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TECHNICAL

D-540

NOTE

THE NUMERICAL CALCULATION OF FLOW PAST CONICAL BODIES

SUPPORTING ELLIPTIC CONICAL S/dOCK WAVES

AT FINITE ANGLES OF INCIDENCE

By Benjamin I%. Brfggs

Ames 1%esezrch Center

Moffett Field_ Calif.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

WASHINGTON November 1960

https://ntrs.nasa.gov/search.jsp?R=19980227791 2020-06-15T22:48:10+00:00Zbrought to you by COREView metadata, citation and similar papers at core.ac.uk

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41.

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IC

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

TECHNICAL NOTE D-340

THE NLD4ERICAL CALC_ION OF FLOW PAST CONICAL BODIES

SUPPORTING ELLIPTIC CONICAL SHOCK WAVES

AT FINITE ANGLES OF INCIDENCE

By Benjamin R. Briggs

SUMMARY

The inverse method, with the shock wave prescribed to be an

elliptic cone at a finite angle of incidence, is applied to calculate

numerically the supersonic perfect-gas flow past conical bodies not hav-

ing axial symmetry. Two formulations of the problem are employed, one

using a pair of stream functions and the other involving entropy and

components of velocity. A number of solutions are presented, illustrat-

ing the numerical methods employed, and showing the effects of moderate

variation of the initial parameters.

INTRODUCTION

The first three-dimensional high-speed flow problem to be solved

exactly was the flow past a circular cone. Pioneers in this work were

Busemann, and Taylor and Maco!l. (See refs. ! to 3.) Reference 4

gives numerical solutions for circular cones using the theory developed

in reference 2. A perturbation technique was devised by Stone, refer-

ence 5, for circular cones at small angles of yaw. The method of Stone

was applied in reference 6 to obtain approximate solutions for cones at

small angles of yaw, and was extended in reference 7 for large angles

of yaw. A somewhat similar numerical _ualysis, the linear characteristics

method, was developed by Ferri for analyzing the flow past yawed circular

cones (ref. 9) and slightly noncircu!ar cones (refs. 9 and i0). Tables,

based on the method of Ferri, are available (ref. ii) for calculating

such flows. Of the various analyses referred to above, the only exact

numerical solutions, based on the full inviscid equations of motion,are the ones tabulated in reference 4. They were obtained by numerical

integration of the equations of motion specialized for symmetrical

conical flow.

There are various ways that an exact (numerical) attack on a high-

speed flow problem, employing the full inviscid equations of motion,

might be organized. One is to specify conditions at the body and then

to integrate the equations of motion outwardly toward the shock wave.

2

This is the manner in which the problem was posed for the integrationstabulated in reference 4. Another way is to prescribe conditions atthe shock wave and then to integrate inwardly t_ward the body. This hasbeen called the "inverse" or "marching inward" :method. The capabilitiesof this method are exemplified in the work of V_n Dyke and Gordon,reference 12, who presented a catalog of solutions of flows past bluntbodies for a large family of preassigned shock shapes. In the work ofRadhakrishnan, reference 13, the marching inwar._ technique was used tofind the conical body behind a circular conical shock wave at a largeangle of yaw. In reference 14, Briggs employed an analogous approachto compute bodies that support elliptic conical shocks.

The work of reference 14 is extended here _soinclude circular orelliptic conical shock waves at finite angles of attack or yaw. Twoimplementations of the inverse method are employed, and each has beenprogrammedfor an electronic computer. The first formulation, whichwas used in reference 14, is in terms of pressure, density, and a pairof stream functions, one of which vanishes at the body. (A discussionof the use of several stream functions in the s_alysis of three-dimensional flows will be found in reference 15_) The second formulationof the problem is in terms of velocity componen_s,pressure, density,and entropy. The body is located by calculatin_ the position of theentropy layer, shownby Ferri (ref. S) to exist on the surface of thebody in the case of nonsymmetrical conical flow. The computations ofreference 13 were based on this latter approach. (An interestingdiscussion relative to the thickness of the aforementioned entropy layeris given by Cheng, ref. 16. )

The flow field behind an elliptic conical _hock wave is analyzedin detail in the present report, both the stres_-fumction and velocity-entropy formulations of the problem being used. The resulting body andsurface pressure is given, along with plots of the variation of thevelocity componentsand entropy in the flow field. A diagram showingthe stream lines in the crossflow is also presented. Comparisonsareshownbetween results of wind-tunnel tests on elliptic cones and numericalcalculations with initial data based on sch!iercn pictures of the shockwaves in these tests. A comparison with the approximate theory ofreference 6 is madeusing a machine result ca!c_lated with a circularconical shock wave at a finite angle of yaw. F_nally, the results of anumber of cases are presented wherein systematic variations of shockgeometry, free-stream Machnumber, and ratio of specific heats were made.These are given in the form of body shape and s_rface pressure.

A38

5

SYMBOLS

a I ,bm semimajor and semiminor axes of elliptic cross section of

coordinate surface of constant 8 = 8o (eqs. (4))

A38

5

a2 _b2

A,B,C,D

f,g

F,G

h i,hm ,h3

k2 ,k' 2

M

n

N

P

r_e,9

S

U_V_W

lll2Vl ,W I

Uoo

x,y,z

r(x,y,z )

_,_

semimajor and semiminor axes of elliptic cross section of

coordinate surface of constant _ = 9o (eqs. (4))

functions defined by equations (3)

stream functions of a three-dimensional flow

specializations of f and g for the present conical flow

problem

coefficients of the metric of the sphero-conal coordinate

system

constants in the sphero-conal coordinate transformation

Mach number

number of points taken in the numerical integration

function defined by equation (41)

pressure referred to D_U_ 2

sphero-conal coordinates (see fig. 2)

entropy, P----p7

velocity components in the sphero-conal coordinates

velocity components in Cartesian coordinates

free-stream velocity

velocity vector

Cartes ian coordinates

ratio of specific heats

coordinate surface that coincides with the shock wave

increments of e and 9 that define the mesh size in the

numerical calculations

density referred to free-stream value

4

Subscripts

b

n

N

0

X_y_Z

conditions on the body surface

nth extrapolation

normal component (eq. (23))

conditions at the shock wave

differentiation with respect to variables

differentiation with respect to variables

free-stream conditions

Superscripts

(n)

!

nth extrapolation

differentiation with respect to the variable G

THE SPHERO-CONAL COORDINATE _YSTEM

The numerical procedure employed here requires that the shock

wave be a coordinate surface. A system of ortaogonal coordinates

r_O,9 which meets this requirement is given i_ reference 17. The

transformation from Cartesian coordinates is

t"

x = r cos e_l-k'mcos29

y = r sin @ sin q0

z = r cos 9_l-kacos_

ka + k'2 : i , 0 __e ___ , 0 __q_ __2_

where x,y,z are the Cartesian coordinates.

the sphero-conal coordinates system is defined by the relation

(I)

_he element of length in

w

ds a = hladr2 + ha2de a + h3_-_dq02 (2)

A3S

where

hl 2 = i

h22 = r2kasin2@ + k'msin29

l-kecosS@

h32 = r2k2sin28 + k'2sin2_

l-k'acos2 9

The coordinate surfaces are:

x 2 + ya + zm = rm for fixed r = ro

y2 Z2X2 - 0

b 12 a i_for fixed e = 8o

x 2 y2_ + z2 = 0 for fixed _ = 90aa 2 b2 a

a12 - l-k2cosa8 Ok2cosa8 o

bl2 = tan2eo a I _> bl

2 l-k'2c°s290

a2 k, acosago

b22 = tan29o aa = > b 2

(3)

Portions of the conical surfaces that are represented by the second and

third of equations (4) are shown in figure I. The shock wave of the

present problem will be taken as a constant e (=8o) surface.

