UNCLASSIFIED

AD 295 992

ARMED SERVICES TECHNICAL INFORMATION AGENCYARLINGTON HALL STATIONARLINGTON 12, VIRGINIA

UNCLASSIFIED

NOTICE: When goverment or other drawings., speci-fications or other data are used for any purposeother than in connection with a definitely relatedgovernment procurement operation, the U. S.Government thereby incurs no responsibility, nor anyobligation whatsoever; and the fact that the Govern-ment may have fornnlated, furnished, or in any waysupplied the said drawings, specifications, or otherdata is not to be regarded by implication or other-wise as in any manner licensing the holder or anyother person or corporation, or conveying any rightsor permission to manufacture, use or sell anypatented invention that may in any way be relatedthereto.

I

F 295 992

A CRITICAL STUDY OFTHE DIRECT BLUNT BODY INTEGRAL METHOD

f DECEMBER 28,1962 DOUGLAS REPORT SM-42603

MISSILE & SPACE SYSTEMS DIVISIONDOUGLAS AIRCRAFT COMPANY, INC.SANTA MONICA CALIFORNIA

-

- , i

/

MISSILE & SPACE SYSTEMS DIVISION

DOUGLAS AIRCRAFT COMPANY. INC

SANTA MONICA, CALIFORNIA February 8 1963A2 -260-W/AT4W-5

Subject: Transmittal of Douglas Aircraft Cc pnyi Inc., ResearchReports Ox-424ige, mEe-6O3, aid S 69

To: ueadquartersArmed Services Technical Information AgencyAir Force Systes CimUnited States Air ForceArlington Nal StationArlington 12, Virginia

Attention: TIPAJoseph BielChief, Accessions BranchDocument Processing Division

1. It is requested that the folloving unclassified reportsbe listed on any apprpriate publication lists or indexes distributedby ASTIA, such as the Technical Abstract Bulletin. Distribution ofthese reports to any interested persons or organizations is desired.

Shanehan, R. J., and G. R. loger. Nco -11us Centered

i~m on of a Dissociated Bt oi Ga lw WTgLOctober 26, 196.

Xerikos, J., and W. A. Anderson. A Critical Study of the DirectBlunt jo& Intepl Nbthod. W.JeS33, December 28, J9W.

Ormsbee, A. I. ressure Distribution on a Mile AcceleratedDeeme 21,a1 .e ehlme.S- ,i

2. Pe tssion is hereby granted to AUI& to forward thesereports to the Office of Tecunieal Servies for the printing ad saleof copies provided they are repodued in theiWr entirety ad the DouglasAircraft Company, In., designations are retained.

Boulaa Airaraft Campmny, Inc.

1. T. Pensford, lef Bginsers dv o lssile Technolog

Baclosures as notedcc: (v/o enclosures)

Al Plant MMWesentative, R AC, A

A CRITICAL STUDYOF THE DIRECT BLUNT BODY INTEGRAL METHOD

DECEMBER 28, 19621 DOUGLAS REPORT SM-42603

Prepared by:

J. Xerikos

Specialist-Gasdynamics ResearchW.A. Anderson1Member Of Applied Research Group

Approved by: Missile Aero/Thermodynamics Section

/ J.W. HindesChief, Missile Aero/Thermodynemics Section

PREPARED UNDER THE SPONSORSHIP OF THEDOUGLAS AIRCRAFT COMPANY INDEPENDENTRESEARCH AND DEVELOPMENT PROGRAM

ACCOUNT NUMBER 82160-176i IS MISSILE SYSTEMS ENGINEERING I MISSILE & SPACE SYSTEMS DIVISION

DOUGLAS AIRCRAFT COMPANY, INC.

_

I

A representative formulation of the one strip blunt body integral method

is treated in sufficient detail to delineate the numerical difficulties

involved in its mechanization as well as to identify limitations in its

application. While the integration of the governing system of equations

for general body shapes comprises a two-point boundary value problem,

the particular case of a flat-faced cylinder is shown to reduce to a pro-

cedure involving a single integration. An investigtion is made of the

extent to which conservation laws are satisfied throughout the shock

Ii layer region. Significant deviations are found to exist depending upon

the method used for computing the distribution of flow properties across1i the shock layer. A procedure has been chosen which insures conpatible

tangential and normal flow property gradients at the shock and body while

significantly reducing mass and momentum defects in most instances. Com-

puted flow field data are presented for spheres and flat-faced cylinders.

Ni

IIII

TABLE OF CONTENS

Section Page

Abstract . . . . . . . . . . . * 0 0 * 0 a 9 * 0 0 0 0 * *iii[ Nomenclature . . . . . . . . . * s . * .. .. . .. * . ix

1.* Introduction . . . . . . . . * & . . . . 9 * * * * * a * . 1

.2. Development of Equations. .. ... ..... .. .. . 2

I 3. Integration Procedure ........... .,. . . . i.i. 1

3.1 Integration Scheme: 04r, 1 /:Z Continuous at r -,r 13

3.2 Integration Scheme: 0o$/o(/' Discontinuous at r-j " . 13

3.3 Initial Values . . . . . . . . . . . . . . . . . . . . . .14

1 3.3.1 Axisymmetric Case . . . . . . . . . . . . . . . . . . . . 14

1 3.3.2 Two-Dimensional Case - Smooth Bodies . ..... . * . .15

4. Convergence Scheme . . . . . . . . . . . . . . . . .. . . 16

4 .1 Initial Value Sensitivity Study .. .. .. .. .. . .. 16

4.2 Program Degrees of Freedom . . . . . . . . . . . . . . . . 25

5. Convergence Scheme - Bodies With Sonic Corner . . . . . . 35

6. Inverse Procedure . . . . . . . . . . . . . . . . . . . . 36

6.1 Iterative Method . . . . . . . . . . . . . . . . . . . . . 36

6.2 "Idnear" Method . 37

6.3 Gradient Method . . . . . . . . . . . . .. . .... 0 . . 38

7. Conservation Relations .. .... ......... . 40

-1

1

TABLE OF CONTENTS

Section Page

T.1 Sphiere Results . . . . . . . . . . . . . . . . . . . . . . 42 .7.2 Flat-Faced Cylinder Results a . . ..*. * . .. . . . 12 "4

8. Applications ................ . . . . . .. 47

8.1 Sphere . .. . . . . o o o . o . o . . . . . . . . 47

8.2 Flat-Faced Cylinder . . . . .. . .. .. . . .. 51

8.3 Nose Caps . o . o . . . . . . . . . .. . o . . . . . . . 58

8.4 Two-Dimensional Results .. . .. .. . . . . . o o 58

9. Concluding Remarks . . o . .. o. . . . . . . . . . . .. 63

References o . o . .. o. .. o. . . .. .. . . . . . 64

Appendix A . . o o . o o . .. . . . .. . . . .. o. ... 66Appendix B . o. o. . . . . . ........... . 7 4

271

|I

~H

J LIST OF FIGMUS

[ Figure Page

1 t,/ ] Coordinate System . . . . . . . . . . . . . . . . 3

2 Definition of Genetric Variables . . . . . . . . . . . . 5

3 Effect of Convex Corner ................. 12

4 Zone of Influence of Body Curvature Discontinuity .. .. 12

5 6 Sensitivity Study: &2Zfo./0 Versus * ........ *18

6 6Sensitivity Study: A/ Versus f.J . . . . . . . . . . 19

7 t5 Sensitivity Study:D Versus . ..

