual profile. It follows that having determined D , b and x . and therefore t v/e can calculate Lg , The corresponding hypersonic parameter is then given by

he

= \ / ^ >

^^8)

e

v/here M is Mach nxmber of the flov/ just outside the boxmdary layer a.t the point x . , whilst the corresponding fractional distance downstream from the beginning of the effective tail is

^ e = ^ ^ ^ - ^ e l ^ / ^ e • ^^5) Viritine

and

/^^S =

he-\

xve b u t

have

xve /\

here

/ ' \

note

\ = t h =

\.' h''

a t Pbe

h

-+ P '\ ^ Xy . - P b Pb ' Xp. • « • , , • • , . . . « v < - 0 ^ .(21)

v/here p, is the pressxire at x . ,

1 2

-downstream of x . i s

A p = P^-P = p ; ^ - 1 U P b e - - \ ^22)

The effect of the displacement on the pressxire frcxi j u s t a f t of the head t o the point x . may v/ith reasonable accxiracy be obtained on the assxxnption t h a t i t i s the sane a.s t h a t i n a simple wave floxv i , e ,

Y P

iS

_{1 de}

P , - P = />P = . ^ . g (23)

and in particular this formula enables us to determine the pressxire u at the point x . for substitution in ecjuation (22), The contribution of the aft-body to the form drag coefficient is finally

r °D ^ = ^ I ^ • ^ • V ^ (^«

V/here Z^, i s t h e distance from the nose of the body t o the

beginning of the t a i l made non-dimensional i n teims of the body l e n g t h Z,

The t o t a l form drag c o e f f i c i e n t i s then

Gj^ = /''-Cj^ : • * - ' ^ ' ^ D ' • ^^^^

P V V^ •: p / g

It will be appreciated that if the calculated change of pressxire over the body due to the boxmdary layer is large then

the calculations of the boxmdary layer development should be repeated with the modified pressxire distribution leading to a new value of the form drag and so on, xmtil successive calculated changes in the pressure distribution due to the boxmdary layer produce negligible changes in the form drag,

3» Results and Discussion

3,1, Mach number and pressxire distribution

Pig, 5 shows the calculated inviscid flow distribution of Mach number over the bo(3y considered for main stream Ivlach

nxmbers of 1 ,5» 2,5 and 5.0, It will be noted that in all cases the local Mach nxjniber v/as somewhat higher than the main stream value over the centre-body and aft-body,

The corresponding calculated pressxire distributions are shox'/n in Fig, 6, The rapid increase in the magnitude of the negative pressxire gradient over the head with increase of main stream Mach nximber is notev/orthy,

3,2, Skin friction results

The overall skin friction results are shov/n plotted in different v/ays in Pig, 8, 9 and 10 treating main stream Mach • nxjmber, transition position and Reynolds number as independent variables and the results are also given in Table I, Perhaps the most striking featxire of these results is the fact that the

difference betv/een the calculated values of skin friction for fully laminar and fully turbulent flow decreases rapidly v/ith increase of Mach number and decrease of Reynolds nximber and is almost negligible in the case M = 5f R = 10 , In Fig, 11 the results for

R = 1 0 , 1 0 and 10 are ccanpared v/ith the corresponding results for a flat plate and it v/ill be seen that whilst the difference in the skin friction for fully laminar arivi. fully turbulent flov/ also decreases markedly for the flat plate the decrease is not cjuite so great as for the body. Indeed the skin friction drag of the body with fully turbulent flox7 is less than that of the flat plate for Mach nxuribers greater than about 2,

Pig, 7 shows the distributions of skin friction over the body in the cases of fully laminar and fully turbulent flow at a Reynolds number of 10 and for M = 1^5, 2,5 and 5.0 and it v/ill be seen that v/hereas in the fully laminar case the skin friction over the head shows a marked increase with M v/ith a

o

small decrease over the centre-body and aft-body, in the fully turbxilent case there is a marked decrease in skin friction XTith increase of M over almost all the body except the forward half of the head,