The components u, v, and w of velocity in the sphero-conal

coordinate system are in the directions of increasing r, 8, and 9,respectively. Figure 2 shows curves of constant 8 and _ as projected

onto the y-z plane. The velocities v I and w I are in the directions

of increasing y and z, respectively, and the velocity ul, not shown

in figure 2, is in the direction of increasing x.

The relationship between Cartesian and sphero-conal componentsofvelocity is given by the equations:

--k_c°s _o)_(L-! 2C0S2_0 )u I = u cos 8 _l-k'acosa9 - v sin 8_kasinae + k,2sin2_

+ wk'Ssin _ cos _ cos @

_k£sin2e + k,2sin2_

l_kmcos2 ev I = u sin e sin qo + v cos @ sin qo _2sin2e + k'2sin2_

! l-k, 2cos_p+ w sin e sin qol

4 kasina8 + kt2si.q_p

w z = u cos _l-k2cosa@ + vkasin 8 cos 8 cos

Skasin£e + k'esina_

kasin28 + k'esin29

THE EQUATIONS OF ME TION

are

The equations of motion for flow of ar ideal compressible fluid

in vector form,

_iv(pV)= o

p(V.grad)V + grad p = 0

V._rad(p/pT)= o

(continuity) (6a)

(momentum) (6b)

(energy) (6c)

The velocity V has been made dimensionless with respect to free-stream

velocity U_, density p with respect to J'ree-stream density p_, and

pressure with respect to the quantitz p_Uc_2. When transformed into

the sphero-conal system, equations (6) become

A _ + s _(pw) + _(2u+ACv+SI_)= o (Ta)

Avu_ +Bw_- (v_+w2) = 0 (r_)

Av _ + Bw _ - ACw 2 + uv + BDvW + --p_ = 0(Yc)

Av _ + Bw _ - BDv 2 + uw + ACvw + -p _ = 0(7d)

A_ _(_/p_ ) + Bw _'(_/p_) -- o (ye)

Note that the assumption of conical flow has been made in the writing

of equations (7) and that the following abbreviations have been employed:

_

l-k2cos2e r2

k2sin2@ + k' 2sin_p ha 2

Ba = l-k'2cosS9 = r___2k2sin2e + k'msin2_ h3 2

C

k2sin 0 cos e

kesin2@ + k'2sina_

D ----

k'2sin _ cos

k2sinae + k'asin2qO

Three-dimensional flows such as those under discussion can be

described by a method utilizing two functions which are a geners.lization

of the familiar stream function of two-dimensional and axisymmetric

flows. This technique was used in reference 14, and a concise description

of the use of stream functions in the analysis of three-dimensional flow

problems can be found in reference 15.

If two functions, f and g_ are chosen such that

D_ = (grad f) X (grad g) (9)

then equation (6a) is satisfied, and

whose intersections are streamlines.

following manner.

f and _ represent stream surfaces

This cam be demonstrated in the

Equation (9) has the three Cartesian components

pu = fygz - fzgy 1

pv = fzgx - fxgz

pw = fxgy - fYgx

where subscripts x_y,z indicate partial differentiation.

equations (lOa) it follows that

From

(lOa)

u:v:w --(fygz - fzgy):(fzgx- fxg_)'(fxgy- fyg_) (lOb)

Equation (lOb) is a solution of the two equations

V-grad f = 0

_.grad g = 0

(!i)

Equations (i!) are valid if 7, the velocity vector, is normal to grad f

and grad g. Grad f and grad g are the norr__is to surfaces f and g,and so V must lie in the surfaces f and g. It can be said, then,

that f and g surfaces contain the streamlines, and that the inter-

sections of these surfaces are the streamlines. Thus it is quite

reasonable to think of f and g as stream functions.

The energy equation, equation (6c), no_ takes the form

A

38

5

Jt : s(f,g) (12)pY

which states that entropy is constant along streamlines. The momentum

equation, equation (6b), can be written in terms of 7, D, and S; that is_

p(_.grad)_ + grad[pZS(f_g)] = 0 (13)

The velocity components u,v,w can be found in terms of the stream

functions f and g by expanding equation (!_). The result, in the

sphero-conal coordinate system, is

C

9

A

38

5

AB

u = _r--_ (f0g_ - leg0)

Bv = _ (fggr - frgm)

A (frge - %gr)w=_-_

(14)

Here f and g are functions of r_8,9 and the subscripts indicate

partial differentiation. Equations (14) are general for three-

dimensional flows and are not limited to conical flows.

For conical flow V; D_ and S are not functions of r. Thus it

is clear from equations (14) that the product fg is homogeneous of

order 2 in r. In order that f will reduce to Stokes' stream function

in the axisymmetric case and that the body be described by f = 0; itis convenient to make the definitions

f(r,8,9) : r_F(e,9)

(15)

Now S cannot be independent of r unless it is also independent of

f. It can be concluded_ then that

s(f,g)= s(o) (16)

for conical flow.

Equations (14) may now be formally written in terms of

that is,

AB_- _ (%% - FmGe)

2B FG9v - p

2Aw = _ FG 8

F and G;

(17)

The components of grad[pYS(G)] are

i0

0

ApTS(r Pe S' )r p S

s, )BPYS _ + G_r p T

} (:8)

in the r,e,_ directions. The equations of' motion in terms of F and

G are :

8 momentum:

_ --6-

: (2__BF)2(GeGqxp-GqoGeq)) - (pY+ZS) (_')G@ + 2B2FGm(Fq)Gs-FeGq_)

+ (2F)2% (:)-B2 _)%+ (2F)_-(B2%_+ A_%_) (:9)

A

38

5

moment um:

(2AF)2G_Gee = 2AaFGs(F_G@-FsG_)+ (2AFGs)2 []@_ - G_ (D + _) + G_ _-_8]

_ (_F)_:o(CGo+ B_:o) + _7+:S _ + a (ao)p --s-

r momentum:

(ABGm)2Fee

(2:)

ii

It has been stipulated that F is to vanish on the surface of the

body. It would appear, then (see eqs. (17)), that the velocities v

and w must also be zero there. This is not generally the case, so

the derivatives Gq and GO must become infinite on the body. The

singular nature of the derivatives of G at the body is evidence of

the previously mentioned singular behavior of the entropy near the body,

since S and G are functionally related (see eq. (16)).

THE INITIAL CONDITIONS

The shock wave is taken to be an elliptic cone of the family

described by the second of equations (4). The value of 8 at the shock

cone is designated @o" The equation of this conical surface is

r(x,y,z): x2 _ _ - z2 : 0 (22)b 12 a x2

The component of the free-stream Mach number normal to the shock wave is

where Mx,My,M z are the x, y, and z components of the free-stream

Mach number. When rewritten in terms of the sphero-cona! coordinates

equation (23) becomes

: sm COCOS_ cos _/(l-x2c°s2eo)(l-w2c°s%)

Y_o 4 k2sin28 0 + k'2sin2_

i l_kacos2eo- sin 8oSin q cos _ sin _ jkasin_e ° + k-7_sin29

- cos _ sinkasin 8oCOS 8o

_k2sin28 0 + k'2sin_

(24)

12

The angles of attack and yaw, _ and _, and the x, y, and z components

of Mach number are shown in figure 3- Note that the axes are fixed in

the shock wave and that their origin is at the cone vertex.