4 Versus IV . . . . . . . . . . . . . . . . . . . 21

9 C Versus %S . . ............... * . . . . . . . . 22

10 Halving Mode . . . . . . . . . . . . . . . . . . . . . . 24

11 Versus -S for . ...... ..... 26

12 C Versus/V : 4 Fixed and Variable . ..... 28

13 /l 1 Versus -' . . . . . . . . . . . . . . . . . . . .29

14 Effect of Fit Distance on 1 ........ .. 30

15 Effect of Fit Distance on % . . . . . * * * * * * . *31

16 Effect of Fit Distance on Downstream Values . . . . . . . 32

17 Control Volune . . . . . . . . . . . . .41

18 Shock layer Mass Balance: Sphere . . . . . . . . . . . . 43

19 Shock layer Mlnentum Balance: Sphere . . . . . . . .

A

LIST OF FIGURES

Figure Page

20 Shock layer Mass and Momentum Balance: iFlat-Faced Cylinder . . . .. ... . . . . ... .a 45

21 Shock Detachment Distance: Sphere ( 9= 1.4) . . . . . . I22 Sonic Point Location: Sphere ( e,= 1.4) . . . . . . . . . 49

23 Shock Wave Curvature at Axis of Symmetry .. . * a . 50

24 Shock layer Pressure Distribution: Gradient Method • • • 52

25 Shock layer Density Distribution: Gradient Method . . . . 53

26 Shock layer Mach Number Distribution: IIGradient Method ... ................ 54

27 Ccmparison of Shock Shapes Derived Frcm sCharacteristics and Integral Methods ........ 55

28 Surface Pressure and Velocity Distributionsfor Ccuplete and Truncated Spheres ......... 59

29 Shock Shapes: Flat-Faced Plate and Cylinder . . . o . . . 6030 Pressure and Velocity Distributions:

Flat-Faced Plate and Cylinder . . . . . .. .. . .. 61

Al Definition of Gemetric Parameters o * .. . . .. . . 67 1A2 Stream Tube Geometry . ..... o o o . . . . . . . . 71

Bi Sonic Behavior of Velocity Gradient Expressions o o o . . 74

B2 Comparison of Exact and L'Hospital Values for 0/10&. . 76

B3 Sensitivity Study: O/V/6& Versus %F . . . . *..... . .79

Bi Sensitivity"Sty: /I4 VersusJ .r . . . o. o..o. 80

Vill

N1I.ENCIATURE

F- Symbol

Speed of sound

h EnFthalpy

M A Mach number

/40 Pressure

9Body radius of curvature

I: Shock radius of curvature

Gas constant

Polar coordinates

[ 5 /Body oriented curvilinear coordinates

Entropy

I Z~Velocity ccmponent

[ Total velocity

Cylindrical coordinates

[ Ratio of specific heats, 40e

Distance from body to shock measured normal to[body surface

CShock detachment distance at axis of symmetry

[ Angle between tangent to body and axis of symmetry

Angle between tangent to streamline and axis of[symmetry

Ix

I

N IATITIITNMNIATMUM

Symbol

1bo Mach angle, sin T

.14 Density

( Angle between tangent to shock and vertical axis

141Stream function *

Subscripts i

Denotes conponent in 4 direction IDenotes ccrponent in 14 direction

K Denotes camponent in Z direction 10 Quantity evaluated on body surface

Quantity evaluated on shock wave

Free stream value :1Stagation value

The remaining symbols and subscripts are defined in the text.

HiiTI

IIII

1. INTRODUCTION

The Dorodnitsyn integral method and its specific applications to mixed

flow problems are well represented in the literature, e.g., Refs. 1

through 7. Owing to the nonlinear nature of the two-point boundary value

I problem involved, however, mechanization of the method involves camputing

problems of a nonroutine nature. The present report discusses these prob-

Ji lems in detail as they arise in the (Belotserkovskii) one strip approxi-

mation. Further results concerning the basic techniques involved as well

[as applications to blunt-nosed configurations are presented. Particular

attention is paid to the determination of flow properties across the

shock layer. For the sake of completeness the development of equations

is presented in detail. The procedure and attendant notation largelyI follow Ref. 5.

II]I

'I

I

II

-I2. DEVELOP!FT OF EQUATIONS

The equations expressing conservation of mass, momentn and energy for a

perfect, inviscid, non-heat conducting gas are, respectively,

0 (2-1)

T) 4 7 r7j 00'~/P~~ (2-2)

-S = (2-3)

where 011

Velocity, pressure and density have been nondimensionalized through use

of mnaximum velocity and free stream stagnation pressure and density.

This choice of parameters has particular advantage when applied to the *

Bernoulli equation. '

For general, curvilinear orthogonal coordinates .1 .4t) where

4(sho h'~ ) 4dti4(')and T #7 ' ,the

vector operations are given by

Ppil z

CI

2

r

[

Since only axisymmetric and two-dimensional configurations will be treated,

Ii a body-oriented curvilinear coordinate system (Fig. 1) is defined such

that

I t .-0 two-dimensional case/_ / axisymetric case

I ,.e.1

FIGURE 1

!II

3

I

Substituting the Bernoulli equation for the momentum equation in the

J-direction, the governing system of equations becomes 1

OW=j (2-4) 1

(2-5) 1

(2-6)

k.Z~' =7(2-T)

ert Z (a)

h~r (b) T(c) 1

(2-8)

A~ai~44 '~'(d)

2H

ya./ 4 '4e~(e)

where2

Using Bqs. (2-6), (2.7) and (2.9), it follows that/ = z'/ - .I- o(2-10)I U

Tl

A modified continuity equation is obtained through use of Eqs. (2-8a),

- (2-8b), (2-10) and the relation /. 9mc:

[? (2-11)

" In addition, Eqs. (2-4) and (2-5) are ccmbined to obtain a modified momen-

tum equation in "divergence" form.*

1 (2-12)

I s

Interand Approximations

Consider an equation given in divergence form, i.e.,

I

'kV, ° BODY

r /

1KIFIGURE 2

More precisely, one takes Pe A Eq. (2-5)' 0 x Eq. (2-4) and adds and

I subtracts SL ' ,,

I

Integrating along an arbitrary line in the /7-direction, e.g., across the

shock layer fram the body, nl 0 , to the shock, 171"' , (Fig. 2), one

obtainsT1

If the functions Z, 4 are approximated in the form

-1

it follows that

Applying Leibnitz' s rule, one obtains an ordinary,. first-order differen-

tial equation:

The integrand approximations employed in the present equations are

Vl

6

tI

[The only formal mathematical restriction on the choice of representationis that the right hand side of Eqs. (2-13) be a specified function of f)

containing 4-dependent parameters which are evaluated at either the

shock or body (for the one strip approximation). For example, one may

v specify

[ or

| In some instances, the specified function of /9 may lead to behavior in

direct contradiction to known inviscid results, e.g., incorrect surface

or shock gradients of flow properties. A more complete discussion of the

validity of the approximations is given under Inverse Procedures.

