C.P. No. 1277

P l

4

MINISTRY OF DEFENCE (PROCUREMENT EXECUTIVE)

AERONAUTICAL RESEARCH COUNCIl

CURRENT f Af’ERS

Numerical Studies on Hypersonic Delta

Wings with Detached Shock Waves

BY

v. v. Shanbhag,

Cambridge University, Engineering Deparfrnent

LONDON: HER MAJESTY’S STATIONERY OFFICE

1974

PRICE 7Op NET

CP No. 1277* June, 1973

NUMERICAL STUDIES ON HYPERSONIC DELTA WINGS WITH DETACHED SHOCK WAVES

- by - V. V. Shanbhag,

Cambridge University Engineering Department

SUMMARY

In this report a numerical procedure is described for calculating

the inviscid hypersonic flow about the lower surface of a conical wing of

general cross-section. The method is based on thin-shock-layer theory

and the cross-section of the wing may be either described by a polynomial

(up to fourth degree) or given as tabulated data. The actual numerical

scheme is an improvement on that used by earlier workers and the computation

time is much shorter. This reduction in computation time has been

exploited to produce a complete iterative procedure for the calculation

of the pressure distribution and shock shape on a given wing at given

flight conditions. (In earlier work graphical interpolation was used.)

The report includes a complete set of tabulated non-dimensional

pressures and shock shapes for flat wings with detached shocks for reduced

aspect ratios from 0.1 to 1.99, and some sample results for wings with

caret and bi-convex cross-sections.

*Replaces A.R.C .34 617

Introduction

It has been shown by Piessiterl, Squire 2.3 , Hillier4 and others

that thin shocklayer theory gives pressure distributions and shock

shapes on delta wings with simple cross-sections which are in very

close agreement with experiments. The use of this theory involves

the solution of a complex integral equation for the cross flow

velocity (w) with boundary conditions at the centre line and at the

leading edge. Once this cross flow velocity is found the pressure

distribution and shock shape follow by direct integration. Most of

the calculated results for the detached shock case have been obtained

by Squire and Hillier for wings with simple cross-sections (flat

wings, diamond cross-section wings, and some circular arc sections).

They converted the integral equation into differential equation

and marched out from the centre line using the first derivative of

w at the centre line as a parameter (a ) . 1 This method was very lengthy

but by obtaining results for a number of values of a 1' they could

use graphical interpolations from these results to obtain results

for a particular wing at given flight conditions. However, the

direct application of this numerical scheme to iterate to find the

actual solutions corresponding to given flight conditions would

require a very large computer time. Also this method can only be used

for the simple sections mentioned above.

In the present report a direct method of solution of the integral

equation is described which produces a considerable reduction in

computer time and therefore it is possible to combine this method

with a direct iterative scheme for the calculation of pressure distribut-

ion and shock shape on a given wing at given flight conditions.

-2 -

2. Derivation of Equations

For steady flow of an ideal, inviscid gas the continuity, momentum

and entropy equations can be written as

Continuity

Momentum

Entropy

These equations must be solved subject to Rankine-Hugoniot jump

conditions at the shock. These are:

Continuity:-

c jq$;;,,

1 =o

Momentum:-

r F

Energy:- C ;(T-<f+&gf =o

I Tangential velocity:-

=b

where the square brackets denote the change in the enclosed quantity

- 3 -

across the shock discontinuity and 3s denotesa unit vector normal

to the shock surface and directed away from the body. The body boundary

conditions require the streamlines to become tangential to the surface i.e.

(3)

In thin-shock layer theory for conical wings the co-ordinate system is

first stretched to

(4)

Where the barred symbols refer to physical co-ordinate system and

unbarred quantities refer to transformed (or stretched) co-ordinates.

In this transformed co-ordinate system the wing semi-span and thickness

become

t, = h/ icEtan&

respectively.

For a shock which differs only slightly from a plane shock Messiter

suggested an expansion of flow properties in terms of C which is the

inverse of the density ratio across a basic shock, lying in the plane

of the leading edges of the wing. In the limit E 4 0 the expressions

tend to basic Newtonian solutions. The basic density ratio across

the shock is given by

I. f’ft Sin’4

( 6)

-4-

where a is the incidence of the plane of the leading edges. The

suggested expansions for flow properties are

Substitution of these quantities into the equations of motion lead

to a consistent system of equations and boundary conditions which are

cv-y)$ +(,-t)bL = -3 az %Y

with shock boundary conditions

(7)

(8)

- 5-

where Ys 0) is the equation for the shock, and V', Ws and

h denote components of velocities and pressure immediately down-

stream of the shock. The equation (8)

acteristics given by 2 = const. and 5

Since the operator ( V-Y)aF -I- (W-Z,

have two sets of real char-

= const. where

is the

total derivative along a streamline to this approximation, the $ = const.

characteristic coincides with the projected streamlines in the conical

plane. Equations (8) also show Wto be constant along a streamline

and therefore it is a function of f only.

Messiter fixed the constant on !f characteristics by putting

f = 2 on the shock. He also showed that solution of equations (8)

depend on one parameter WC F) and by considering body boundary condit-

ions he showed that

WC?9 ==b (11)

for the detached shock case.

- 6 -

The solution of equation (8) leads after much algebra to

This is the fundamental equation for the determination of ~(1).

The pressure on the body and the shock shape are given by

(14)

Messiter also showed that the appropriate boundary conditions for

WCf)are W(0) = 0 and WC-n> I= i+n. The first condition

corresponds to zero cross-flow on the centre line. The second condition

was chosen to give a singularity in the shock curvature at the leading

edge since a similar singularity occurs in certain two-dimensional blunt

body flows. Ey equation (9) there is a similar singularity in the span-

wise derivative of WCZ) at the edge and this leads to some difficulty

in the numerical solution of equation (12).