Expressions are required for evaluating velocity components, density,

pressure, and the stream functions just behini the shock wave. The

oblique shock relations (see, e.g._ ref. 18) along with equation (24)

and equations (17) are utilized in the writing of these expressions. In

the following paragraphs the equations that apply only at the shock will

be clearly indicated so as to distinguish them from such general relations

as may appear in the analysis.

The velocity components u and w lie in a plane tangent to the

shock, just ahead of the shock. They therefore undergo no change through

the shock, so that at e = _o

u = cos eoCOS _ cos _ _l-k'acosa_

A

385

+ sin eosin _ cos _ sin _ + cos D sin _l-k2cosaeo (25)

-- cOS eoCOS c_ cOS

k'asin q0 ccs

_kasin2eo + k' asina@

I ]-k' 2cos2_- sin eoSin q0 cos _ sin _I 2 -a-_---- _,_

@k sir @o + k sin_

- sin _ sin _I (1-_2c°s2e°)(l-k'2c°s2_)/

4 k2sin2e o + k'asin2_

(26)

The velocity v

relations to be

and the density and pressure are found from the shock

(7-1)MW + 2v = - (27)

(7+l)Ml_

(7+1) MNa= (28)

(7-1)MN= + 2

13

P :(29)

where M N is represented by equation (24).

If equations (27)_ (25); and the first of equations (17) are

combined_ the result is

A

38

%ov - _ = -_m_ (30)

If the second equality is taken in equation (30), and if equation (24) is

inserted for MN/Y_o an expression is obtained which can be solved for

Gg_ valid; of course; only at 8 = 8o; that is_

G9 =_-_ os 8oCOS _ cos _ 4]-k2cos28o

/l-k2c os _ o

- cos 8osin 9 cos _ sin _l_k,2cos29

k±sin @oCOS 80",- cos _ sin _ ", (31)

_l_k'2cos2_ /

at 8 : 8o. The form of the functions F and G at the shock is still_

to some extent_ arbitrary. If it is asked that G be equal to <0 on

the shock_ then this can be accomplished by setting

F : _ in 8oCOS _ cos _ _l-k2cos_o

- eos 8osin _ cos m sin _!I l-k2c°s_9on,_---_-

J l-k cos

k2sin 8oCOS @o k

- cos _ sin _ /2 2COS <D

(32)

The streamlines, upon which entropy is constant, converge near the body

and come together at B. Thus all values of e:_tropy exist at B, a

singular point. The point E is also singular by reasons of symmetry.

The entropy therefore undergoes rapid change i_ the vicinity of the body,

thus leading to the notion of an entropy layer. The streamline emanating

from the shock at C impinges upon the body a_ A, and follovs the body

surface around to B. The velocity v is nornal to the body at points

A and B, and so it is zero there. The body ca_ be completely determined_

then, by plotting the points on AC and BD wh_re v vanishes and by

noting the points where curves S versus q0 (e = const.) cross the line

S(_ = 0o, _ = O) = const.

THE NUMERICAL ANALYSIS

The conical shock problem is solved here by two methods_ one using

the stream-function formulation of the equations of motion and the other

using the velocity-component formulation. The two procedures are pro-

grammed for electronic computers, and for convenience they are termed the

"stream-function program" and the "velocity-entropy program." Descriptions

of the computer programs are given in the following sections_ along with

a brief discussion of their specific applicatT"on and limitations.

A

38

9

The Stream-Function Program

This program has been set up for an Electro-Data 205 Electronic Data

Processing Machine. The program is in most ways identical to the one with

which the calculations were performed in reference 14_ with the extenstion

here to include angles of attack or yaw. Equations (19) to (21), along

with equations (25) to (28), (33) to (36)_ ant (38) to (42), define the

p rob iem.

For a given case, values are prescribed ]or

shock-wave geometry

free-stream Mach number

adiabatic exponent

angle of attack

angle of yaw

3C

17

A

38

5

increment in @, a positive quantity

n number of points

The increment _ is found from the tabulation below_ which is

incorporated in the computer program.

___ am Range of

0 0 £/2n (Z_q0/2)_<qD_<(£/2)- (Z_/2)0 #0 _/n -(_/2)+ (_/2) < $ S (_/2)- (@/2)#0 0 _/n (_/2) _<_ S _ - (_/2)

In practice 15 or 20 points are used, but as few as i0 and as many as 60have been used in some cases. The size of Z_9 is chosen so that 8 or

i0 increments are taken to get to the body. Cases have been run with as

few as 2 and up to 50 or 60 steps to the body. No particular difficulty

is encountered in using the very small increments. This is in contrast

with the calculations of reference 12j where an inherent instability limits

the number of steps to the body.

The calculations are arranged according to the outline below.

Step la.- Compute initial values of p, F_ G, Fo , and G_

at the shock wave using the input values of 8o, bz/al, etc.

Step i.- Compute the derivatives pq_, F@, ,G_, F@Q, GS@.

F_, and G@_0 numerically. Compute S and S'/S. Solve the

equations of motion for PS, Fee _ and GS@. Calculate the

"externally iterated" value of P, designated Pit' using theformula

Inl ]Pit = Pit - T L o "

(At the shock, Pit = P') It is termed "externally iterated"

in that is is computed at each point using only the informa-

tion at hand, and it is not used in the subsequent extrapo-

lations. The pressure p is computed with the relation

7p = Pit S . Pressure, density, and other pertinent flow-field

data are now read out of the computer.

Step 2.- Extrapolate p, Fe, and Ge to the coordinate Qn+l

using the formulas

18

f l)Take these values of F$ n

+

previously calculated results

F and G with the equations

and Gin + z)

F(n) and G (n)

along with the

and extrapolate

F(n + i) = F(n) 4,-+2 }

Step 3.- If any values of the stream function F have

become negative at e = 8n + z go to step 4. Other_ise

repeat steps i, 2_ and 3.

Step 4.- Some values of F are negative at @ = en + z"

Recall the flow data for G = 0 n and extrspolate to the body

with the following equations. The subscript b indicates

results on the body.

F(n) }

meb - Fin)

0b = 8n - AOb

(h_)

(45)

(4_,)

A

385

Yb

zb -

sin _ sin 0b

cos eb Jl - k 'a 2cos O

os<] cos k' cos cp

19

A

38

5

Pb=p(n) Obp n) 1= p p(n) GO ]

Note that Yb and zb are actually ratios (y/x)b and (z/x)b,

and are, as such, projections of the body coordinates r,

_b, and _ onto a plane x = i. These coordinates, along

with Pb _ are now read out of the computer and the case isfinished.

The order of procedure discussed in the foregoing paragraphs can be

seen graphically in figure 5, which is a simplified flow chart of the

computing program.

Note im step 3 of the foregoing outline that the integration is

stopped as soon as the stream function F changes sign anywhere on the

coordinate ellipse On + i. The body is then located by linear extrapo-lation of the data at On to the place where F vanishes. In cases

where the ellipse On is markedly different in shape from the body, the

extrapolated distance, _@b, may be excessively large, resulting in an

inaccurate representation of the body. This situation occurs for very

large values of _ or _, or for small values of bl/al_ or for small Mach

numbers. The computer program was not readily amenable to modification

to overcome this difficulty, which in general prevented close analysis of

the entropy layer on the body. A number of cases were calculated using

this program, and limits on parameter values were generally established

from comparison of results with calculations using the velocity-entropy

program. The latter program has been set up so as to overcome the

difficulty discussed above, and it is described in the following section.