Applying the preceding considerations to the modified continuity and

momentun equations yields (noting that h = z 0):

a 70 (2-~14)

1 (2-15)

7

~1

Eqs. (2-14) and (2-15) may be expressed entirely in terms of the dependent

ithrough use of the following auxiliary

relationships:

I. -I

II

;r-" L e--6"/ )][ "/)w i A'dfr" 3

V/o &r " , a 4-)' J4 "(<(Y-) -1oblique

4mrelationsj

£ = ' - I7o -

CKI

I TI, 6,

Ebcpanding Eqs. (2-14) and (2-15) and introducing a geometrical relationship

between X and one obtains the governing system of equations in final

form:

(2-16)

7_______ (2-17)

d4,_ 4 {#4/V (2-18)I ___

I" where

- "- 1. . 'z . j

I

I11 do

1-J

- - ~ ~

p L(~4~)~c~ ~ J

-Y

7

(,, -~/~JJ sVM~~ ....p4jv~E~~4K4 ~JY6 -J/ U" rJ ft

P [#(' - ~ ~~JV4f'j ~ I

pii171TI

Ii[1

10 II

IIII1 3. I~ITEGRATION PROCEDURE

In order to initiate the numerical integration of the system of nonlinear

total differential equations, knowledge of the dependent variables f, (and e at the axis (euo) is required. The initial values

and :Yll$ are indeterminate at the axis and require special treat-

ment. One may deduce immediately that ) w 4e()- ; however, the

shock detachment distance J_,b)=4r remains unknown and must be deter-

mined fron an auxiliary condition. A two-point boundary value problem

[ therefore exists which must be solved by means of an iterative approach.

(A bar will be used to distinguish the "precise" or final value of e as

opposed to an estimated value during iteration.) The specific approach

taken depends on nose geometry and its influence on the surface -elocity

gradient.

A few preliminary remarks are in order before discussing the two types of

I integration procedures. Application of the integral method to smooth

bodies (i.e., having continuous slope) is facilitated by the choice of a

natural coordinate system such that the body surface coincides with a

coordinate surface. In particular, this choice insures the vanishing of

the normal camponent of the surface velocity which leads to certain ana-

[lytical simplifications. Employing body-oriented orthogonal curvilinear

coordinates, as in the present case, one is only practically limited by

[ the difficulty in analytically describing arc length, radius, slope and

curvature along the body surface. However, when a slope discontinuity

(convex corner) is encountered, the 4-derivatives became infinite at this

point and an "area of cmission" exists which prevents continuation of the

integration. There will be a class of physically meaningful solutions

(for the region from the axis to the corner) obtained when the corner cor-

responds to the body sonic point (Fig."3).

I[ 11

nI-l1

S "

SS

no solution solution valid for

FIGURE 3

A mechanism in the integral method which tends to further limit its appli- -.

cability in certain cases is the direct transmission of body information

to the field inmediately normal to the body rather than, for example,

along a characteristic direction in the supersonic regime, e.g., Fig. 4.

Zone of Influence

integral method I I

rI

characteristics

FiGURE 4

1112

Thus a change in body slope or curvature imposes an immediate perturbation

across the adjoining field. This effect should be less pronounced in the

transonic region where the "normal" mode of information transmittal is

more nearly realized in the physical case. It may be reasoned that this

effect will be suppressed as the integral approximations are improved

[ through further subdivision of the shock layer. However, by rewriting Eq.

(2-16) (which is a geometrical relationship independent of the integral

Ii approximations) as

Iit can be seen that a curvature discontinuity, e.g., hemisphere-cylinderwhich involves a change in curvature from unity to zero, results in a

discontinuity in dci,' < at the same value of 4.. Further, this discon-

-_ tinuity is introduced into the exprcssions for o/KkP and

(which contain &971(S ). Caution must therefore be exercised in interpret-ring flow field results in cases involving rapid changes in body geometry.3.1 Integration Scheme: Continuous at a

Assuning that the body slope is defined by a continuous and monotonically

jdecreasing function, both physical and mathematical reasoning calls forcontinuity of the surface velocity gradient. The denominator of the ex-

pression for I.//, however, vanishes identically at the sonic point.

The auxiliary condition for determining 4 therefore becomes

in order that A7 be finite. Further justification for this proce-

dure is given in Appendix B.

3.2 Integration Scheme: 1A.. &k? Discontinuous at J-%

When the sonic point is known to coincide with a convex corner, a dis-

continuity in the velocity gradient at this point is consistent with

13

requirements of inviscid theory. The previous auxiliary condition is

therefore no longer valid. One must now vary C until

There is a class of bodies for which a definitive choice of method cannot

be made, e.g., sharp-cornered spherical caps. For example, if the sonic

point occurs below a convex corner (on a smooth portion of the body), it

will not be possible to attain the requirement .J'. . Further,

if the corner lies below the intersection of the "limiting characteristic"*

and the body, the slope discontinuity can in principle still affect the

upstream subsonic and transonic flow regions. In practice, however,

there is no feedback scheme in the one strip integral method to properly

account for this effect. 1

3.3 Initial Values -I

3.3.1 Axisymmetric Case

Direct substitution of the initial values into Eqs. (2-16), (2-17) and I(2-18) results in indeterminate (o/o) e xpressions for and

~//- Q)*. Application of L'Hospital's rule yields the proper starting Ivalues,

g1

*Ref. 13 defines the limiting characteristic as the "locus of points each Uof which has only one point of the sonic line in its zone of action."

14

[1

S vhere - ---

Also, by symmetry

33To Dime nsoa Case

No indeterminateness arises in this case (primarily due to the absence

of the radius terms); therefore, no special formulation is required. It

is interesting to note the relationship existing between two-dimensional

and axisymmtric initial derivatives, i.e.,

(# ~D *~(~)(3-1)

[J2) / ) (3-2)[

I

I,15

II

I

4CONVERGENCE SCHEME - SMOOTH BODIES

As indicated earlier, the determination of the correct initial values of

eO) - involves the expression for the surface velocity gradient which 1for convenience may be expressed as &,t-' A/, Assuming that

/VL, =0 ',)) L'Hospital's rule states that if there

exists a neighborhood of S=-, such that D(V)* except for a - X iand and ODK") exist and do not vanish simultaneously, then

$,, ,°V,,< -2'.;,, -e whenever the limit on the right exists. How-4-..4J-

ever, in the present case, for 6 r' , =/A/ * )- O and Y/o

displays singular behavior in the neighborhood of . One must also

note that since S is a function of 6, it is in effect a "floating"

singular point. I

Basically, the iterative process involves successive refinements of 6?

such that IV is reduced in magnitude to the extent that the singular

behavior of ,V/ is confined to a small neighborhood of a. At this

point, extrapolation techniques may be applied with little loss in accu- TIracy in order to continue the integration beyond 4"I 9.

4.1 Initial Value Sensitivity Study 1

In order to establish the feasibility of a given prediction-correction

scheme, it is necessary to establish the sensitivity of the integration

to the choice of 6. Fram the standpoint of physical measurements, it

is reasonable to expect that a carefully conducted experiment will yield

shock standoff distances (in terms of nose radii) accurate to two signifi-

cant figures. In the present integral method, for a given /W,, d', into- 11grand approximation, etc., there exists a precise value of e- Z (mbich,

rounded off to two or three figures, is physically acceptable) which will

cause the solution to be sufficiently well behaved in the sonic region

1 H

such that the integration may be continued into the supersonic regime.