It appears that the increase in skin friction v/ith M o over the head in the laminar case results Isurgely from the greater importance that the positive velocity gradient there plays in the

laminar than in the txirbxilent case. Further, it is a consequence of the large velocity gradient over the nose that the laminar skin friction coefficient of the body is somev/hat higher than that of a flat plate. The cause of the very marked fall of the skin fric-tion over the major part of t?ie body with increase of M v/ith fully txirbulent boxmdary layer is rather more difficult to assess closely, i\n appreciable fall may be expected since it is

inherent in the basic flat plate results on x-rfnich the calculations are foimded but v/e see from Pig, 7 that the calc3ulated reduction is considerably greater than that for a flat plate. Referring to Pig, 6 \re see that over much of the boc3y the local Mach number is sonewhat higher than the main stream value and so v/e may

expect the flat plate reduction of skin friction xvith M to be somewhat magnified. Further the local values of p.u. (to v/hich the local skin friction coefficient is proportional) faJLls x^ith increase of M. in the range of Mach nxxmbers covered, '.Te might therefore expect the skin friction in the turbulent case for the body to decrease v/ith Mach number somewhat more rapicUy than on a

flat plate but the actual rate of decrease is nevertheless sxir-prisingly high. However, relevant to this point are the

correc-tions that will need to be made in the extreme cases of high Mach nxunber and low Reynolds nxmiber for secondary effects of the

boundary layer on the pressxire distribution reacting back on the boxmdary layer development and also for the effects of rotation. The possible effects of these corrections are discussed in S]i,3 Vi*iere it is concluded that they may appreciably modify the picture

at high Mach nxjmbers bringing the body and flat plate skin friction values for fully txirbulent flow into closer a.greement,

3,3. Form drag results

The results of the form drag calcjUlations are shov/n in Pig, 12 and Pig, 13 illustrates a 'carpet' plot for the case M = 5 . The results are also given in Table II,

The general variations of f o m drag v/ith transition position and Reynolds number are as described in Ref, 1 but the overall increase with M is at first sight sxirprising. This

APP

eajTS

X In the momentxxm equation the terra du./do i s m u l t i p l i e d by

(H+2-Mf) and H for the laminar boxmdary l a y e r i s almost txvice as

great as H for the txirbulent boxmdary l a y e r a t any given Hach number,

appears to be related to the fact that xvhereas the boxmdary layer causes an increment in pressxire over the head xvhich increases fairly rapidly v/ith Mach nximber in the range covered, the corres-ponding pressure increment over the tail increases far less rapidly xrlth Mach number. The reason for this can be seen from Fig, 2

and 4 where it x-/ill be noted that the quantities {dC /SKj) and

(3C /3XTT) for the head are considerably greater than the

corres-ponding quantities (öC /i^K-) and (öC /i^x^) for the tail, Pb -MD

Changes of Mach number result most directly in changes of the hypersonic parameters K-, and IL, and also in seme changes in the position parameters XL^ and x^, and the net effect is that the positive form drag contribution of the head increases more rapidly xvith Mach nunxber than does the negativo form drag contribution of the tail,

3,4, Profile drag results

The profile drag coefficients for the cases considered are shown in Pig, 14 and are also given in Table III. The corres-ponding curves for fully laminar and fully txirbulent flox7 over a flat plate are shoxvn in Fig, 14 for comparison. The general trend of the results is very much the same as for the skin friction

coefficient results shov/ing in particular the relatively small effect of transition position at a Mach number of 5*0 and Reynolds nximber of 10 , The contribution of the form drag is in general

small although in the extrem.e case of fully laminar or fully txirbulent flow at M

o of the profile drag,

txirbulent flow at M = 5 it can be of the order of 10 - 15 per cent

3,5. Secondary effects and the effects of rotation

It is clear that v/here the caiculated displacement e of the profile is not small compared with the radius of cross-section further iterative calcxilations may be necessary to ensxire that the calculated development of the boxmdary layer is sxiffic-iently consistent vri.th the finally determined pressxire distribution,

In the case of the fully txirbulent boxmdary layer, M = 5 »