The Ve_odity-Entropy Program

This program was set up for an IBM 704 electronic computer for the

purposes of studying the entropy layer on conical bodies, and as an

independent check on the stream-fumction program. The velocity formula-

tion of the equations of motion, equations (7), is used here. This

formulation is numerically less involved than the stream-function formu-

lation. 0nly first derivatives must be taken numerically in the velocity

formulation_ for example_ whereas second derivatives are required in the

other. The velocity-entropy program is the more versatile of the two in

that the range of _ is arbitrary and need not be precisely _/2 or

radians. It is the more difficult of the two to use, however, in that

the body is found by an analysis of the entropy_ which cannot easily be

programmed for the computer. It is not intended as a production program

and only a few cases have been analyzed in detail with its use.

2O

The initial shock-wavedata are found fro::aequations (25) to (29)

_ith prescribed values of @o, bl/al, M_, 7, _, _, A@, and n. The number

of points n may be in the range 7 _< n _< 50. (The lower limit 7 is

d_ctated by the numerical differentiation sche:ae that is employed here.)

The range of _ is also specified along with 8o_ bl/az, etc., by pre-

scribing values of LI and L2. These two quan Sities are angles and are

put into the relation _ = (L2 - Ll)/n. The values of _ are then

taken at intervals _ in the range (L1 + _@/_-) _ _ _< (Lm - 0/2). For

= _ = 0 it is usual to take L 1 = 0 and La :_ 90 ° , but different values

may be used if desired. An outline of the com)uting procedure is presented

next.

Step la.- Compute initial values of

the shock wave using the given values of

S = p/p 7.

u, v_ w, p, and P at

eo, bl/am, etc. Compute

Ste 2 i.- Calculate numerically the derizatives _, vQ, w<0, Pq0,

S_. Calculate ue, ve _ we, O@, and S_ from the equations of

motion. Compute Pit using equation (43). Compute pressure p

using the formula p = Pit7S. Read current Talues_ at @ = @n_

of u, v_ w, p, p_ and S out of the computer.

_.- Extrapolate velocities_ p_ and S to the next @

coordinate using the formulas

u(n + _) = u(9 _ SOu_n)

v(n + l)= v(n)_ Asv_n:i

w(n + I) : w(n) _ Aswan: (49)

A

38

5

Step 3.- If any values of the velocity v have become posi-

tive at e = 8n+ l go to step 4. Otherwise _epeat steps i, 2,

and 3-

Step 4.- Some values of v (n+l) have become positive. Recall

the flow-field results for the station e = On and continue the

integration using one of the two options A and B. The option

choice is specified along with the initial d_ta @o, bl/al, etc.

21

Option A.- Calculate a new value for _@. The new _0

is some fraction_ say i/5_ of the original value. The integra-

tion is continued now (steps i and 2), and instead of using all

n data points at once, several smaller segments are used. The

segments may overlap, and the integration proceeds on each

segment until all values of S are greater than the value cal-

culated to exist on the body_ or until a pro-set number of

steps have been taken, or until the calculation breaks dow_ as

evidenced by negative S or p values. The size of the small

segments, the number of points between the first points of

succeeding segments, and the size of the new _@ are all

specified with the initial parameters.

Option B.- This process starts at 0 = 0n in the flow

field and uses a reduced value of _0, as in option A. It

depends on the entropy variations being of the form shown in

figure 6, a situation that prevails for elliptic cone shocks

at _ = _ = 0 in the region 0 _ 9 _ 90o , or circular cone

shocks at finite angles _ in the range -90o _ _ _ 90o •

Observe in figure 6 that the derivative S_ is negative over

most of the flow fieldj but that zero values begin to appear

as the integration proceeds. The points of zero slope are

labeled S z and $2. If the data to the right of point $2 on

the On + 2 curve are used in extrapolating to the coordinate

On + 3_ numerical difficulties arise due to extreme values of

S_ in this region, and due also to the fact that the velocity

v passes through zero near S = 0. Divisions by very small

values of v can introduce large errors in the subsequent

calculation of S_ p_ etc., from the equations of motion. These

same difficulties appear in data to the left of points Sz. The

technique in option B is to continue the integration from 0n_

but to retain, at the extrapolation step, only the data lying in

the region between points Sl and $2 on the entropy curve. The

integration is continued in this manner until erroneous negative

values of p or S occur within the range _(Sl)to _($2), or

until all values of S are greater than the body value.

The body is found with either option by converting the _ - 8

coordinates of places where the entropy curves cross the line of

constant entropy that represents the body entropy to y_ and Zb,

using equations (47).

No flow chart is presented for the velocity-entropy program. It is

identical in concept to the one shown in figure 4 for the stream-function

program.

As has been indicated, the numerical data near the body are subject

to the ravages of division by zero and abrupt changes in slope of quanti-

ties whose derivatives must be obtained numerically. These problems occur

to the largest extent in the region nearest the entropy singularity_ and

have a physical basis in the fact that the entropy does indeed change very

22

rapidly in a thin region near the body, and _ost rapidly near and at thesingular point. There does not seemto be aay general numerical proceduresuitable for machine programmingby which ths calculations can be performedwith assurance in this region. Results can usually be obtained over thefirst portion of a body, but only with difficulty over the region endingat the singularity, even with the devices of options A and B. In a givencase it is usually necessary to recompute several times, using both optionsA and B, and with various lengths of segment size in option A in order toget a reasonably complete picture of the entropy variation in the regionnear the entropy singularity.

PRESENTATIONOFRESULTS

Preliminary Calculations and Comparisons

Preliminary calculations for elliptic cDnical shock waves at zeros_uglesof incidence are given in reference i_. The main purpose of thesecalculations was to study the stability and _onvergenceof the numericalprocess, and the effect on the results of variations of 7 and }/6o. It_as found that the calculations are stable a_d convergent for small incre-ments _. Thebodies that support elliptic conical shocks are themselvessome_¢hatflatter than elliptic cones. The change in body shape and surfacepressure with changing Machnumber is as mig% have been anticipated. Thelimiting case of 7 = i, M__ _ corresponds to the so-called Newtonianplus centrifugal theory. A machine result c_Iculated with 7 = 1.001 andM_= i0,000 is comparedwith the simple New_)nian theory, _hich states thatthe surface pressure is proportional to the _quareof the sine of the localbody angle. Laval, reference 19, has calcul_ited pressures on conicalbodies in the limit 7 = i, M_= _. Hepresents a result for the case of@o= 30°, bl/al = 0.6, and _ = _ = 0, the s_e case used for the comparisonin reference 14. These results are brought _ogether in figure 7-

Initial comparisons of calculations usi_ the two computer programsdescribed in the present report were madewi_h the case of @o= 30o,b_/al = 0.6, M_= 6, 7 = 1.4, and _ = _ = O_ The computed body shape

and surface pressure are shown in figure 8. Observe that the results of

the two computer programs differ by a small _nount. Another comparison

of the two computer methods is shown in figure 9 for the case of a cir-

cular shock wave, _o = 30o , at an angle of y_w of 20 ° , and _ith 7 = 1.4

and M_ = i0. This is the case reported in r_ference 13. Again, there is

a slight discrepancy between the results of the two machine procedures.

The velocity-entropy calculation agrees clom_ly with the result of refer-

ence 13, except on the leeward side, which i_ remarkable when it is

realized that Radhakrishnan' s work was carried out on a desk computer.

The differences in the machine calculated results can probably be ascribed

in part to differences in programming proced_res, and to the numerical

difficulty associated with the body extrapolation process of the stream-

f_Iction program. The differences are not great, however, but the velocity-

entropy program is presumed to give the more precise results.