The degree of precision called for was found to be in excess of eight

decimal places.* Figs. 5 through 7 explicitly illustrate the behavior

of the most sensitive function involved, i.e., / /14 , /a& and ., when4 is perturbed plus or minus one digit fran a reference value in the

second through eighth decimal places. The saddle point nature of the sin-

gularity at the sonic point mentioned by previous investigators (e.g.,

Ref. 6) is clearly evident. Note that for a poor guess of 4, the neigh-

borhood for which o'44(S' is significantly affected extends back to the

stagnation point, e.g., for &- u-!. /, a deviation of approximatelysix percent in ( occurs. It should be emrhasized that the

figures presented require a knowledge of 9 and hence represent an[I "a posteriori" view of the convergence procedure.

Initial attempts to satisfy the downstream sonic condition consisted of

a series of "single pass" integrations with a running plot being kept of

/Van %S in order tn make successive predictions of 6 (Figs. 8 and

9). The stopping conditions were either or > 7c , /

Values of NO and - *were determined by means of extrapolation poly-

namials (of degree - - ) applied successively to 4 (to determine

S ) and IV (to determine IV #) J. A positive value of //W indicated

that 6' was too small while a negative value indicated that 6 was too

I large. It quickly became apparent that a slope prediction technique

alone could not cope with the nonlinear behavior of the functions during

early stages of convergence.

A value of standoff distance established for / sufficiently small will

be biased by choice of the parameters f and 7. Therefore, if f or ?

are to be changed during the iterative process, a high degree of refine-[ ment of 6 is not called for prior to the change. Regarding the parameter,

This necessitated the use of a double-precision mode of operation (UM700O-FORTRAN). Unfortunately, double-precision input and readout capa-bilities were not available; therefore, the exact number of figuresrequired beyond eight could not be established.

17

A61

.60I/ -

A.58 M \\ \\ 7 1.4

+,-n

.\ 111 al 1

.J s =S. 3=.

f ESTVTYSUYdSY/SV

1 .02t

M. 01 V1.4

[ .014 /N

1 .012

I .0S

I E SENSITIVITY STUDY - N VS S

"am

.020-

.021 .

.016-

.014

D

.002

-. 002, I6 SENSITIVTY STUDY - D VS, S

nou 7

207

.1494-

Mm -. Y-__1.4

.149

IOU:

.21

.1491

.1496--

.149 *1SPHERE

K..=5.8 y' 1.4 I

.1494-

I-.1493-

.1492-

.1491-I

390 40 40 [.1490

.60 A6 6 6 6 .70 He VS S* 1

~22 11KH

I

one must obviously compromise between fitting from too far (loss of

accuracy in extrapolation) and fitting from too close (loss of accuracy

owing to erratic behavior of functions). Further details are given under

J Program Degrees of Freedom.

[The convergence procedure employed was initiated by establishing upperand lower limits for 4 (denoted by 6 and <, respectively) and mak-

ing an initial guess for d A halving mode was followed until

two values of /V'were recorded. Fig. 10 displays the principal opera-

tions involved. The tests indicated were applied after each integration

step. It should be noted that the curve fitting procedure does not always

yield values of and IV, particularly during the early stages of con-

vergence. In addition, for some combinations of ; and step size, it is

not possible to attain a value of aS between stations corresponding toIi <~7 c~and '

After obtaining two values of /, a linear prediction was made:

If Z , was used; otherwise it was discarded and the halv-

ing procedure was once again applied. This test suppresses the "overshoot"

tendency of the slope technique in the nonlinear range. When all /Vhad

the same sign, the two smallest values were used. If both positive and

I negative values were available, the smallest value of each was used.

[ Typically, the reduction of /V 'proceeded slowly until / O )

owing to the fact that the halving technique was being used in this range.

rAt this point the linear prediction came into play, resulting in order ofmagnitude reductions in a single iteration, e.g., decreases in /VWfrom

O9(/dto 0,/ and from O('/"5 to O(/OJ for a given itera-

tion have been noted. No particular advantage appears to be gained from

'23

Inp ut s ~S tartI n e r tIntegration

-H1

Snew upper limit Tes

6 new lower limit Ts~- E~,I(yes -

etc.s

Fit Acrss Tes

FIGURE 10 Halving Mode

24

I

reducing ,V below ' • Figure 11 displays the variation of the

I versus X for 6 -e.

No attempt was made to use polynomial extrapolation formulas of higher

degree for estimating new values of 6. One disadvantage of increasing

I the degree is the associated requirement for more values of /V, i.e.,

polynomial of degree M? requires *7P/ determining conditions.

For a given body shape and specific heat ratio, if a large number of runs

I corresponding to different values of 14, are to be made, one may expedite

matters through use of polynomials of the form

TM

using the first few cases to determine the constants. Using a sixth

degree polynomial, it was found that five reliable figures for 4 could

be recovered, therefore reducing the number of iterations required.

4.2 Program Degrees of Freedom

Mechanization of the integral method introduces additional parameters

which influence the numerical results. Their values are set primarily by

an empirical approach using the computer as an experimental device. A

brief discussion of their influence is given below. The data are taken

from sphere calculations v -,4)

Integration step size: Table 1 indicates the AS effect on 5 and S

I for a fixed stopping condition 4 - "5 " The fourth-order Runge-

Kutta method of integration was employed in compiling these results.

IUse of other numerical integration schemes was not considered in thepresent investigation.

ji2

SPHERE F 1

M.=~5.8 1=1.4 -

3-*

-sso

dX F1JI +21s

II

I LZ S _ _ _ _ _ _ _ -_ _ _ _ _ __--

.005 .21592940 .14823571 .13692880

.01 .21592940 .14823570 .13692880

.02 .21592943 .14823565 .13692875

1 .005 .76711752 .72000300 .71176106.01 .76712370 •71997798 .711763371 .02 .76713051 .71991991 .71171965

rTABLE 1

Near-sonic stopping condition: For those cases where M/O, a criterionis required to stop the integration procedure short of the sonic point(defined by D(,) -0 ). One may fix this point in terms of a physicallocation v or may specify it in terms of a certain percentage, Iof the sonic velocity. Although the former method requires an approximate

knowledge of %S, a "linearizing" effect is introduced (see Figs. 12 and13) with reference to the variation of /V # versus 6'. Figs. 14 through1 16 illustrate the influence of 4 on the standoff distance and sonic

point location as well as the quantities /( , /4/o/s, // , f,j o'4Z10 evaluated at the downstream location S "'I. The latter plots are

intended to indicate the increase in accuracy gained by fitting from apoint closer to the sonic point. An asymptotic behavior is indicated but

camputational restrictions prevent determination of the final asymptoticvalue. *-he stopping condition was found to be most convenientfrom a computing standpoint in spite of the disadvantage of fitting fromdifferent terminal values of a [depending upon the value of standoff

j distance chosen which in turn controls the functional behavior of C's).More specifically, a randomness is introduced in the function Al/(;

j which can seriously impair slope prediction techniques. In addition, a

/A.S influence is introduced unless a local adjustment of step size ismade such that A7 s precisely at the end of a given step. Table 2

indicates the effect of 7 on 4 and .

27

.14971

.1493- SPER

.1492

.14921 .