- 7 -

3. Numerical Solution of the Integral Equation

The analysis up to this section was similar to chat of Messitor and

Squire. Squire solved the integral equation by differentiating once

again and then using Runge-Kutta procedure for integration of the

resultant differential equation. This procedure takes a long time

for computation of W(f) for a single value of the parameter a 1'

Againthe step size for Runge-Kutta type of integration must be extreme-

ly small ( 0.001) so that large storage was required. The

Runge-Kutta procedure was used up to a certain point (i.e. wet) 7 1. + 0*7st)

and for the remaining part manual graphical extrapolation was used.

By a suitable choice of al it was thus possible to get a set of results

for a range of c? and C where C is thickness ratio of the wings with

diamond cross-sections. These results were then used to produce

a set of charts which could be used to find the pressure distributions

and shock shapes for any given wing wit;1 diamond cross-section at

given flight conditions. A similar method was used for caret wings,

and for wings with biconvex cross-sections. In general this method

cannot be used for general cross-sections.

In the present evaluation of the integral equation (12i, a

different approach

Let us assume

i th station. Then

:‘c was used . This approach is as follows.

that the solution has been obtained up to the

at the ith station

*bi

c dy ) 4 .n

gi- - - WC’b)L - :(;I -2

bi + J cWCS~-S]lL

b; f i

(161

This method was originally suggested by Dr. R. Hillier, but he

only applied the method to the case WC+., >t ,

- b -

Similarly at i + 1 th

station

LO 1-O +

wcz,). -'b 1 L w czbli+l - 'bi+i

‘6;

If we take the step length to be sufficiently small the intez,rals

can be evaluated usin:: the trapezoidal rule and the equation

- 9 -

become

+ A5 I

1.0 a E wc ‘bJi+f

where

Ml = Z&+, - 2,. L cav

Equation (20) was solved by a marching process for a given startin:;

parameter al, until w(t) > 1 + 0.75t and then a

parabolic type of extrapolation was used to find the value of 9

which satisfies WC-h)= 1 + R . The whole process was iterated

to get correct value of al (i.e. correct VJCf) function) for

given boundary conditions of cross-sectional profile, Nach number

and incidence.

Equation (20) was solved subject to boundary conditions

WC02 = 0 4nd WC-h) = It JL . Near the

- 10 -

origin (i.e. Zb = f 50) a fifth power series solution

was assumed for wCJ:, i.e.

This is similar to Squire's treatment for analytic cross-section

case. Here a?, a3, a 4 and

as are related to the cross-sectional

shape of the body by the followin:; expression

C

a e -a -ZZ5(

3 a:+24~+2) + 25-

Ql 3s; (3a,2+449 + 3)

2 ad% -- S c al= + lQ1 + 2)

9,

+ a: a3 ( 6 3~: + 8a, + IO)

4F

In the case of an analytic cross-section in the form of a polynomial

up to fourth degree it is easy to calculate the slope (

- 11 -

to equate coefficients of powers of Z to calculate a2, a3, a4, a5, in

terms of al. (This is the parameter which is to be determined by the

iterative procedure mentioned above, > This procedure was used by Squire

and Hillier for delta wings with diamond and circular arc cross-sections,

But the real problem in the general case arises as follows; first,

if the given profile was a polynomial of more than fourth degree and

secondly, if the profile was given in the form of a table at finite number

of discrete points. To overcome this problem,we approximate the cross-

sectional shape by a five point Lagrangian formula then by differentiating

this formula with respect to Z , an expression for cl ( y> 012 bodY

can be

obtained at any 2 . This expression for ( - can be expressed as a

third degree polynomial in 2 as follows. 3 2

s K,Z + K3Z + ‘(tz +I<,

where

K,’ -

( z1 +Za+Z3+Z4fZ5 -q)

K, = -35 y. 5 iar 1 r Cziazj)

i--i.

5 Kq = 4x

Y i fit Zi -zj)

;= 1. j#‘

i=i

- 12 “.

‘L’ha,rc coefficients were used to calculate a*, a3, a4, a5 at the origin

for any given Q1,

After the, initial polynomial, axpanriwn for WC*) (t being

thr xuaning variable) has baan found,the direct solution of the equat-

ien can be! undarteksn, The actual otsp by step procedure ie boot

understood By naeinp that wqis the ramr function of tb 08

wCf2 kr, that of $ t So if a ro1uti.m haa been obteimd up to

ta particular value of the independent variable (say tf 1 then

wt&# %& 9 WcqI and f are knsm for all VL3lU@N ef %b md

d 1~88 than, or equal ta tf .

Nsw Nupp8NQ, at tf; w’tp 4 gf,Ln ttti 8 caaa wt2 can identify

t with $ and s%nee xb - WCf 1 L ?j , VW&,) ia knawn,

Th~reQwrs! equation (20) em be written as,

- 13 -

In this equation WC f )i+l and W c Lb Ii+ i are unknowns.

But if we know W<f)i,, which is equal to CZb )i+i, and as

wCf’i+l 4 fi,i ’ wCZb) i+i will be less than Lb i+~

and can be interpolated from previous values. The method of bisections

was used to evaluate the correct value of WCFI;+l from a

first approximation (which was the linear extrapolated value from

the previous step), so that equation (24) was satisfied. Once the

correct value of wcfli+l was obtained, the solution was carried

for the next step.

On the other hand, if WC-.*) f f

then % is identified with

‘b and since in this case J L zb it can be interpolated from

already computed solutions at previous t values. Re-arranging the

equation (26) we get a cubic equation in WCZbl i+i i.e.