6 °

and R = 1 0 the calcxilated value of the ratio e/r at the end

o o

of the body xvas indeed of the order of xinity, although i n a l l other

. ^ TECHNISCHE HOGESCMOOL

Kanaalitraat 10 - DELFT

cases i t v/as considerably smaller. This case was therefore used

as a test case to assess the effect of a second approximation, The skin friction distribution v/as therefore recalculated using the pressxire distribution as given at the end of the first stage of allov/ing for boxmdary layer effects. The results are sho'//n in Pig, 15 and it v/ill be seen that the change in skin friction in going from the first to the second stage is by no means insignificant ajid it results in an overall increase of the skin friction coefficient of about 10 per cent. This effect would therefore increase somev/hat the small difference given by the first stage be-tween the profile drag xrith fully laminar and with fully txirbulent boxmdary layers at high Mach numbers and low Reynolds numbers although it xvould not alter the general trend of a marked reduction of this case inherent in the flat plate data which are basic to these calcnilations,

Hov/ever, as xve have already noted, at the highest Mach nxirabers for this body the effect of rotation which has been ignored for these calcnilations would also make an appreciable effect on the pressxire distribution causing in general an overall rise in pressure. This xvould therefore have a similar effect on the calculated skin friction a.s the change in pressxire due to the presence of the boxmdary layer and the effects may v/ell be of comparable ma-gnitude, We may in consequence expect that when full alloxvance is made for these corrections the fully txirbulent profile dra.g values for the body v/ill be much closer the flat

plate value at the Mach nxniber of 5.0 than is indicated in Pig, 14, Nevertheless it is lilcely to be smaller than t he flat plate value, a possibility xvhich is in interesting contrast to subsonic results,

In the fxirther and more comprehensive calculations v/hich are contemplated for a range of fineness ratios it is intended to make proper allowance for both secondary effects and the effects of rotation,

4, Conclusions

The calculations of profile drag for the particular body considered indicate tha.t the overall difference in profile drag between fully laminar and fully turbulent flov/ decreases rapidly v/ith main stream Mach number and rather more rapidly than does the

correspending difference for a flat plate. Indeed at Hach numbers greater than about 2 the profile drag of the body v/ith fully txirbxilent flow is less than that of the flat plate. If tills is confirmed by more comprehensivo calculations as a general. result it Viill be of significance in assessing the value of

striving for 3.a.rge extents of laminar flov/ at high Mach nximbers, However, the elf'octs of rotation in the floxv introduced by the nose shock and secondary effects on the boxmdary layer development due to the modification of the external pressxire distribution caused by the boundary layer arts of sippreciable significance at the hig]iest Mach number considered (ivl = 5 ) and it is lilcely that when allowance is made for these effects that the drag coefficient of the bocSy for fully txirbulent flov/ v/ill be much closer to that of the flat plate than is indicated by these calculations (Pig, 14). A fxirther comprehensive series of cal-culations covering a range of fineness ratios is planned in v/hich allowance vrill be nade for these factors,

No, 1. 2.

3.

5.

Author Yoxmg A,D, Tsien, H, Ehret, D,M,, Rossow, V.J., Stevens, V,I, Praenlcel,L,E, Yeji Dyke, M,D, Title

The calculation of the profile drag of aerofoils and bodies of revolution at suioersonic speeds,

College of Aeronautics Report No, 73 (,1952) Similarity la:v/s of hypersonic floxvs,

J, Llaths, and Physics Vol,25, No,3, Oct. 1946,

Jin analysis of the applicability of the hypersonic similarity lav/ to the study of flov/ about bedies of revolution at zero angle of attack,

N,A,C,A, T.N, 2250, 1950,

Cailculations of the pressure distributions and boundajry layer development on a body of revolution,

R,A,E, Report Aero, 2482, 1953. Practical calculation of second order supersonic flox/ past non-lifting bodies of revolution,

-18-No. Author Title

6. Rossow, V,J, Applicability of the hypersonic similarity rule to pressxire distributions which include the effects of rotation for bodies of

revolution at zero angle of attack, N.A.C.A, T,N. 2399.

7.

8.