.149w" _ _ _ _ _ _

.004.00.1.6

4 vs N*

FIGMR 12

IIII Io'l'

I1-

SPHERE

- .14968826

~SF DETERMINED BYIL CATION AT WHICH

I

SlF.69.6 .76.65 .60I i io6 -0 -

f SN E-

i FKWU 13

29

10-I-

71

i-

-I

SPHEREm. 5.8 ,-1.4

1 -IF .14968826

10-7-

10o-8

EFFECT OF FIT DISTANCE ON I1

F0GUR 14

.7212-

1 .7210-

.70

1 .7206-

S* SPHERE

.70 M. - 5.8 y 1.4

.7198

.6. 14 S .0

EFFECT OF FIT DISTANCE ON 5'

FWKn 15

V 31

.53102

.531002

.53M-0

.37234 SPHER

ds IS

.37218

.37210 T

.60 .A2 .64 SF -96 .83

.2M]6

EFFECT OF FIT DISTANCE ON DOWNSTREAM VALUES

32 MGM 16

1 .38620

I dli 386 10

.386001

u6" .6O 2 464 A6 JO87

40.740, SIR

SPHERE

)l-140.738- M. - 5.8 y- 1.4

Ii 40.736-

II 40.734

40.7321 I I

.60 62 44 .46 .68 9SF

[ .54536

1 .545321d x

.54528

.54524 I5

.60 .6 .64 .66 76s .70SI:

(CONT.) EFFECT OF FIT DISTANCE ON DOWNSTREAM VALUES

[ FIGURE 17

l

9

.975 .21592946 .14823571 .1369288095.~ .21592940 .14823570 .13692880 1•90 .21592835 .14823543 .13692869

.975 .76717822 .72006109 .71182377

.95 .76712370 .71997798 .71176377S.90 .76682185 .71952513 .71135662 "

TABI= 2

Degree of curve fit: A quadratic expression was used for determining

both % and /. Other investigators (e.g., Ref. 7) have employed

cubic expressions. A parametric study of the influence of the degree

of the curve was not made for smooth bodies; however, results pertainingto this topic are given later for the flat-faced cylinder case (see -I

Applications).

341

i

TI

H

I F'34

I

II

[ 5. CONVERGENCE SCHE - BODIES WITH SONIC CORNER

As stated previously, the condition to be satisfied for bodies with a

Ji single convex corner at which sonic velocity is reached is simply given

by

The variation of 4* with i- for a typical body displays none of the

:adically nonlinear behavior of the smooth body case. Through the use1.IjC~j1 wasof a quadratic interpolation, the criterion - 5

satisfied in a relatively small number of iterations. For the special

case of a body with zero curvature up to the corner, i.e., flat-faced

cylinder, cone with detachud shock wave, no iteration is required in a

strict sense since the relation is precisely linear. Further details

are given under Applications.

[EII:

ILtr

6. INVERSE PROCEDURE "1

6.1 Iterative Method

Solution of the governing system of total differential equations yields

, ' , & and their derivatives. Since r , ,. 9

and e/e6.ej have been previously given, one may now determine

4, 6 ' , f' , ' p' , , as functions ofkJ . If one chooses

to follow a formal inverse procedure in calculating shock layer proper-

ties, 2. , , 4 , cS and 4 must be evaluated. Hence ,)-!

Z6 J )and a'X';) are specified through use of Eq. (2-13). Finally

and and are determined from the relations

-(6-1)

___ ___ _;F) _ (6-2)

'/'0W )(6-3)

7.0 (6-4) f

Since the pressure relationship obtained by substituting Eqs. (6-2)x (6-3)

and (6-4) into Eq. (6-1) is implicit in nature, an iterative solution is

36H

II

required. Unfortunately, the solution has separate roots corresponding

to " §/, and switches roots along normals ( %r= constant lines) cross-

ing the sonic line. he calculated flow field in this region is always

discontinuous since the expression does not allow a smooth transition

from a subsonic body point to a supersonic shock point. In addition,

I 6r for the case of a flat-faced cylinder and the system of equations

becomes underspecified, i.e., only the Z and e linear relations plus

[Bernoulli equation are available for determining ^. , 4 . Thissituation also arises when the system of equations is written directly in

I cylindrical polar coordinates (e.g., Ref. 8) since only two approximating

functions are required.

I6.2 "Linear" Method

I Ref. 5 employs separate linear approximations for 3 and

which leave the differential equations unchanged but lead to a simpler but

overspecified algebraic procedure for determining the flow properties,

i.e.,

I

1 fu

1 44

1 37

1I

For a sphere calculation, this situation does not occur if polar coordi-

nates ( ) are used since the two terms collapse naturally to a

simpler expression, i.e., using 17-r-R, ,r-R and rFXW6i ,S

reduces to (- ,# Re P1199jF1A . Ref. 9 represents

linearly thus decoupling the P term. Mh-s decoupling is not possible

when representing c /) linearly; therefore, while the original gov-

erning equations in either J/; or ; & coordinates are equivalent,

the integral approximations employed are not equivalent. iThe foregoing considerations indicate that use of a formal inverse pro-

cedure in the present one strip calculation leads to coputational dif-

ficulties. Ref. 17 (two strip approximation) appears to circumvent this

difficulty by abandoning the formal procedure and employing quadratic

expressions involving values of the flow variables on the body, shock and

the intermediate center strip. "1

6.3 Gradient Method

The derivatives of the flow variables in the .r and /2 directions at the

shock and body may be expressed entirely in terms of quantities yielded

by the basic integration for f, ( and e • This suggests a flow

field determination based upon a knowledge of these gradients involving

cubic approximations for three variables with the remaining variable

being recovered from the Bernoulli equation and the entropy function.* iAn empirical procedure involving different canbinations of cubic fits led

to the use of the following system of equations (see Appendix I for the

derivation of the gradient functions):

*It should be noted that an approximate higher order scheme involvinggradients is discussed in Ref. 5 in which the one strip differential 11equations are modified with correction terms in an attempt to obtain

improved body and shock data.

38

II

Sand ' are determined using expressions of the form

IIT

I eis subsequently determined fr the Bernoulli equation with thecubic representation (denoted by a prime) being employed in order tor fix the sign.

Idat. The usefulness of the graient method illo of course., be dimin-

ished for those cases for %hich the shock and body values beccme lessI asc ,te.

3

iI

I 2h ai oftheforeoin prcedre s torecvermmxrituinfrmaionfra

71

7. COBSERVATION FMATIONS

A measure of the mathematical consistency of the gradient approach is

afforded by determining the extent to which conservation laws are satis- "

fled throughout the shock layer region. It has been noted (Ref. 10) that

significant deviations in conservation of mass and momentum can occur in

flow fields obtained using the integral method.

Considering the control volume shown in Fig. 17, the conservation equa-

tions are given by:

Mass :

1/7

Momentum:

(7-2)

400 1I !B

I

I j Energy:

I (7-3)

I. "

' 'I------------- S --rX

i FIGURE 17

For an adiabatic flow process involving a thermally and calorically per-

fect gas, h= (b nondimensionalized using I" ); therefore,using Eq. (2-7), the integral form of the energy relation reduces to the

[ form of the mass conservation equation.

[ Defining

4II 41

I

values of In and 4 have been computed for a number of cases. The 1results are presented in Figs. 18 through 20. The deviationi noted are

primarily a consequence of the inverse procedure employed to recover the

flow variables.

7.1 Sphere Results

Fig. 18 presents a comparison of the deviations from mass conservation

resulting from application of the "linear," iterative and gradient inverse

methods for A= 3, 6, 9. A single ( 11= 6) curve is presented for the

iterative method. Me break in this curve corresponds to the "double-

valued" region mentioned previously. From a practical standpoint, the 1deviations occuring in the range corresponding to low supersonic Mach

numbers in the field (.8 <C <.9) are particularly significant in that

starting solutions for characteristics are derived from the flow field -

results in this region. Suppression of errors which could subsequently

propagate into the downstream field is therefore necessary in this range.