23

The variations of entropy amd velocity components with the coordimate

for the case shown in figure 8 are presented in figures i0 to 13. The

stream_lime pattern in the crossflow is shown im figure 14.

Stocker has calculated conical bodies by a two-stream-function

technique wherein the stream functions are independent rather than depend-

ent variables. _o report has been published by Stocker, as yet, but the

results of ome of his calculations_ as given in reference 20, are sho_a_

for comparison in figure 8. His results agree well with calculations using

the velocity-entropy program.

Mauger_ reference 21_ has applied complex variable theory and the

method of Garabedian (ref. 22) to compute bodies that support conical

shock waves. He presents a calculation for an elliptic conical shock

_ave, and the results of the same case as computed by the method of

Stocker. These are shown in figure 19, along with a calculation by the

velocity-entropy program. The agreement is very good. Note that pressure

has been plotted against the angle _ where _ = arc tan(Yb/Zb).

An interesting facet of the present work is the inclusion of

calculation of a so-called "externally iterated" density. (See eq. (43)

and also ref. 12.) Pressures were computed at the place _ = 0 on the

body for the case shov_ in figure $ using both iterated and uniterated

densities. The results differed by about 1.5 percent at that point.

Comparisons With Experimental Data

Various wind-tunnel experiments have been performed on elliptic

conical bodies (see, e.g._ refs. 23 and 24). The shock supported by an

elliptic conical body is not an elliptic cone, and conversely an elliptic

conical shock wave will be supported by a body that is not elliptic in

cross section. Thus no precise comparison can be made between results of

tests on elliptic cones and the calculations that assume an elliptic

conical shock wave. Some calculations were perfo_ned_ however_ assuming

that the shock waves of references 23 and 24 are elliptic cones. The

resulting bodies and pressures are shown with the experimental data in

figures 16 and 17. Note that pressures are plotted against the angle

where _ = arc tan(yb/Zb). T_o attempts were made in calculating the

conical body to compare with the actual cone tested in the _ = 6

experiments of reference 24, and both results are showm in figure 17.

Comparison With an Approximate Theory

A case was calculated for a circular conical shock wave with

0 o = 30 °, M_ = i0, 7 = 1.4, _ = O_ and _ = i0 °, using the stream-function

program. The resulting body was almost circular in cross section, and it

24

was used as the basis of a computation using the first-order yawed-conetheory given in reference 6. Relevant parameters are, in the notation ofreference 6, M_= i0, 8s = 26.5° , ¢ = 0.1575, and _ = _ = 0.1745. Thesurface pressure, calculated with respect to body axes, is

Pb = 0.217[ i + 0.584 cos(_ + 90o)] (5o)

This expression is converted to shock-fixed _xes by use of the relation

+ 9o°) = + - + 9o°) (51)

where

= tan-!0sCOS( + 90D) (52)

The machine-computed surface pressure and th_ approximate result are shown

in figure 18. These calculations are slightly inconsistent in that the

machine computation was carried out using 7 : 1.4, whereas the tables,

reference 6, use the value 7 = 1.405.

Systematic Variation of Parameters

A number of cases have been calculated _ith the stream-function

program, wherein one of the initial parameters was systematically varied

while the others were held fixed. The accurmcy of these calculations was

then checked by comparing a fe_ cases with cDmputations performed using

the velocity-entropy program. Each of the t_o programs has presented

difficulties which limit the quantity and qumlity of obtainable results.

The stream-function program produces accurate results only where the

increment hSb is small over the whole extrmpo!ation region. This, forpractical reasons, places lower bounds on H_ and bl/al and upper bounds

on the angles _ and #. This problem has been avoided in the velocity-

entropy program, but the analysis of the entropy data is very time-

consuming, and so the number of cases that c_n reasonably be considered

is small. The results presented here agree _losely by the two computer

programs.

There are many ways in which the initial parameters can be

systematically varied. No attempt is made here to do more than sample a

few combinations, however. The cases presented are intended mainly to

illustrate numerical techniques. The parameter combinations that have

been chosen are summarized in tables I to IV, along with the numbers of

the figures where the results are plotted.

C

_5

CONCLUDING REMARKS

A

38

5

The inverse method has been applied to the problem of calculating

bodies that support elliptic conical shock waves at finite angles of

incidence. The entropy layer at the surface of such bodies has been

investigated numerica!ly_ also. T_o machine computing procedures were

employed_ and it is presumed that discrepancies of results are due

primarily to differences involved in the programming of the two procedures.

The application of two stream functions_ one of which vanishes on the

body_ is a powerful technique in the solution of the three-dimensional

conical flo_ problem. Future attacks on the problems of more complicated

three-dimensional flows may employ two stream functions with advantage.

The requirement that the shock wave be an elliptic cone makes it

difficult to compare calculations with results of actual tests on conical

bodies. A logical extension of the present work would be to develop

techniques whereby the present assumed shock shapes could be perturbed

so that a body of prescribed shape could be found in a few tries.

The major difficulties that have been encountered in the machine

computations could be overcome to a large extent if the segmenting

processes of the velocity-entropy program and the two-stream-function

formulation of the problem were incorporated into a single computing

program. Thus good accuracy would be assured_ even for very thin conical

bodies_ and the plotting of final results could be accomplished in a

reasonably short time.

Ames Research Center

National Aeronautics and Space Administration

Moffett Field_ Calif._ July 26_ 1960

26

REFERENCES

i.

.

o

4.

.

.

.

.

.

0.

ii.

12.

3D

Busemann, A. : Drucke Auf Kegelf_rmige _;pitzen bei Bewegung nit

Uberschallgeschwindigkeit. ZAMM, vol. 9, 1929, p. 496.

Taylor, G. I., and Maccoll, J.W.: The Air Pressure on a Cone Moving

at High Speeds. - I. Proc. Roy. S,c.. series A, vol. 139,

Feb. 1933, pp. 278-311.

Busemann, A.: Conical Supersonic Flow _rithAxial Symmetry.

Luftfahrtforschung, vol. 19, no. 4, J_me 1942, pp. 137-144.

(Ministry of Aircraft Production R.T._'. Tr. 1958).

Staff of the Computing Section (under the direction of Zdenek Kopal):

Tables of Supersonic Flow Around Cones. Tech. Rep. i, Center of

Analysis, M. I. T., Cambridge, 1947.

Stone, A. H.: On Supersonic Flow Past _. Slightly Yawing Cone.

Jour. Math. and Phys., vol. 27, no. i Apr. 1948, pp. 67-81.

Staff of the Computing Section (under tl_e direction of Zdenek Kopal) :

Tables of Supersonic Flow Around Yawing Cones. Tech. Rep. 3,

Center of Analysis, M. I. T., Cambrid_e, 1947.

Staff of the Computing Section (under tl_e direction of Zdenek Kopal) :

Tables of Supersonic Flow Around Conel at Large Yaw. Tech. Rep. 5,

Center of Analysis, M. I. T., Cambrid_e, 1949.

Ferri, Antonio: Supersonic Flow Around Circular Cones at Angles of

Attack. NACA Rep. 1045, 1951. (Supe_'sedes NACA TN2236)

Ferri, Antonio: The Linearized Charactc:ristics Method and Its

Application to Practical Nonlinear S_ersonic Problems. NACA

Rep. 1102, 1952. (Supersedes NACA TN 2515)

Ferri, Antonio, Ness, Nathan, and Kapli_ a, Thadeus T.: Supersonic

Flow Over Conical Bodies Without Axia[ Symmetry. Jour. Aero. Sci.,

vol. 20, no. 8, Aug. 1953, p. 563.