Bolton-Shaw,B,Vir, The raj^id accxirate p r e d i c t i o n of pressxire Z i e n k i e w i c z , H,K, on n o n - l i f t i n g o g i v a l heads of

a r b i t r a r y shape a t s u p e r s o n i c s p e e d s , E n g l i s h E l e c t r i c Co, L t d , Rep,

No, L , A . t . 0 3 4 , 1952.

Yoxing, A,D, Skin f r i c t i o n i n t h e lajninar boxmdary l a y e r i n c c m p r e s s i b l e f l o w ,

College of Aeronautics Rep, No. 20 (1948) (j'llso Aero, Quarterly, Vol, 1, Aug, 1949, pp,137-164).

TABIE I

OVER/JLL SiaN FRICTION COEi^'i^'lCIENTS - C

T r a n s i t i o n P o s i t i o n

loog

» • 6 6 | ^ 1 1 0 ^ 1 f M o 5 . 0 2 . 5 1.5 5 . 0 2 . 5 1.5 5 . 0 2 . 5

' ^'5

R = 106 o , 1.768 1.740 1.624 1.796 2 , 2 1 4 2 , 5 5 8 1.825 5.095 1

1 3.977

R = 107 o 0.559 0,550 0 , 5 1 4 0.652 1,013 1.320 1.024 1.970 2,629 nP^^Q^ R = 10Ö 0 0.177 0 . 1 7 4 0,162 0,265

0.536 1

0,771 0.627 1.285 1.751

T J ^ L E I I

FORM DRAG COEFFICIENTS - C.^ _x 10'

4

Transition Position R = 10»! o '6! R = 107 o R =10^^ o 1.971 0 . 1 4 8 0,038 1,029 -1,201 -1.471 2.769 -0,210 -0.568 0,601 - 0 , 0 0 4 0,039 0,207 - 1 . 0 5 0 - 1 . 1 8 8 1.839 - 0 . 3 5 0 - 0 . 3 0 7 i 0,341

1

I -0,003 i -0,011 i 0,089 j - 0 . 6 1 7 j - 0 , 6 4 2

I 1 .225

I - 0 , 2 2 6 i I -0.177 T/.BLE I I I PROFILE DR:\G COEFFICIENTS - C.^ x 10-o Transition Position M R = 1o6 ! R = 107 ! o i o ! R = 10O o 5.0 3.5 1.5 5.0 2.5 1.5 5.0 2.5 1.5 1.965 1.755 1.628 1.898

1 2.094

1 2,411 i 2,102 3.074 3.920 I 0.619

i

I 0.550 ! i 0,518 j 0,673 I i 0,908 j j 1,201 i I 1,208 ; 1.935 i 2.593 0.211 0 , 1 7 4 0.161 0 . 2 7 4 0 , 4 7 4 0.707 0.750 1,262 1.733

-20-GEOMETRY OP TEE BODY CONSIDBRED

The forebody is a circulox are ogive and hence with origin at the nose its equation readily fclloxvs,

vis,-,. j 2

x^ + r^ - 2x L„ + 2r I ^ - ^ / = 0

o H o \ D 4 .

^ . • j 2 ' ,

The r a d i u s of the c i r c u l a r ogive i s ! 5— + T1 '

Tlie nose angle 0^ i s given by

S:m0s = ^ li—'^lj'' .

' '• i i

For the p a r t i c u l a r body chosen 1^ = Z/3, Lr^D = 2 , 5 .

The centre-body i s c y l i n d r i c a l and for the p a r t i c u l a r body chosen

i s of the sajiie l e n g t h as the head. i , e , Z/3,

The b o a t - t a i l i s formed from a paxabolic exc ajid i s such tha.t i f

i t s length had been continued t o zero c r o s s - s e c t i o n a l radius i t s

l e n g t h v-fould be •^ . L , , where K, i s i t s a c t u a l length. In

5 D

consequence the base r a d i u s b i s equa.l t o ~- .-^ , The equation

of the b o a t - t a i l p r o f i l e i s then

D L 4 2;

^o "^ 2 9 ^B? '

• j

PARABOLIC BOAT TAIL OGIVAL HEAD'

b.