In this respect, the gradient method provides a substantial improvement

over the other techniques. Conservation of momentum deviations are also jsignificantly reduced through use of the gradient method (Fig. 19).

7.2 Flat-faced Cylinder Results

As indicated previously, the need for two of the approximating functions

vanishes in the case of the flat-faced cylinder. A decision was made to

retain these relations in determining the flow field since the apprcxima- fJtions consistent with this "linear" inverse method appear to have more

validity than the gradient method on the basis of results shown in Fig.

20. Since the accuracy of the gradient method is directly related to the

4[242 !1

1 .0 /71 .08 SPHERE// /F .0 11.4

.0 -- GRADIENT/ /- -LINEAR M = /[ .06 ITERATIVE M9

1 .05 /

.04 /

1.03- d-

1.02- 9

" Ieo1'~~~e 06~. A .

0- lolA

I -.0 SHOC LAYER MASS BALANCE4NPERE

43

.10/ / *

.09I $!*

.08 , /.07- SPHERE/ /1

7 =1.4/ /M INVERSE METHOD

.06 -GRADIENT

.0 1771 ITERATIVE ii

.04-M9~/

.03-//

.02 7

.01- Ir21

0 IsJOU .93

Um

1 .10

r .

FLAT-FACED CYLINDER[M= =6 7 =1.4.05- INVERSE METHOD

GRADIENTA LINEARA.4

62.

IA K

SHOCK LAYERt MASS AND MOMENTUM BALANCEI FLAT-FACED CYLINDER

NOI2

validity of shock and body data, particularly "second-order" quantities

such as R. and a4,1/kS , the deviations in 1 J) may be interpreted as

an indication of shortcomings in the basic solution for this body shape

and at the given free stream conditions. 1I

TI

46IHF'i

[1HI

11I

8. APPLICATIONS

The success of the one strip integral method in predicting shock shapes

and surface pressure distributions has been established by previous in-vestigators, excepting those cases where A44 is too close to unity or

I where rapid changes in body curvature are involved. Unfortunately, little

information is available for determining the accuracy of the distribution

I of flow properties across the shock layer [sonic line data (Ref. 11) and

interferonetric ballistic range data (Ref. 12)].

I In view of the preceding considerations, the results presented do not

include experimental data but are intended to typify the information

yielded by the integral method.

8.1 S-Phere

The variation of shock detachment distance with free stream Mch number

for 1.4 is given in Fig. 21. A sixth degree polynomial of the formI ' _ _,

Iwas fitted to the numerical data in order to provide accurate detacmentI distances for intermediate Mach numbers (which are usable in the present

formulation to approximately five decimal places as initial E values).I Afew cases were run for # 1.4 (e.g., for = 1.2 and/ Al= 20,o= .073666) which displayed trends in accord with the results of other

I theoretical methods. A plot of sonic point location versus A4 is alsoincluded (Fig. 22). For A/Vl> 4, a crossplot indicated that the relation-

ship between c and 4 becomes nearly linear. Fig. 23 presents the

I 47

.40.

j=0.128125 0.001546 +0.672177 -1.122906 I(MW -) (M.- 1)2 (M. 1)3

.35 1.373536 _ 1.000109 * 0.3066713(M . 1)4 ( - 1) (/ -1)6

.25- -_ _ _ - - -_ _ -

.20- _ _ _ _ __ _ -

.15-_ _ -

_ _ - _ _ -

tA. 1000 [12 3 47 910 h 3

SHOCK DETACHMENT DISTANCE-SPHERE (Y. 1.4)

48 F

, ,

I

I ___

.86-

I

IS .82-

1 .78-

I.74

I ,

.70 M. 1000

12 3 0 2 0 40

SONIC POINT LOCATION-SPHERE (. 1.4)

FIRE 22

42l

U I

4c~

a u

u

-1

o6 0b

so RWRI 2

rvariation of shock radius of curvature at the axis with /,. The limit-

Iing value of /? as A -= is significantly higher than the correspond-

ing result obtained by assmning a concentric shock wave, i.e.,/,4 -"Z.

The distribution of flow properties across the shock layer was determined

by means of the gradient method. In particular, the variations of.1I,) and /1 /) are shown in Figs. 24, 25 and 26, respectively,

for ",.= 6 and 0= 1.4. The density plot displays the small .4 varia-

tion in the stagnation region which is the physical basis for a number of

approximate constant density solutions.

Fig. 27 gives a comparison of shock shapes derived through use of the

Ii method of characteristics (with an integral method starting line) and

the integral method alone. The characteristic separating the sphere

(0 38) region of influence and the region additionally influenced

by the afterbody is indicated in order to illustrate the "illegal" dis-

[turbance imposed in the integral method by change in afterbody geometry.

Typically, as the free stream Mach number is increased, the integral

Ir:ethod shock wave tends to be farther below the true shock location (as

defined by characteristics and experiment) for A*/- > 2. It is interest-

ing to note that for a ecplete sphere the integration into the super-

sonic regime continues smoothly until J. approaches - ,

-I,1m w/o'o)' . Below this point the normal to the sphere cannot inter-

sect the shock wave. For example, for A1/ 1000, the integration pro-

[ceeded up to 4 = 3.13, i.e., 179.30 around the spherel

1[ 8.2 Flat-Faced Cylinder

The parameters which control a given integration are

P I:

r 51

C-4I

w

IX GAIMH

x I

C W 4

52 ROURE 24H

ICIjj l

I:- zLu

Q 2

Lu >.I CIn

I"I;ciC

IFIGURE 25 53

0

cim

Ile Iilis

z TC4fC11NE

on'

- d d IC;I

FIGURE 26 f

Fz

o z

IA

II

MA um

FE \

-Iw'0,

141'

I0:

a" A mA qm

I:A4-iI

For conical or wedge nose caps terminating in a sonic corner

< )the latter two parameters become fixed and

become directly proportional to the independent variable , .* As a

result, no coupling exists between r and the geometrical parameters as 1the integration progresses. The location of the body sonic point which

serves as the only characteristic dlmension remaining in the problem

therefore becomes purely a function of o. Hence, a scaling is Implied

which allows one to fix the sonic corner radius, 4'#, at a prechosenvalue eploying a maximu of two Integations. The first integration

establishes the ratio

and the second integration serves to adjust to a convenient ref-erence value using W-M "K* (in the present case, o was chosen as

unity). The latter operation was found to be more convenient than one 1involving a storage of all quantities to be scaled.

Since the accuracy of the integral method is directly related to the

approximations governing functional behavior in the n-direction, the

validity of mploying results derived from local transonic solutions

(Ref. 16) within the fronswork of the compVbing procedures was chocked

in the following manner.* The scaling proprty of the flat-faced cylinder fcse provided a means for determining the consistency of using the flnc-

tional form suggested by second-order transonic theory for the surftce

velocity extrapolation formula at the comer, i.e.*,

*reformulation in terms of the coordinate system employed S.s necessaryto handle oases for 0 < W . In this instance, %f would be replacedby a corresponding variable.

56 1

II

I -

For 0 </ * / , this expression allows the upstream surface velocity

Igradient to approach an infinite value at the corner (as dictated byinviscid flow theory).rThe success of the scaling procedure was influenced by the choice of the

I exponent /7. For example, a value of n- J yielded a sonic corner

radius of 1.0000001 while a value of /7 - 4 (given by second order

theory) gave a "less precise" scaling, 1/ '= 1.000004. A weak depend-

ence upon /7 therefore exists. In addition, numerical determination of

/7 in an inverse fashion employing the last valid points computed near the

corner yielded a value of 0.4996. Finally, employing a cubic fit which

is in direct contradiction to the requirement that

as - gives the poorest scaling, ,j- 1.0004.