+ (RHS) c a?

*b i+l =b iti 7 =o (25)

where

- 14 -

Equation (25) was solved by Newton-Raphson method for wczb)i,l

So, to summarise, i+ Wet) 7 t equation (25) was solved

whereas if WC&J 4 t equation (24) was solved.

Solving the above equations step by step, cross-flow velocity

distributionsof one of the following two cases (i.e. case A or B)

are obtained.

If the cross-flow velocity distribution was as in the case A the

whole set of calculations were repeated with a new value of a 1 equal

to half of aie and conversely if the distribution was as in the

case B , the new value of al was taken as twice the value of a 1 . W

This process was repeated till we get both cases A and B .

The correct value of al and the corresponding Wcf.) distribution

lies in between these two cases. After obtaining this upper and

- 15 -

lower bounds of the cross-flow

process was continued with new

% =a + new %

This process was repeated till we get the VUc$?I distribution

such that EOMEGA 0~ wtC-L> is within one percent of

the correct value of R given or 1 + R given respectively. After

obtaining the correct cross-flow velocity distribution, the non-

dimensional pressure coefficient, c P'

shock shape, CL, CD, etc.,

were calculated.

A different approach, namely

velocity distribution, the iterative

al parameter such that

ai z Qlc + 9iw was also tried

new c2

but it was found that the first procedure converges slightly faster

than this second procedure.

There are two main difficulties in the integration procedure.

One concerns the outer boundary condition given by WC-n)= 1+n.

This boundary condition was chosen by Messiter to coincide with the

singularity in the shock curvature. Squire (2) has found that

near the point where W(L) = 1 + IL, w(t) < pi-t) 42

+ . . . . . . . . . . . . . . . . . . . So the solution of the equation was stopped

when WCt) > l+o-'lst a nd then remaining portion of the curve

was obtained by a parabolic type of extrapolation with the vertex

of the parabola having co-ordinates (9 extrapolated, 1 + Q extrapolated)

consistent with the WCt)values calculated so far.

- 14 -

The other difficulty arises when w(t)= t since at w(tJ=t

the equation becomes indeterminate. A trap was therefore included

such that when WCt)curve crosses the line w(t]=t , as found

during the solution of equation (24) the value for that step was

obtained by 6 point Nevil type of extrapolation from previous solutions.

This is best explained in fig. 2 curve (a).

If WC*) as extrapolated above falls below the W(t)= t line as in

the case of case (B) fig. 2, this indicates that WC+) is increasing

rapidly and the w(f)value is influenced by the square root singularity

at the leading edge. So u\/(t) value was re-calculated using a parabolic

type of extrapolation (stipulating similar type of singularity as that

at the leading edge).

- 17 -

4. Result

Using the above programme, the pressure p , pressure coefficient

cP and shock shapes were calculated on flat delta wings. Sample

calculations are also given for a caret wing and for a circular arc

cross-section wing when the cross-section was given in the analytic form

as well as in the form of a table ( si , at 51 points. For flat

wing the functions, p in the pressure coefficient and the non dimensional

shock shape are functions of Z/Land -fL . These functions have been

calculated for the range 0.1 s J-L s 1.99 and are tabulated in tables

Ia and Ib and plotted in Figs. For these calculations the programme was

modified to read -fl, directly, together with number of steps into which

wing span has to be divided, which determines step length, and the starting

value of a 1'

The number of steps used when -(r. < 1.0 was 200 and 12 >/ 1.0

was 400. The accuracy of the result was tested by doubling the number

of steps in the same case and it was found that there is no variation

of results up to four figures. A typical solution for flat delta wing

takes about 5 to 8 seconds on Cambridge University IBM 370/165 computer

with FORTRAN Gl compiler.

An interesting result shows up if we plot the correct al parameter

against .fL for the flat delta wing, fig. (4). In the region between

a= 0.5 and 0.51 al jumps from al > 1.0 to al 4 1.0, This

can be explained by the sketches of the flow field (fig. 3). If al< 1.0

we getLL9t)G'knd the flow field 1s as shown In fig- (3a) and of W@K tthe

flow field will look li.ke fq, 3(b) and so iit. cert;ln the flow field will

jump from (a> to (b) or vice verse, depending on whetherfiis increased

- 18 -

or decreased.

Table II gives p , c P

and shock shapes for a caret wing.

Table III compares results for circular arc (biconvex) cross-

section when the cross-section was given in the analytic form with

that when the cross section was given in the form of a table at

51 points. Both the results compare very well. The results of

calculations were compared with experimental results of Squire (ref. 5)

in fig. 5, which shows a good agreement.

Although the programme converged successfully from any starting

value of al for a variety of shapes, such as flat wings, caret

wings, biconcave wings and thin biconvex wings and also a wavy

type of cross-section(sketch a),some difficulties were experienced

on more extreme shapes. In particular it was very difficult to get

converged solutions for the shape shown in sketch (b) and for very

thick biconvex wings. The difficulties appeared to be caused by

the fact that if the initial value of al was too far from the correct

value then the computed cross-flow, w-p was completely unrealistic

and the iterative procedure did not converge. To overcome these difficulties it was necessary to do a preliminary series of computations using the basic programme (i.e. without iteration) for a range of values of al .

By plotting these results, it was usually possible to find values

of al which appeared to be in the correct range. The iterative

procedure could then be used to complete the solution. However,

it should be pointed out that on caret wings al is usually small

particularly near design, whereas for thick wings a 1 can be large.