Ness, Nathan, and Kaplita, Thaddeus T.: Tabulated Values of Linear-

ized Conical Flow Solutions for Solution of Supersonic Conical Flows

Without Axial Symmetry. PIBAL Rep. No. 220, Polytechnic Institute

of Brooklyn, Dept. of Aero. Eng. and _pl. Mech., Jan. 1954.

Van Dyke, Milton D., and Gordon, Helen I_.: Supersonic Flow Past a

Family of Blunt Axisymmetric Bodies. NASA Rep. i, 1959.

Radhakrishnan, G.: The Exact Flow Behilid a Yawed Conical Shock.

Rep. No. 116, The College of Aeronautics, Cranfield, Apr. 1958 .

A

8

5

27

A

38

5

14.

15.

16.

Briggs, Benjamin R.: Calculation of Supersonic Flow Past Bodies

Supporting Shock Waves Shaped Like Elliptic Cones. NASA TN D-24,

1959.

Yih, Chia-Shum: Stream Functions in Three-Dimensional Flows.

Reprint No. 158, State Univ. of lowa_ Reprints in Engineering,

July-August 1957.

Cheng, H.K.: Hypersonic Shock Layer Theory of a Yawed Cone and OtherThree-Dimensional Pointed Bodies. WADC TN 59-335, Oct. 1959.

17. Kraus, L.: Diffraction by a Plane Angular Sector. Ph.D. Thesis,

New York Univ., May 1955.

18. Ames Research Staff: Equations_ Tables, and Charts for Compressible

Flow. NACA Rep. 1135, 1953.

19. Laval, P.: Ecoulements New%oniens Sur des Surfaces Coniques en

Incidence. La Recherche Aeronautique, no. 73, November- December

1959, pP. 5-16.

20. Van Dyke, Milton D.: Some Numerical Solutions in Hypersonic Flow.

Colston Papers, vol XI, Proc. llth Symposium of the Colston Res.

Soc., April 1959.

21. Mauger, F. E.: Steady Supersonic Flow Past Conical Bodies. A.R.D.E.

Rep. No. (B)3/60, Fort Halstead, Kent, England, May 1960.

22. Garabedian, P. R.: Numerical Construction of Detached Shock Waves.

,our. Math. and Phys., vol. 36, no. 3, Oct. 1957, PP- 192-205.

23. Chapkis, Robert L.: Hypersonic Flow Over an Elliptic Cone: Theory

and Experiment. Hypersonic Research Project Memo. No. 49,

Guggenheim Aeronautical Lab., Calif. Inst. of Tech., May 1959.

24. Zakkay, Victor, and Visich, Marian, Jr.: Experimental Pressure

Distributions on Conical Elliptic Bodies at M_ = 3.09 and 6.0.

PIBAL Rep. No. 467, Polytechnic Institute of Brooklyn, Dept. of

Aero. Eng. and Appl. Mech., March 1959.

28

TABLE I.- MACH _JMBER VARIATION AT F_<ED VALUES OF bl/al

AND WITH _o = 30o AND AT _{0 INCIDENCE

bl/all Hach no.0.6 i0, 20, i00

•7 l 5, i0, 20, i00

.8 1 5, i0, 20, i00•9 3, 5, i0, 20, i00

Figure

192O

21

22

TABLE II.- VARIATION OF bl/al AT FIXED VALUES OF 8o AND

WITH M_ = i0 AND AT ZERO _INCIDENCE

30,

degb i/al Figure

i0 0.8, 0.9, 1.0

15 0.7, 0.8, 0.9, 1.0

30 0.6, 0.7, 0.$, 0.9, 1.0

40 0.8, 0.9, 1.0

50 0.8, 0.9, !.0

2324

2526

27

TABLE III.- ELLIPTIC CONICAL SHOC]{, 8 o = 30°, AND

WITH M_ = i0, AT FINITE INCIDENCE

_, 5, Figuredeg deg

O, i0 0 28

0 O, 5 29

TABLE IV.- CIRCULAR CONICAL SHOCKS WIT_ M_ = i0 AND WITH

VARIATIONS OF 5 FOR FIXED V\LUES OF 8 o

80,

deg

i0

15

3o4o

5o

_ F[guredeg

O, 2.5

O, 2.5, 5

O, 2.5, 5, i0, 20

O, 5, i0, 20

O, 5, i0, 20

3O

31

32

33

34

29

A

38

N

x

x

/

f-O

t-O

i-0

1-0

c;

+_

4o

oo

o

_c_

4j

oo

o

o

!

o

30

vI

A

3$

5

Figure 2.- Velocity components in the sph,_ro-conal and Cartesian

coordinate systems.

31

A385

Mx

z

I/Shock wave

Figure 3.- Cartesian components of the free-stream Mach number.

32

_=O

Shock Wove

/

//

/Streomlines

E B

A

38

Figure 4.- Streamlines in the coni_:al flow field.

5C

33

.k

33

5

} START

Load specific values of 8o,

bl/al_ a, _, Moo, _', AS, andn for the case at hand.

No

Compute and print out body

coordinates and pressure

using equations (46-48).

This case is finished. Go toSTART for next case.

Calculate shock wave data

using equations (25)-(28),

(33)-(36), and (38)-(42).

Compute _-derivatives

numerically. Calculate PS,

FS_ , G88 using equations(19)-(21).

Compute Pit and p. Print

out p,/o, F, G and S, andother flow-field data.

i

LExtrapolate _n-data to

coordinate _n+l using equa-

tions (44)-(45).

I

Figure _.- Flow chart for the stream-fu_ctiom program.

3_

Points SIEntropy ot the body 7

II

Singular point7/

/

//

/

S

A

38

0<_<90 ° for elliptic conical shock, e=/_= 0

-90<_ <_90 ° for circulor conicol shock, a=O, /_>0

¢

Figure 6.- The variation of entropy with the coordinate _ in the flow

field behind elliptic conical shocks at zero incidence, or circular

shocks at positive angles _.

35

.5O

.45

.40

Cx

oo

if)

.35

.3O

.25

Shock wove, 80=:50 ° , Moo =10,000,

y=l.O01, bl/o I =0.6, ,,=/3=0

,---- Moch ine colcu

b

ef. 19

imple Newtonion theory_

lotion

0 15 30 45 60 75 90

@, deg

Figure 7.- Comparison of surface pressure in the limiting case of

Moo _ _ and 7 _ i with simple Newtonian theory and the theory of

reference 19-

S6

zb

I0

9

8

7

6

5

4

0

,, Shock wove, 8o=30 °

," bl/al=0.6, Moo=6

7= 1.4, a=/_:o

\

, \

programBody

--Stream- function.... .Velocity -entropy

0 Stocker's result

.3

.2

.I program __t

L

Yb

A

38

5

Figure 8.- Comparison of body shapes for au elliptic conical shock wave

as calculated by the two computer programs _nd the method of Stocker.

.6

.4

z b

.2

0--.6 -.4

Stream-funct=on program

----Velocity-entropy program

0 Ref. 13

," Shock Wave, 8o=30 °

_ MOO=10, 7"= 1.4,

-.2 0 .2

Yb

.4

(a) Body and shock wove shapes

37

.6

.6

.,13Q.

(.o(/)

ct

G)

2=" .23

03

8o; 30? Moo= I0,

7,= 1.4, a=O, B=20°

Stream-function

---- Velocity_entropy

0 Ref. 13

\

program

p rag ro rr

0 _"--"_" _--'_-90 -60 -30 0 30 60 90

_, deg

(b) Surface pressure

Figure 9-- Body and surface pressure for a circular conical shock wave

at a finite angle of yaw.