1

r

y4^:i^—

f

\ . \ .

^ \

\

CIRCULAR ARC HEA

D

t

N X «N T 9 2 .3/^L 2 B ACTUAL BODY c. PARABOLIC BOAT TAIL

EQUIVALENT DISPLACED BODY

d EQUIVALENT DISPLACED SHAPE

FIG. 2. C O L L E G E O F AERONAUTICS " I REPORT No. 81. 6-0 5-O 4 - 0 3-O 2'5 Po l O o O « 1 ^ / . /y ^ • ~ ^ C H ; ^

V

^NGE /

y

^ , , — OF

y

^ < H -SCAL loo: / / / ^ ^ ^ 1 — Mo E C H -/ / /

y

- ' • — o ^ 3/1-0 ~

''^0

_f / A y " — — •H

vf

/ / /

y

- ^ — •

H

— •

7

/ ^ / / /

y

/

7

y

/

Y

/ / / . > X -I-s .. / f

1^

/ / 4g, " —

A

/ / / ^ i 2 -60 7 0

—1

l

/ r / /

y

o ?o oo

}

f /

y

y

/ / /

y

^ — 2.'6 -—

*-po'^ K^ FOR HEADS

O • in V^ 0 tf) O

o

s

1

1

1

S!03 CO—— / /

f/

/

7.

Ii

o

o

/

f/

/ /

7

/ / / J

A

/ / /

V

/ /

v/

/ /

n

/ f '// / / / / /

y/\

/ / /

A

A

7/

/ //J

y

/ ///

w

₇

0/

y

o I o I o I o _I Ó _I Ó

o

a.

U

K„ GRAPH FOR CENTRE SECTION

FIGS. 4a. & 4b. COLLEGE OF AERONAUTICS REPORT No. 81. u - s Cp. 0 1 O

i

V

7

y' --•^ _ i ,'/ + UBIAINbU r-HUM Kth Ï / //, / , / / " /^ //^ ''^Z / / y ^ - v t / / /

y

V

/ / ' X -^ ••B = / //

7/

/ / /

y

^ Mo ,/

J

/ / J / / y" - - + ' 1 D / L / / / / / / /• y ^ / / / / / / / / /

y

^ DY / / / /

7

A

lOO <E' / / / / ' / <:^ 5ü y o ^^-SOl / / / / '

A

io / / o / /h

V'

o / / y /" y - ^ / / 0 / y / y ^ / '0 /

y

y ^ ^ / • z' y^ y ^ ^ • ^ ^ y y^ ^ ^ ^ --' , ^ ^ ^ ^ ^ ^

-c

0-Ö 0 - 4 Po 0 - 3 0 2 O loo: • 8C

è

if

7

C B =

't

^

i

//

1

lo 'I^QO

it

1

li

1 / 1 / // / innx--9C / / / / / //

7

A

f/

/ :' s< —^ ^ '/ DLN /

Y

f^

y ^ // y ^ ^ /^ J ^y 1-0 KB -^ 2 0

Cpb~KB FOR TAILS. (PARABOLIC) Cpj^—Kg FOR TAILS (PARABOLIC^

" • B "

I O O X B ~ PER-CENT TAIL LENGTH. AFT OF BEGINNING OF TAIL.

FIG. 4A.

l O O X . —PER CENT TAIL LENGTH

6 0 S O M, 4 - 0 3 0 2 0 l O ^ / /

y

' M o - 5 / -/ -L M C - 2 ' 5 ^^l^=_l_2 • ^ 0-2 0-4 V 0-6 0-8 l-O 0-9 0-8 0-7 P/Pn 0-6 0-5 0 - 4 0-3 O-2 O l O l O O ^

v\

l\\

\ \ ^ \ \ \ \ \ \ \ ^ \ \ Mo=l-5 Mo=2-S \ - ^ rviQ—o-v.' "~~-— —'

K

0-2 0-4 -y^ 0-6 0-8 l O

DISTRIBUTION OF M, OVER BODY

£/D = 7 - 5

p / P n - X FOR ^ID^I'S

FIG. 7

COLLEGE OF AERONAUTICS REPORT No. 81.