Fram a practical standpoint, the preceding arguments become somewhat

academic if the integration is not continued past the sonic corner since

J upstream flow field results are not significantly affected by the fitting

procedures.

I Owing to the steepening of the velocity gradient near the corner, a step

size reduction was introduced as the corner was approached. This effect

became particularly critical as the Mach number was decreased. The reduc-

tion criterion was based on a test of 0/h//b rather than owing to

the rapid divergence of the velocity gradient expression within the span

of a few integration steps. Sequential interval reductions fromAS = .01

I to 4%S = .0001 (based on /= 1.0) were employed in some cases.

j The scaling relationship which exists for flat-faced slabs or cylinders

serves an auxiliary purpose, namely, that of program checkout (excepting

I7

those terms which vanish when 9--oa). If the "two integration pro-

cedure" does not place the sonic point precisely at a chosen locationI

then an error exists in the program.

8.3 Nose Caps

Having determined the flow field for a smooth nosed configuration, one

may proceed to generate results for a family of nose "caps" provided that

they terminate in a sonic corner. Modification of extrapolation proce-

dures near the sonic point is necessary, however, to allow for the proper

functional behavior in this neighborhood, e.g., infinite upstream co Igradient.

The sphere provides a useful example for illustrating the preceding con-

siderations. Representing the sphere shock standoff distance at a given

. and dr by , each integration for a given 6 < o will produce

results corresponding to a spherical cap. A few iterations are necessary

in order to determine the value of c correspond!ng to a specified corner

location; however the convergence is relatively rapid. A comparison of

results for complete and truncated spheres is shown in Fig. 28. he INewtonian value is also included for comparative purposes.

8.4 Two-Dimensional Results

Flat-Faced Plate

1hploying the step size reduction procedure used in the axisymretric case

near the corner, no difficulty was encountered in verifying the E -scalingproperty for the two-dimensional flat-faced body, i .e., only two integra- [ftions were required to fix the sonic point precisely at the corner of the

body. Figs. 29 and 30 give a comparison of shock shapes and surface pres-

sure and velocity distributions for the flat-faced plate and cylinder.

Referring to Eq. (3-2) for 9-o, H

SB II

II

1 .030 M.: 6 y= 1.4

I .6

.,,.,NEWTONIAN

.024-

- 4

. TRUNCATED SPHERE SPHERE

(INTEGRAL METHOD) (INTEGRAL METHOD)

v 1o Po

.016-

.3- P / %

.012-

I *4

.2-

.008.0.1 -

S0<S*

.004

S*c (Oc* = 300) S* (0* = 41.250)

0.0 .1 .2 .3 . .g .

SURFACE PRESSURE AND VELOCITY DISTRIBUTIONS

FOR COMPLETE AND TRUNCATED SPHERES

FIGURE 28 59

AX ISYMMETRIC 2D

.48

606

040

ILAII--

I w 44a

W6U

LI.

ph-

1 -9-

II

IFor Zd > , therefore, the shock curvature at the axis is greater

for the axisymmetric case. This inequality was satisfied for the cases

computed, e.g., for ",,= 5.8, 2(.49662) >.87025.

Circular Cylinder

On the basis of preliminary evidence, the convergence procedure initially

devised for the axisymmetric case was inadequate for handling two-

dimensional bodies, i.e., circular cylinder. i

The principal factor involved was the singular behavior of (,.0 for jthe same range of 4 in which the velocity gradient displays erratic

behavior. Setting tests which restricted shock curvature* (see Fig. 10)

resulted in successful runs at discrete Mach numbers (e.g., A,= 4, 5).

However, the iteration displayed generally erratic and inconclusive be-

havior. For the converged runs, the shock wave displayed points of

inflection in the supersonic regime which could not be ascribed to known

physical behavior. I

It is worth noting that the WSR cylinder results (Ref. 2) were presented (primarily for the two and three strip calculations with only surface pres-

sure data given for the one strip version (indicating the possibility of

inadequacy of the one strip results). In addition, no other reference

was found which presented two-dimensional integral method results of any

description.

*For the axisymmetric case, Ref. 5 proposed a 404 test to handle [1situations where the initial d guess is extremely poor.

62

IIII

9. CONLUDING REMARKS

In the present report emphasis has been placed on the discussion of a

representative formulation of the blunt body integral method which in-

cludes sufficient detail to delineate the numerical pitfalls involved

as well as some practical limitations in its application. Particular

importance was placed on the flow field determination in order to provide

an accurate description of the flow properties across the shock layer.

The gradient method was employed (for smooth bodies) since its applica-

tion insured compatible tangential and normal flow property gradients at

the shock and body while minimizing deviations in conservation of mass

and momentum in the shock layer.IIn the course of the investigation, a previously unreported feature of

the method came to light, namely, that for the flat-faced cylinder case,

the two point boundary value problem reduces to one involving a single

integration of the system of governing equations. This feature obviously

affords a significant reduction in camputing time.

I Regarding the practical utilization of the method, rational flow field

data have been generated for a range of blunt-nosed axisymetric configu-

[ rations employing a combination of the integral method and the method of

characteristics. Application of the latter method was found necessary

in order to maintain downstream accuracy.

I

1 63

I

REFERENCES ]1. Doroditsyn, A. A. "On a Method for Numerical Solution of Same Aero-

Hydrodynamic Problems," Proc. Third All-Soviet Math. Congress,Vol. 2, 1956.

2. "A Contribution to the Solution of Mixed Problemsof Transonic Aerodynamics," Advances in Aeronautical Sciences.Vol. 2, Pergamon Press, 1959.

3. Belotserkovskii, 0. M. "Flow Past a Circular Cylinder With a Detached ]Shock Wave," Dokl. Akad. Nauk, Vol. 113, No. 3, 1957.

4. "On the Calculation of Flow Past Axisymmetric

Bodies With Detached Shock Waves Using an Electronic CamputingMachine," PNM, Vol. 24, No. 3, 1960.

5. Traugott, S. C. An Approximate Solution of the Supersonic Blunt BodyProblem for Prescribed Axisymmetric Shapes. The Martin CompanyResearch Report RR-13, August, 1956.

6. Vaglio-Iaurin, R. "On the PIK Method and the Supersonic Blunt Body 1Problem," IAS Paper No. 61-22, January, 1961.

7. Holt, M, and G. H. Hoffman. "Calculation of Hypersonic Flow PastSpheres and Ellipsoids," IAS Paper No. 61-213-1907, June, 1961.

8. Gold, R., and M. Holt. Calculation of Supersonic flow Past a Flat-Head Clinder by Belotserkovskii's Method. Brown University,Div. of Appl. Math., AF0SR TN-59-199, March, 1959.

9. Holt, M. "Direct Calculation of Pressure Distribution on Blunt Hyper- Ifsonic Nose Shapes With Sharp Corners," Journal of the AerospaceSciences, Vol. 28., No. 11, November, 1961.

10. Feldman, S., and A. Widawsky. "Errors in Calculating Flow Fields bythe Method of Characteristics," ARS Journal. Vol. 32, No. 3,March, 1962. !I

1. Kendall, J. M., Jr. Experiments on Supersonic Blunt Bo Flows.Jet Propulsion lab., C.I.T., Progress Report 20-372, Feb., 1959.