On complicated shapes such as that shown in sketch (b) it was found

that possible values of al lay in a very narrow range and that with

- 19 -

values of a 1 outside this range the computed curves of WC 51

were completely unrealistic. Thus it may require a few preliminary

runs to find appropriate range of al .

ta) cb)

Acknowledgements

The author would like to thank Dr. L.C. Squire for suggesting the

problem and useful discussions.

- 20 -

No.

1

Author(s)

A. F. Messiter

2 L. C. Squire

3 L. C. Squire

4 R. Hillier

5 L. C. Squire

References

Title, etc.

Lift of slender delta wings according to Newtonian theory. AIAA 5.3, pp.427-433. 1965.

Calculated pressure distributions and shock shapes on thick conical wings. Aeronautical Quarterly, ~01.18, p.185. ky, 1%'.

Calculation of pressure distribution on lifting conical wings with application to off design behaviour of wave-riders. AGARD Conference Proceedings, 30. 1968.

Some application of thin shock layer theory to hypersonic wings. Ph.D. Thesis. Engineering Department, Cambridge University. 1970.

Pressure distributions and flow patterns on some conical shapes with sharp edges and symmetrical cross-sections at M = 4.0. ARC R & M 3340. 1963.

TABLE fa -------_--------__--------------------------------------------------------------------------------------------- . z p” d;shibuL;otis an flat &lC> 1J;mgs $0~ JL 5

---,,,,,,,,,,,,,,~--,,,,,,,,,,- 2i

-----_-------__----_---------------------. _“---_^-----------_-------

0.1 0.2 0.3 0.4 0.5 i.s 0.7 c.f 0.3 0.51

--------------------^_______^______^___ -------------c-------------------------------~-~----------------------

0.36C9 (1. jl4b 0.00 - 0.1001

0.05 - 0.1085

0.10 - 0.10%

0.15 - 0.1114

0.20 - 0.1139

0.25 - 0.1171

0.30 - 0.1213

0.35 - 0.1265

0.40 - 0.1328

0.45 - 0.1403

0.50 - 0.1495

0:55 - 0.1603

0.60 - 0.1735

6.65 - 0.18(B

0.70 - 0.2102

0.75 - 0.2361

0.80 - 0.2702

0.05 - 0.3172

0.X - 0.3077

0.95 - 0.514O

1.00 - 1.558t

0.0257

0.0255

a.0248

0.0237

0.0221

0.0200

0.0173

0.0142

0.0099

0.0040

- 0.0012

- 0.00~1

- O.OlG5

- 0.0309

- 0.0464

- 0.0669

- 0.0752

- 0.1363

- 0.2008

- 0.3267

- 1.1435

0.1095

O.lO$t

o.roy

0.1087

O.lOeo

0.1072

0.1060

0.1045

0.1026

0.1000

0.0066

0.0323

o.oe67

0.0791

0.0691

0.0540

0.0330

- 0.0004

- 0.05J.31

- 0.1852

- 1.31Cl

0.1733

O.i734

0.1735

0.1736

0.1738

0.1740

0.1744

0.1747

0.1750

0.1749

0.1748

0.1739

0.1725

0.1700

0.1656

0.1569

0.1461

0.1256

0.08~8

- 0.0135

- 1.2360

0.2275

0.2270

0.2283

0.2271

0.2282

0.2302

0.2327

0.2350

0.2391

0.2426

0.2466

0.25oC

0.2554

a.2600

0.2637

0.2GG9

0.2672

0.2&!G

0.23c5

O.?G63

0.3224

0.3226

0.3234

0.3249

0.3271

0.3293

0.3332

0.3376

0.3421

0.3477

0.3533

0.36M

0.36C9

0.3776

0.3e70

0. j;C

0.4111

0.4244

0.43$

0.456b

0.3389

0.3333

0.3404

0.3423

0.3451

0.3406

0.357

0.3582

0.36+2

0.3716

0.37~6

0.389

o.3wl

0.4113

0.4255

0.4409

0.4$5

0.47c2

o.yx4

0.5261

0 . 3512

0.3516

0.3530

0.3553

0.35G6

0.3623

0.3GTQ

CJ. 3747

x3:23

0.3'X 3

0.4017

0.4137

0. h274

0.4431

0.4611

0.4Ul7

0. 5054

0.5323

0. y3c6

O.&G2

0.363+ 0.3141

0.3711 0.3140

0.3739 0.3153

0.3777 0.31xl

0.3r31 0.3203

0.3971 0.3233

0.3715 0.3253

0.4070 0.3306

0.41PQ 0.335s

0.430? 0. $50

0.4459 0.3456

0.4632 C.3jlB

0.4032 0.3iC6

0.5064 0.3657

0.5334 0.37:&b

0.5651 0.3933

0.6026 0.3335

0.6474 0.4043

0.6971 0.4163

- l.lLO2 -0 .ct40 - 0.7739 - 0.7263 - 0.62G8 - O.G756 --------------------____________________------------" -- -..- -- --_------------------,-------------

~nL*L nf P* I lC cc. YL i F * 4.i 2 = ---,,,,,-,,,,,,,,,,L_,,--_,,,,,,,,-----------------.--,--0------------- -_------_------.--_------------------.__-----------_ - o.2101 - 0.0650 0.0396 0.1293 0.2160 0.3!:62 0.37C6 0.4003 0.44eo 0.32&

“-------_--_----^------------------------------------------------- --_---_---_--__----.----------- - -m---- - - - . - - - - - - - - - - - - - I

T n ?3 I. E II

Txi- ,cp, A. and Ken-dhensloml Shock shay for Caret-LYng.