xr^VlO

C ^ ' ^

^oT

0'2 0-4 -^ 0'6 0-8

a. TRANSITION AT lOO %

i-o 3-0 4

(.C, «roVlO

2 0 l-O / ^

1

f

^ \ M —2-5 Mo-S-O \ 0'2 0-4 ^ 0-6 0-8

b. TRANSITION AT 0 %

1-0

q X T Q DISTRIBUTIONS AT R Q ^ I O *

- ? / D = 7 . 5

REPORT No. 81. CpXlO Kanaalstraat 10 - DELïï ^ \ ^ A N ~~~~~--.^..Êr66 S. AT oX

\ X

AT l 0 0 7o Ro = , 0 ' _ _ _ _ _ _ M, CpXiO ^ x _^, ^•""--.v^ANS. AT O X -^^.AT___66V^ AT iOO % 1 Ro=IO J •— M^

VARIATION OF TOTAL SKIN FRICTION (c^ COEFFICIENT WITH MACH No.

FIG. 9. COLLEGE OF AERONAUTICS REPORT No. 81. 4 3 CpXlO^ 2 1 " " " " - - - - ^ . . ^ M ^ l - S 2;5 ^ " ^ 5-0 Ro= : : ^ : \ lO^ CpXlC? C p X i O 2 0 4 0 TRANSITION POSITION % 6 0 2 0 4 0 6 0 TRANSITION POSITION 7-O 2 0 4 0 TRANSITION POSITION "/-6 0 8 0 8 0 8 0 IOO - ^ ^ ^ _ _ ^ ^ Mo = 15 s-o R o = l ( )^

C:^^;;;;^^

IOO

III2I^

— — -~-~~~_Mo = _ _ J 1-5 •o Ro= :|0« IOO

VARIATION OF TOTAL SKIN FRICTION COEFFICIENT (Cp) WITH TRANSITION POSITION -^/D = 7 5

4 i

-CpXlO

LOG|o RQ

LOG|o R o

LOG,o R Q

VARIATION OF TOTAL SKIN FRICTION COEFFICIENT ^Cp) WITH REYNOLDS NUMBER ( R Q )

FIG. II. COLLEGE OF AERONAUTICS REPORT No. ei.

SO 4 0 CpXiO 3 0 2-0 l-O BODY OF REVOLUTION

t

LAMINAR: REF a"l ' TURBULENT: REF I.J ^'"*'^ PLATE

2 M f 3 0 2-O CpXicf I O ~~ - * « ^ , ^ ^ ^ ^ ^ • ° * ^ - ^ = = < : : : ^ ^ F URBULENT !o='0^ LAMINAR I Mr

COMPARISON OF SKIN FRICTION ON BODY WITH THAT ON A FLAT PLATE OF THE SAME SURFACE AREA.

CopXlO^ (TOTAL) (TOTAL) CopX IO'' (TOTAL) IOO IOO IOO

FORM DRAG COEFFICIENT AS A FUNCTION OF TRANSITION POSITION. • ^ / D = 7 . 5

FIG. 13.

COLLEGE OF AERONAUTICS REPORT No. 81. O

9

Ol

+

Q:

o

3

I

a

FORM DRAG COEFFICIENT AS A FUNCTION OF TRANSITION

POSITION AND R.N. f / D = 7 ' 5

C . X IO

FLAT PLATE

FIG. IS.

0-6

COLLEGE OF AERONAUTICS REPORT No. 81.

- O l

a. AFTER BODY PRESSURE DISTRIBUTION WITH AND WITHOUT THE EFFECT OF BOUNDARY LAYER

3'0

2-0

l-O

= io6

M o - 5 0 Ro='0'

WITH MODIFIED PRESSURE DISTRIBUTION -INVISCID FLOW PRESSURE DISTRIBUTION

0-2 0-4 0-6 0-8 l-O

THE EFFECT OF THE RECALCULATED PRESSURE DISTPIBUTION ( D U E T O B. L. DISPLACEMENT E F F E C T )