38

.9O

.86

.82

"I

O

.7 8

.74

.7 02

f

J f

e = 50.0 °

= 24.0 °

= 25.6 °

= 23.4 °

=25.2 °

A

38

5

0 15 50 4 5 60 75 90

(_,deg

Figure i0.- The variation of the velocity component u In the flow field

behind the shock for the case where @o = 30o , M_ = 6, bl/a i = 0.6,7 = 1.4, _ = _ = O.

39

.02

A

38

5

O_ j j1_f

• "_ 23 .4 °-.o2 ,:,........",, ',,' 2:5.6 °

',,\ .... 2:5.7 o\ .... 24.0 °

-.04

> --.06

,4--

01

> -.08

-.10

-- .12

-- .14

,,r--8=26° _ -

,---e=28 °/

,'-- 8 =30 °/ Shock wave

--"160 15 :50 45 60 75 90

8, deg

Figure ll.- The variation of the velocity component v in the flow field

behind the shock for the case where @o = 30o , _ = 6, bl/a I = 0.6,

7 = 1.4, _ = _ = O.

4o

28

.I

00

>

.24

2O

16

12

O8

O4

0

Shock wave\k

=28 °26 °

/15 50 45 60 75 90

Figure 12.- The variation of the velocity conponent w in the flow field

behind the shock for the case where 8o = 30 ° , _ = 6, bl/a I = 0.6,

7 = 1.4, _ = _ = O.

41

.048

3

I /Body I '

,,.046 _--"'" r_\" j' I/ ,l if"8=23.2 o I I

I

.044 I

23.4 ° I I

.042 _ ," 8=23"6°/ ,

O= 30°- _ I=280-" I

..... , /•040 24 o _____--_-____._

00

_.o3a

w Shock o

.036

\.034

.032

.030

-- Extropoloted portions

>

0 15 30 45 60 75

¢, deg

9O

Figure 13.- The variation of entropy in the flow field behind the shock

for the case where _o = 30°, M_ = 6, bz/a I = 0.6, 7 = 1.4, _ = _ = O.

42

hO

9

.8

Body,•7 S = .0461

.6

Zb .5

.4

.3

.2

.I

0

wove

Streomli'les,S = .045

= .042f _

j.-- - . 040

=,038

.036

=.034

"= .032 --

.2 .3 .4 .5 .6 .7

Yb

Figure 14.- Streamlines in the crossflow between shock and body for the

case where eo = 30° , 9_o = 6, b_a I = 0.6, 7 = 1.4, _ = p = 0.

A

38

5

A

38

5

43

.6

.Shock wave, 80 = 21.8

,5 __../.-" bl/al=O.8, Moo=lO '

-_ _ 7: 1,4,a=/9:o

% \. \,

.5 -- Ref 21, Stocker"__._ _zb Ref 21, Mauger and "_

velocity-entropy ./".2 program -- _"

l/0 ,I .2 .4

,18

(/)u)

a.

03

.12

.5Yb

0

.5

(a) Body and shock wave shapes

.6

.-Ref 21 and velocity-entropyf

_" program results nearly

nticol.

0 15 50 45 60 75

_, deg

(b) Surface pressure

9O

Figure 15.- Comparison of body shape and surface pressure for an elliptic

comical shock wave as calculated by the velocity-entropy program and

the methods of Stocker amd Mauger.

44

.25

.2O

.15

zb

.10

.O5

Moo = 5.8, 7'= I._'

a= 3=0

-- Ref. 25

\\'_#_'_ I 80= 11"87°__b I/a,= .791

',\

.O8

- .06

.04I1)

O

L.

if)

.02

0 .05 .I0

Yb

(a) Body shapes

.15 .20

! I

Moo=5.8, 7,=1.4, a=/£'=O

I I

,-- Ref. 257 0

,/---I eo =11"87Ibl/a I= .791

10 15 50 4.5 60 75 90

_,deg

(b) Surface pressures

Figure 16.- Comparisoms of calculations with experiments on am elliptic

cone of unit length having semiaxes of lengths 0.104 and 0.208.

A

385

25

.5

.4

.5

Zb

.2

.I

0

n_

Ref. 241/ \_.__

80= 18.62 ° "_// \_bl/aa =.720._

/

/

8o=17.47 ° "_/"bi/a i = .677_

.IYb

¢: 1.4

=0

.2

.5

.5

z b

.2

.I

.4 _ Moo =5.09,

_ ,,,'_ a =,8=0

Ref 24 "/ _k

8o= 24.71 ° _/,_

bl/a I=.860 t

'll\\

.3 0 .2

Yb

7"=1.4

.3

(a) Body shapes

.25

_-.20

S..15C)£)

w,-

(.n .10

.05

\

C

/

/

/'/

"<22

0 9O

Mco:6, 7,- • / [ "-- ---.,

Ref. 24 --/ J

8o = 18.62 (, {//i

bl/a I = .720)

I

15 50 45

_/, deg

!

Moo: S.09, 7: 1.4, a :/9:0

--Ref. 24 I

8o= 24.71bllai= .860

"-----Jr .......

_ 7Oo:,7.47°t/-

bl/al =.677 )I

60 75

(b)Surface pressures

Figure 17.- Comparison of calculations with experiments on an elliptic

cone of unit length having semiaxes of lengths 0.225 and 0.402.

48

.5

zb

.4 --

.8 __"_. F---Shock wave, 80=30 °,

bl/a I =0.7, 7 = .4,

\,

.3

.2

0

.40

/ // , /

Body _ /M_o=5 -j /

:20 -J /

-->'O0--J• I .2 .3 .4 .5 .6

Yb

(a) Body end shock wove shapes

o._ .55

_ .3O

_ .25

o')

_. 0.7, T 1.4

.2O

0

\

I80= 30 °, bI/al :o=B=O

-- M_=5

",<..15 30 45

f5, deg

q_

60 75 90

(b) Surface pressures

.7

Figure 20.- The effect of Mach number variation on body shape and surfacepressure; bl/a I = 0.7 series.

A

38

5

C

8

49

.7

.6

.5

A

38

5

zb.4

.3

.2

.2 .3 ,4 .5Yb

(a) Body and shock wave shapes

.6 .7

.32

r_c_ .28

.24r',,

(.}0

.20if)

fl LMoo=5L---->IO

0 15 30 45 60 75 90

8,deg

(b) Surfoce pressure

Figure 21.- The effect of Mach number variation on body shape and surface

pressure; bm/a I = 0.8 series.

5o

.7st-- ....

/

/

.5

| !

Shock wave, 8o= :50°j

bl/al=0.9, 7=1.4, a=/_=O

.4

zb

.:5

.2

.I

\

20 "/

_100" !

t' /.I .2 .:5 .4 .5 .6

Yb0 .7

(o) Body and shock wave shapes

_..28

.24

.20

if)

-_ 5///

>_10/

i 0 ! I

8o=:50, bl/al :0.9, 7"=1.4a=_3=O

I I

0 15 :50 45 60 ?5 90

8, deg

(b} Surface pressure

Figure 22.- The effect of Mach number variation on body shape and surface

pressure; bl/a I = 0.9 series.