Xbch ihm3er = 3. 37 ; Incidcxe = 23.c degrees ;

b = 0.1318 , h t: - O.lCC2j , A = C.62039 , C = - 0.745537

*""""""""""""I""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

z/a r"t C-, ShOC?i Sk=

.I""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

O.Rl

o.e5

0.93

0.95

1.00

0.6041

o.Go75

0.6116

0.6163

0.6214

0.6275

0.6345

0.6427

0.6517

0.6632

0.6753

0.6897

0.7079

0~7281

0.7536

0.7846

oZ227

0.&716

0.9365

1 c210 .

- 0.0993

0.2743 0.2%9

0.2749 0.2&x

0.2755 0,2t?65

0.2763 0.2053

0.2771 0.2152

0.2781 G.2842

0.2792 0.2828

0.2&15 0.2812

0.2620 0.2792

0.2838 0.2769

0.2957 0.27111

0.2@?0 0.2707

0.2309 0.2669

0.2$2 C.2623

0.2983 0.2570

0.3032 0.2508

0.31333 0. 2434

0.3171 0.2347

0.3275 0.2240

0.3410 0.2106

0.12% o.ly6 .“““““““““““““““““““________1____1___1__”””””””““““““““““”””””””““““““““““““““““““““”

TASLE III

Canmision of 1% distributions and Cp distributions on a

Biconvex wing when the profile is given in the analytic

form as well as in the form of a table at 51 descrete pi?.t,s.

Equation of the cross-sectional profile :-

-2 Y

m-s = 0.047856 ( 1 - v"-- ) 2-2

X bX

Mach-ITumber = 3.97 ; Incidence = 23.8 ; Aspect-rat?0 = 2/z

bega = c.53Fg6 ; C = 0.409627

Iterated value of omega in the calculations is,

Case I Azialytic cross-section = 0.5333581

Case II Wbular cross-section = a5337565

-_---------------------------""-----------------------------------------.

E P distribution Cp distribution

A "-------c----_---_--_______c____________-------------.

Case I Case II CAse I Case II ------_--^--_-------___________________^--------------------------------.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.9

0.95

1.00

O.llEkIll

0.116995

0.113907

0.108g64

0.101e67

0.032615

0.081213

0.0673C3

0.051077

c.o3l~cl

0.003eo4

- 0.0163~~1

- 0,0456&g

- o.oi3322

- 0.12CG95

m 0.1 Gy33g

- 0.2XG76

- 0.303636

- 0.4c&~6

- 0.523507

- 1.4cfl570

0.110757

0.117773

0.114793

0.10957

0.102En

0.033m9

o.oeeOce

o.oGEcye

0. c)~?r“Y ,-",,

Q*C337?7

0.012035

- 0.013911

w 0.043045

- o.:l-(yy4

- O.llf?J,73

- O.lG57?1

- 0.224650

- G.3CO503

- C.4cl5~5

m 0. jD345

" 1.4g;r)cQ ,

0.415261 0.4153%

0.4150?9 0.415223

0.414617 0.414746

0.413614 0.413956

0.412678 0.412@+3

0.411205 0.411491

0 4c9373 . 0.40?%2

0.407159 c.4074c4

0.404550 0.424234

C.4Cl475 C.W17$

0.37?%5 O.??CjO2

0.393eD9 c.3$166

0.3Ow.63 o.~Phc7 i -

0.3"3538 c 3--p? . ‘J

0.3'(7027 0.377':16

0.359317 c. jE?Tjh

0.35m7 o.:yf?!+~5 ._

0.3477e5 o 47tc126 . _

0.331c51 0.331m

c.302937 0. 3fJZ2:@k

c.162641 0.1G383o

APPENDIX

Canplter Prcpame (FORTRAN)

CALCULATION OF PRESSbdE OISTRIBUTIDN ON DELTA WING OF GENERAL CdDSS-StCTlU\ Al HYPE-RSONIC SPtEliS

RtAL b’.‘,Lli, 11’dLtiT OIKEI~S:CI~~ ZHAK~!D5),YBAK~105)rZ~lO5~rY(IOS~1OEKFU~l~,STD~EE~~~ DIMEhS!Ufd AK(4),STGkt-A(Z),hAPZYI(2),AKZl2),CUFP(4) uIME&SI(id PSTAKL(Zl),PSTAHD(21) DIMEI\SILI~~\ kT(405) ,SUMWT(405) rEITA(405) rUtK1 (405) uOuRLt PKtCISIDN Z~~AR,Y~AR,L,Y,~T,EITA,SUI~~‘T,DEK~,AK,DER~D DOUBLE PKECISIUN KHS,tbMtGA,APPkI1,APP~I2,CC,CCD,B~2,BBl,BBO UUUBL~ Pi(tCISIUN tiZYll,WZYi,kWZHI1,WL~I,DABS,DS&lKT,EFti DOURLt PKtCISIO’J AZ,A3,A4,~5,~1,BL,B3,~4,~5,DELTA OL)UBLE PKECISIDN WAPZYI,AKZ,AINCKT,AA,AB,AC,OMX,U~Xl,PSTARL,PSTARD

DUUBLE P&ECISIO:r PRtSUK,SHOCKS,UYBYDZ,RHSl,CL,CD,CLBYCD COMMdN HT,EITA,SUMWT,II,I~~CKT,T,TT,ZBP,ZYIP tXTEdNAL t-CTZB ,FCTZYI,SINT,F

IO CO\TI%UE READ (5,LO,END=lCJOO)ANALTC

iD FUAkATIF15.7)

If O”;E 15 INTtRESTED IN CALCULATION OF LCItFFICIENT OF LIFT C:lEFFICIEr~T OF DdAG,CL/CD THEN GIVE FUR YOCLCD AVALUE OTHER THAN 2 AND IhlCLUDE CORRECT EXPRESSION FOR DY/DX