12. Sedney, R., and G. D. Kahl. Interferametric Study of the Blunt BodyProblem. BRL Report No. 1100, April, 1960.

13. Hayes, W. D., and R. F. Probstein. Hypersonic Flow 7heory. New York:Academic Press, 1959.

14. Gerber, N., and J. M. Bartos. Tables for Determination of Flow Vari- [able Gradients Behind Curved Shock Waves. BRL Report No. 1066,Jan. 1960.

64 [

I

rREFERENCES15. Hakkinen, R. J. Supersonic Flow Near Two-Dimensional and Axially

Symetric Convex Corners and Curvature Discontinuities. DouglasAircraft Company, Inc. Report No. SM-27747, July, 1957.

16. Friedman, M. P. "Two-Dimensional and Axisy~metric Rotational FlowsPast a Transonic Corner," Journal of the Aerospace Sciences(Readers' Form), Vol. 29, No. 4, April, 1962.

[ 17. Belotserkovskii, 0. M. Calculation of Flows Past Axially SymwnetricBodies With Detached Shock Waves (Computational Formulas andTables of Flow Fields). Moscow: Computing Center, Academy ofSciences, 1961.

6IIII

III

I

I

1]

APPEMDIX A

FLOW VARIABLE GRADIE1TS AT SHOCK AND BODY iThe governing equations expressed in body oriented curvilinear coordinates

are given by 1(Al)1

f// _ "*2 2y~i IV 410(A2)

70(0_)0 ' L414 22 (A3) 1

R"O (A)

,IShock Gradients

The oblique shock equations serve as auxiliary relations in obtaining flow

variable gradients immediately behind the shock wave. In addition, geo- Umetrical considerations (Fig. Al) enable one to express 4-derivatives in

terms of 4- and (-derivatives, i.e., letting aj -.- /,

or ii

(A5H

<, N

IIIo

r/SHOCK

Ii FIGURE Al

[ Use of this operator in Eqs. (Al), (A2), (A3) and (A4) results in an

algebraic system of four equations for the gradients 041 )i(CP/C4, )[ ~ ~One subsequently obtains:

I (A

)(AT)

67

-~ I (A9) 1

Old. c

where

. 1

-/ !]dA

4)/ &ie

68I

,, I1

II

I 7

-(OW449OW~JI7he data required to evaluate these gradients are 4'W,, 99.9~ 6

(nt tha and

I ne may now express the 4'-derivatives in terms of the /-derivatives

throu& use of the original equations.

II ((I{()

1 69

I

(A12)

Z3) I/Ar

[Alternative expressions may be determined through use of the equations

10and =(~sf.

Body Gradients

Expressions for andp.(40 , and are obtained by setting /7 TJ

and J t 0 in the governing equations.

(A14)

where ,-,, is assmned known. Note that the coefficient of the sur- IIface velocity gradient vanishes at the sonic point indicating that similarbehavior in the integral method is independent of the formulation and Iapprcmimations employed.

70 1]

I[

The remaining normal derivatives are determined employing a procedure

outlined in Ref. 15. Consider a function /(-&4) with dYr e) and

t(4 ' defined as coordinates tangential and normal, respectively, to

the streamline direction. One may therefore write

&e e I - a_

I- Asstuming that ( j/' &) ((= shock wave angle at intersection ofri shock and streamline r= constant), the expression reduces to

rrI

FIGURE A42

From continuity considerations

or

./'

~I

I71

.1

{~.A1

At the surface of the body, Z O, 0

I1Setting

r 4K.f41'., 7 _) 1 iiTherefore

H

H

I1,

72

III F'nally

I

The tangential derivatives at the surface are

I - - - °- 6)

I

I Letting Y represent any flow variable, the angle between the curve

= constant and the X-axis is given by

Hence, this angle may be determined at the shock and the body employing

the derivatives given above.

I71 73

APPENDIX B

BEHAVIOR OF THE SURFACE VELOCITY GRADIENT EXPRESSION IN THE NEIGHBORHOODOF THE SONIC POINT 1A computing scheme was devised which was intended to eliminate the need

for curve fitting and extensive 6 refinement required to cross the sonic 1point. It was based upon the following assunptions:

a. Application of L'Hospital's rule to the equation for the surface

velocity gradient yields a well behaved expression, exact at

r='S for 6=.

b. This expression approxnates the correct value of in

some finite neighborhood of overlapping the region in which

the value calculated by the conventional expression is still valid.

c. The expression is less sensitive to the initial value, 6 , than Ithe original gradient equation.

The scheme involved switching to the L'Hospital or "L" value Just before

dZ . begins to behave erratically and switching back on the other side

of the sonic point, e.g., Fig. Bl.

N

I'%1 L VALUE

S [FIGURE Bl

7H474 !i

Switching expressions resulted in a temporary instability in the inte-

gration which recovered in the supersonic region. Characteristically,

the integration process displayed strong "recovery" ability in that an

I erratic value introduced by stepping too close to 4 introduced only

a temporary perturbation which was rapidly damped such that downstream

1i results were relatively unaffected.

Mathematical Considerations

The denominator of the surface velocity gradient equation is given by

I '~' (1)

I where =/-./ =0 for di=q Wq a

Therefore

Assuming that the surface velocity gradient has a finite, nonsero valueat a."miS # , the preceding considerations indicate that application of

I i

1 75

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

/z 1/ * 0Ci 1

ded

-- 0

R- -R -.

76u n

II

L'Hospital's rule yields an expression which is well behaved in the neigh-

borhood of the sonic point. More explicitly,

_z ______0 (Bi4)I

Proceeding in an admittedly nonrigorous fashion, Eq. (B4) was treated as

a quadratic expression for 04/6%/S (valid, hopefully, at ,- and

I =s). Subsequent evaluation of this expression indicated that one root

gives the origin value correctly and the other root gives the sonic value

correctly.

Carrying out the outlined scheme entailed a considerable anount of work in

differentiating and expanding the Al expression. Owing to its length (as

I well as its limited applicability) the resultant equation is not included

herej however, a plot of the variation of the two roots 4 and Z is

given for a typical case, Fig. B2.

I The proposed scheme was not adopted in view of the following considerations:

[ a. 7he 4. value did not approximate with sufficient accuracy

in a large enough neighborhood of the sonic point.

b. he "switching transients" were more severe in some cases than

erratic functional behavior due to stepping too close to %S

I71I7

c. The 4 value was sufficiently sensitive to the value of d to

require almost the seine degree of e refinement. This sensiti-

vity is indicated in terms of o o4' and .// in Figs. B3

and B4, respectively. The practical advantage of utilizing the

L expression is thus lost.

d. he sheer bulk of the / expression cortributed significantly

to conputing time expended.

781

.16-

.14 I

.12 SP H E R EI1 ~.10- KEYU 1 1.

.0 6 - ® i -

.06-

.04 /

I .02 IiIN1 ~~-.0 2 -'

> s . *

I ~.04 -

.1 -o: SENSITIVITY STUDY -dN/dS VS S~\

*1

/.08- - . .

.06-

.04- ' N,/'

.02- m.-.\ *\'5 ,6

-.02 *-.04 \ I

.0"\\\,\,/ ii

-.086-

-. 10

-.14.I

SENSITIVTY STUDY - dD/dS VS, S1

so