A

38

.2O

.18

.16

.14

.12

zb

.I0

.08

.06

.04

.02

0 .02 .04

Shock wave, 8o: 10°

Moo = 10°, 7" = 1.4,a=p:0,(not shown)

\

.06 .08 .10Yb

(a) Body shapes

Body,._bl/al =0.8

_-- - =0.9\,/

\ ,\

\

.12 .14

51

.05

_ .05

0

_ .02,

0 15

8o=10 °, Moo=10, 7=1.4

a=_ =0 /_bl/al =0.8

/- =0.9/ r= 1.0

, I/

50 45 60

_, deg

(b) Surface pressure

75 90

Figure 23.- The effect of the variation of bl/a I on body shape amd

surface pressure; eo = i0° series.

52

.36

.32

.28

.24

.2OZb

.16

.12

.O8

.04

I I I

Shock wave, 8o=15°

Moo =10, )"=1.4,a=,8=0

" _(not shown)._ Body,

_ //_1/01 = 0.7

" "_ ,---=0.9

A

38

5

0 04 O 8 .I2 .I6 .20Yb

(a) Body shape

.12 I

..Q

" .10 i\

i "_- bl/al =0.7 --= =o.8,_ '_ ,'._ 7=0.9

o"®S.08 -------.'_

_.06 i(,/')

8o=15° Moo=lO,7= 1.4,

.24

I

.04, I' I

0 15 30 45 60 75 90_, deg

(b) Surface pressure

Figure 24.- The effect of the variation of bl/a I on body shape and

surface pressure; eo = 15° series.

A

3$5

.9

.8

.7

.6

.5

zb

.4

.3

.2

0 .I

I I I

Shock wove, 80=50 ° ,MG0=I0, 7=1.4, a=_=O,

_-- (notlshown) lBody,

f4-- bl/O I =0.6

....... 0.8

\/ _-- --0.9

.2 .3 .4 .5

Yb

(o) Body shope

.6

53

.5

J_

_.4

L.

.32L..

80=30 ° , Mc0=I0, 7=1.4,a =B =0

"_ ....... bI/ol--0.6

=0.7

_ --O.B...... 0.9

0 15 30 45 60 75

4, deg

(b) Surface pressure

Figure 25.- The effect of the variation of bl/a I

surface pressure; Oo = 30o series.

9O

on body shape and

54

Zb

1.0

.8

.6

.4

,2

Shock wave, eo = 4 0 °,

Moo=lO, T =1-4, a =_ 0

(not shown)

Body,

//----bl/al = 0.8

_/- ..... 0.9

r-- = ,.o

A

38

5

o .2 ,4 .6

Yb

(a)Body shapes

.8 1.0

.5

c_

Q)

_.4

{3.

13

o3.5

i Ieo = 40°,M oo= IO,T= 1.4,

= 0 r---bl/al = 0.8¢_-, F 0.9

0 15 50 45 6C 75 90

, deg

(b) Surface pressures

Figure 26.- The effect of the variation of b_a 1 on body shape and

surface pressure; eo = 40° series.

A

38

5

12

I0

.8

Z b .6

.4

.2

! i

Shock wave, aO=50 °

Moo =10, 7=1.4, a=/£=O(not shown)

_ Body,

bila I=0.8

, .... 0.9

...... 1.0

55

0 .2 .4 .6

Yb

(a) Body shape

.8 1.0

.7

-1

u_ .6U)G)

(3.

(J0

o .5

10

l I !

80=50 °, Mco=lO, 7=1.4,a=/_=O

/--_-b I/a I = 0.8, -- .... =

/

1.0

/

15 50 45 60 75 90

¢, deg

(b) Surface pressure

Figure 27.- The effect of the variation of bl/a I on body shape and

surface pressure; eo = 50 ° series.

56

1.0/ Shock wave, 80=3) °

_._. bl/al=0.6, Moo=6

.e _ ""N)'="4' _=0

.6 "?'_

.2

Z b

0

-.2

-.4 /

-.6 , /

-.8

-I.00 .2 .4 .6

Yb

(o) Body shapes

.8

.6

.2O

"c

i

eo:3o':bt/al:O.6 , M(_: 6,

7=1.4, _'=0

,-'a=O/

//

"-'--'--7'

0 30 60 90 120 150 180

_), deg

(b) Surface pressures

Figure 28.- The effect of _ on body shape and surface pressure; an

elliptic conical shock wave.

A

38

5

BC

57

A

38

5

1.0

.8

.6

zb

.4

.2

0--°6

Shock wave, 80--

bl/a 1:0.6, Mm = 10,)'=1.4, a=0

-.4 -.2 0 .2

Yb

(a) Body end shock wave shapes

.4 .6

.6

¢¢.4i-

(/}

t.=Q.

_.2

(/)

,8=0_ )'= 1.4, a:Oeo:30 °, bl/al=0.6, Mw=6,

0-90 -60 -30 0 30 60 90

_, deg

(b) Surface pressures

Figure 29.- The effect of _ on body shape and surface pressure; an

elliptic conical shock wave.

58

.2

z b

.I

0u2

Shock wave, 8o =10 °, ',

a= 0 _f___

I

-.I 0 ,I

Yb

((]} Body and shock wave shapes

,2

A38

5

D6

--- .04

t,.,.

t',,

0

"_ .ozZ9

0

8o=1O °, Moo:lO, _':l.4,a=0

- 90 -60 -30 0 30 60 90

(_, deg

(b) Surfoce prossures

Figure 30.- The effect of _ on body shape and surface pressure;

circular conical shock with @o = i0°-

A38

5

.3

.2

z b

.I

0-.3

Shock wave,80=15° .... ,

.2 -.I 0 .I

Yb

(a) Body and shock wave shapes

.2 .5

59

.12

.o8

.o4

(/3

80 = I 5°,Mco=lO, T=l.4, a =0

0-90 -60 -50 0 50 60 90

4, deg

(b) Surface pressures

Figure 31.- The effect of _ on body shape and surface pressure;circular conical shock with eo = 15°.

6o

.6

4

Zb

2

O--.6

also)

.4 .6

(a) Body and shock wave shapes

A38

5

.6I I r

2

.4_

80=30 °, Moo=lO, 7"= 1.4_ a--O

//----I/_= 0 I

lJ'-'---'_ : 2.5°/ \/ r-Jr - = 5.0 i

/\/ I o_, ,,\,-_ : ,o.o

0 _-90 -60 -50 0 5D 60 9(

_, deg

(b) Surface pressures

Figure 32.- The effect of _ on body shape and surface pressure;

circular conical shock with 7o = 30° .

61

k

38

5

I.o Shockwove, _o=4o,_, I

Moo=IO_ 7'=1.4, _

Zb.5 a=O /_

i-I,0 -.5 0 .5

Yb

(a) Body and shack shapes

1.0

.8

._.6{3.

P

(3

03

.2

/"

/ /

(/

0-90 -6(

I

/ B : 20°

// =10 °

//=5 °

_ //=0° I

\

8o=40 °, Moo= I0, _,= 1.4, a=O

-50 0 30

_, deg

6O 9O

(bl Surface pressures

Figure 33-- The effect of } on body shape and surface pressure;

circular conical shock with 6o = 40 °.

62

1.2

.8

Zb

.4

.4 .8 1.2

Yb

(a) Body and shock wave shapes

1.2

.8

£_.

2

O0

//9 =20 ° 80=50 °, Moo= I0, 7 = 1.4

/ /= I0 °/ _ =0

' / =5 ° I--/./. = 0

"--%i

-90 -60 -30 0 30 60 90

, deg

(b) Surface pressures

Figure 34. - The effect of _ on body shape and surface pressure;circular conical shock with 6o = 50o.

NASA- Langley Field, Va. A-385