RiAD (5,3cI) NUCLCD READ (5,l)O)NSC

30 FCIKMAT (13)

ABOJf NSC REPRESENTS NUMBtR UF POINTS LP-TO WHICH POWER SIKIES SOLUT IOFl IS USED i-13K h(T) NEAK OKIGIl\

IF (ANALiC.tU.l.) GO TO 40 60 TU 60

40 REAj (5,20) COFP~4),CUFP(3),COFP(2~,COFP(1) hRITt (6,501 COtP(4),COFP~3),CDFPlZ~,CCFP~l)

50 FUK4Al (‘l’,’ CUEI-FICIENTS, A=‘,F15.7, ’ B=‘,Fl5.7, ’ C=‘,F15.7, 1 ’ 3=‘,l-15.7)

COFP(i),COFPl3),COFP(L)1COFP(l) REPRESENT COEFFICIENTS A,B,CvD RESPtCTIVELY OF ThE EQUATION OF THE CROSS StCTIGNAL PROFILE YhAK= A*ZBAR+*4 + B*ZBAR+*3 + C*ZBAR*+2 + D*ZBA& + F

C

GO TU 100 60 K~AO (5,;IU) NP

hRITc (6,701 NP 70 FORYAT (‘1’1’ NUMBER DF POINTS IN CROSS-SECTIONAL PROFILti=‘,I4)

READ (5,fiO) (ZCAR(I),l=l,NP) KtAD (5,tiO) (YEAk(I),I=i,~P)

80 FbqMAT (4D15.0) hRITE (6,YO)(ZBA~(l),I=l,NP) GiRITE (6,~0)(YBAR(I),I=l,NP)

90 FDRMAT (4f20.73

kEAD (5r20) XBAR,YSPAN,HMOK WRITE (h,IZO)XBA~tdS~AN,HMOR

120 I-DRMAT (’ XBAR=‘,F15.6,’ 1 F15.6)

SEMI-SPAN=‘,Fl5.6,’ MAXIHUM D&DINATE=‘,

ALPl=~ALPHA/.IB0)*3.141593

130

140 150

WKITE (6,130) EPS,SQEPS tORMAT (‘EPSYLON=‘,E16.7,‘SQU~E ROOT EPSYLON=‘,E16.7) TALP=TAN(ALPl) SNALPH=SIN(ALPl) IF (ANALTC.EU.l.) 60 TO 150 DO 140 I=l,NP Z(I)=ZbAR(I)/(SQtPS*TALP*XBAK) Y(I)=YBAR(I)/(EPS*TALP*XBAR) CONT INUt [JMEGA=~SPAN/ ( SbEPS*TALF%XBAR 1 CUNIL=HMUK/(XtiAK*BSPAN*SUEPS) TOLlO=HMOK/(XBAK*EPS*TALP)

READ (5,SU)NNXU INCRT=UMEGA/NNXU READ (5,20)GOtAl ILIMI I=OMEGA/INCRT

16b hKITi(6,160lI~~CKT,COEAl;NSC,OMEGA,ILIMIl,CG~I~ I-URMAT ( ’

1 1NCKIMENT DT=‘,F12.6,’ COEFFICItNT Al=‘,F12.6,’ i~Sc=‘,I.i,

’ OMtGA=‘,FI2.6,’ ILIMIT=‘tI6,’ PARAMETEk C=‘,F12.6) IL=l+ILItJaIT UO 1YD IM=l,IL T=(IM-l)*INCRT IF (IM.Ec;. IL 1 T=DMtGA IF (ANALTC.EO.l.) GO TO 170

CALL LGR (Z,Y,T,NP,AK,DERFD) GO TU 180

170 180 190

CALL ANSLUP (T~AK~OERFD~COFP~SO~PS~TALP,XBAR) DERl(IM)=DEKfD(l)

CONT INU~ KKKKK=O KKKk=l KKKI=l KK IK=l COEA2:O.O CALL SCLOCK

READ (5,2C) ACGNTY If (ACONTY.EC.2.) GU TO 200

KEAD (5970) STDREA(l),STUREE(l),STOREA(2),STOHEE(2) KKKK=2 KKKI-2 KK IK=2

200 UELTA=~CDEA1~~LOG~COEAl~+~.O-COEAl~/~l.O-CU~Al~~~2 IF (COEAL.EO.COEAl) GG TO 710 COiAZ=CUtAl CALL KC-LOCK (ITIME) IF (ITIME.GT.lOOO) Gir TO 730 WdITE (6,210)C.OEAl ,DELTA

130 READ (5,201 MACH,ALPHA,GAMA WAITE (6,110) MACH,ALPHA,GAMA

210 tORMAT (’ VALUE OF Al =‘,E20.7,’ DELTA=‘,El8.7)

110 i-il?MAT(’ MACH NUMHER=‘,F10.4, ’ INCIDENCE=‘,F10.4, ' GAMA='rF10.4) II=1 WT(l)=O.O

. ,ALCULATIOti OF COEFFICIENTS OF POWEK SIRIES SLILUTI~X'I NEAK UKIGIN

s T=U.O

IF Ir,:~ALTC.EQ.l.I GO TO 220 CALL LGi< IL,Y,T,NP,AK,UERFDI

017 TU 230 22.l CALL A’iSLOP IT,AK,UEKiU,CU~P~SQEPS~TALP~X~ARI 2 3') CdhiTlNClt 2 4 3 Al=LOtAl

A2=tbK~1~/I1Al+1.C~/IAl~~Al-il~~~~-I~2*ALOGIAlI~/~Al-l.~~~*3I~ A~=(AKI2)~Al~~2I+A1~*3+~A2~~2/AlI ~4=AL~~Al**4-Al~A~+A~~~2~/IAl~Il+AlII+(Al~~3~*(AKI3II/(l~AlI

;-(L~~,Z~+~)/(A~+QL)+(~~A~~A~I/A~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

l"~~~:2)*(3~AI~*2+b~AI+lO.Ol/Al~~2+(A2"94~~(3~Al~~2+lO~Al+l~~/(3* 2A1~~311/~3*A1~~2+4~A1+3~

ol=l.O/Al b.?=-AZ/(Al**3)

b3=2*(AL~+2I/lA1~~5I-A~/~Al~*41 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ t5=14~IA2+*4)/iAl*~Y)-21e(AZ**2)~A3/(Al~~b)+3~(A3~~2I/(Al~~7)

1 t~*AZ*A4/(Al**7I-A5/(A1986) 250 IF III.UE.NSC) b0 TD 270

II=II+l I=(II-l)*lNCiXT wT(JI)=Al~T+A%*T*92+A~~T*~j+A5*T~~S SuMw;(lI~=S~~kTIII-1~+I~~~T~I~kT(II~~WT~II-l~~/2~ tlT&(JJ):~l*T+B2+~T*~Z~+~~~~T~~3~+~4~IT~~4~+b5~lT~~~~

c c IF OirE IS INTERESTtD IN THE PRINT OUT OF ALL THi k(T) PRINJ C OUT, KtMUVE THt C FROM I-IKST COLLJPN II\ THE FGLLCkING TWG CARCS C hKJTL (6,26UIT,kTIIII,SUMWT(IIII,tITAIII~,UtRl(II~ C 260 FURMAT (Fl0.6,4E15.6)

r L

GO TO i5u 270 11=11+1

T =III-2I*INCRT TT -(II-lI*IhCKT 1~ IwT(II-lI.GT.(l.O+0.75aT)) GO TO 630 IF (II.GT.(l+ILIMIT)I GU TU 660 IF IwT(II-1l.LT.T) GU TO 350 1~ (hT(II-l).EO.TIGO TO 430 JJ=O

283 JJ=JJ+l IF (JJ.Gr.(II-1)) GO 10 430 1~ (hr(JdI.tO.TT) GO TO 290 IF I~T(JJ).GT.TTI GO TO 300 ~0 T;; 2bL

' 290 ~ITA(III=IJJ-i)*INCRT 60 ro 310

303 tITA~II~=IJJ-2~~I~CRT+INC~T~ITl-WT(JJ-1~~/(WT(JJ~-WTfJJ-l~~ 310 ~nS=GtR!(II~-OER1(II-l~-kT(II-lI-l.O/(WTIII-lI-T~-(O.5~I~CRTI

1 *I1.O/IwT(II-1~-TI**2)+O.5*~EITA(II)-EITA(II-l~~~(~l.O/~TT 2 -EITA(III)**2~+(1.0/(T-EITA(III-llI**2lI

bB2=-Z.*TT+?HS bE!=rf~~~+1.0-2.~TT~RHS bBU=ITT**2)*R11S- l-T-II&CRT/ZI A~PWI~=~.*W~I!I-~I-~T(III-~I IF (APPWI1.GE.(l.0+0.75*TTII GO TO 630 JK=l APPWI2=1.O+OMEGA

320 JK=JK+l IF (JK.Gl.20) GO TU 430 CC=APPWIl**3+bB2*APPwIl**2+BBl*APPWIl+bbO CCD=3*APPWI1**2+2.*b~2*APPWIl+bBl

IF (UAbS(APPkI2-APP~111.LE.0.000001~ GCi TO 520 APPWIZ=APPWIl APPWIl=APPWI2-(CC/CCD) IF IAPPWIl.GE.(l.O+TT)) GO TO 330 IF (APPWIl.LE.WT(II-1)) GO TU 430 GO TIJ 320

330 IF (WT(II-lI.GT.(l.O+O.;*(TT-1NCKT)Il GO TO 630 IF IABSIITT-UMEGAI/OME6A).LE.0.05) GO TC 630 tOME(IA=T WRITE (6,340) COEAl,tOMEGA

340 FORMAT I ’ COEFFICIENT Al=‘,Fl5.7,’ CURVE RIStS STEEPLYtCAStBtElMti lA=TI=',Fl6.7) . KKhK=KKKK+l .

GO TO 650 350 wZbI=wT(II-1)

IK=I I-l CALL NtVIL I6,WZbI,O.O,INCRT,IK,WT,IER~ tITA(II)=TT hZVIl=nT(II-1) AINCKT= (WT(II-1I-kT(II-2))/2. JK=O KHSl=OERl~II-II-OE~l(II)*WLBI+WZbI~l.O/IWZ~I-WT~II-lI~-O.~~I~C~T~

1 Il.O/IWTIII-lI-T)**ZI IF IIWT(II-lI-WTIII-2)I.LT.O.O) AINCRT=CAbS(AIhCKTI

360 JK=JK+l KZ=l AKZIl)=WTIII-1) kAPZYIIl~=IO.5*INCKT~~((l.O/(WT(II-lI-TTI~~2~~(l.O/(kT(II-l~-T 44

1 2))-DEK1(II-1I+DtR1(I1) IPP=2

KIQ=l 371.l kZYIl=kZYIl+AINCRT

IF IkZVIl.tQ.TTI GO TO 420 IF IhLVIi.GT.TTI GO TO 490 kk.ZbIl=kLYIl CALL NEVIL (6,WhZbIl,O.O,INCRT,IKlkT1IEH) ~t~S=hWSI-WWZHI1-l.0/(HWZBI1-WZYI1)+O.S+(WZYll-WT(II-lI~~(l.O/

1 (WwZBIl-WLYIl~**2+l.O/(WZBI-WT(II-l))c*2I aCPZYI(IPP)=(G.5*INCRT)/((WZYI1-TT)**2)-RhS

380 IPP=2 IF IkA~ZYI(1I+WAPZYII2).LT.O.O) GO TO 3YO kAPZYI(l)=kAPZYI(Z)

AKZ(lI=wLYIl IF (KIO.EQ.21 GO TO 400 60 TO 370

390 KZ=KZ+l AKZIKZI=WZYIl

. 4 . l

IFTUABSTAKZ(2)-AKZT1)).LE.O.OOOOOlI GO TO 410 KZ=l KIU=2

400 PINCQT=AINCRT/2. WZYIl=AKZT I I GO TO 370

410 wZYIl=WZYIl-1AINCRTI2.1 4;!0 wT(III=wZYIl

GO TO 530 430 JK=O

EFC=TT IK=I I-l

C

: IF TWTTIII.EO.TTI GO TO 590 IF TkT(III.GT.TTI GO TO 570

C ZYIP=TT c ZRP=wTTIIl C GO TO 5BO C 570 ZBP=TT C ZYIP=EITA(III C CALL QG6 TLYIP,ZBP,SINT,SLUPEI C GO TCJ 600 C 580 CALL QCIh (ZBP,ZYIP,SINT,SLOPEI C SLOPE=-SLOPE

CALL YtVIL (6vEI-G,O.O,INCRT,IK,WT,IEi?I C EFG=ZBP wTllII=EFG C CALL NCVIL T6,EFG,O.U,INCRT,II,WT,IERl SUMhTIII~=SU~wT~II-1~+INCRT*~WT~II~tWT~II-l~I/2.O C DYBYDZ=-EFG-l.O/TEFL-ZbP)+SLOPt IF TTT.GT.WlTIIII GO TO 450 C GO TU 610 JKK=O C 590 WRITt (6,595) II,JK,TT,WT~III,SUMWT~I1I,EITA~III,DER1~II1

440 JKK=JKK+l c 595 FUKMA1(14,17,F12.7,3El5.7,E30.7I IF TWT(J<KI.GT.TTI GO TO 460 C GU TO 270 IF TWT1JKKI.EO.TTl GO TO 470 C 600 DYBYDZ=-WT(III-l.O/TwTlIII-TTI+SLOPE

GU TO 440 C 610 CUNT I IJUE 450 EITA(III=TT c WRITE (6,615) II,JK,TT,~T~IIl,SUM~T~II~rElrA~II~,DY8YDZ,DERl~Il~,

GO TO 530 C 1 SLUPt 460 EIlAlII~=~JKK-2~+1NCRl+I~~CRT*~TT-WTIJKK-l~~/~WT~JKK~-WT~JKK-lI~ C 615 FUKMAT TI4,17rF12.7,6E15.7)

IF (WT(IIl.LT.(TT+INCRTII GO TO 480 C GO TO 530 C

470 EITATIII=(JYK-II*INCRT C IF TWTIIII.LT.TTT+INCRTII GO TO 480 C

GO TO 530 C 4PU II=II+l C

T=III-ZI*INCKT C TT=T II-lI*I?dCRT C IF (WTTII-lI.GT.(1.0+0.75*7)) GO TO 630 C

IF (II.GT.jl+ILIMITII GO TO 660 c qc..O JK=O C

EFG=TT C IF ONE IS INTERESTED IN THt PRINT OUT OF ALL THE W(T) PRINT IK=I I-l C OUT, RtMOVE THE C FKOM FIRST CULUMN IN THE FOLLOkING TWO CARDS

CALL NEVIL Tb,EFG,O.O,INCRT,IK,WT,ItKI C WTT I I I=EFG c WRITE (6,620) II,JK,TT,wT~III,SUMWT(IIII,EITA~III~DER~~III SU~I*T~II~=SllMWT~II-1~+INCRT+~WT~II~+WT~II-1~~/2.O C 620 FUKMAT (14,17,F12.7,4tl5.7) IF (tFG.GT.TTI tiU TU 510 C

WTlIIl=~S~K~~2.0~~WT~II-l~-WT~II-2~)/(SORT~2.O~-l.OI C SU~hT~II~=~U~~T~II-1~+l~~C~T~~~T~II~+WT~II-l~~/Z.O C WRITE (6,SUOI TT,wT(III GO TO 270

500. FORMAT 1’ PAKAElJLIC EXTRAPOLATION IN INTERVAL AT kT=T,=‘,F12.6, 630 KKKK=KKKK+l lE15.6) AA=2*twT~II-lI-WT~II-2))-INCRT

tl:sMEbA= TT AB=wT~II-2I*(kT~II-2)-L.O+Z*(II-2)*INCKTI -WT(II-lI*TWT(II-ll- WKITt (6,640) COtAl,EOMtGA 1 2.0+?*( I I-3I*INCRTI -7*INCRT

KYKK=KKKY+l AC=~II-3I~~INCRT*WT~Il-lI*~wT~II-lI-2.OI -(II-2I*INCRT*WT(II-2I* GO TO 650 1 TWTTII-2)-2.0) -1NLRT

510 JKK=O EOMtGA= (-/\tr+I,S09T(At’ncL-4+AAsAC))/(Z.O*AA) GO TU 440 WRITL (6,640) CUEAl,LUMT(,A

520 CUNT INlJt 640 FORMAT 1 ’ CtltFFICItNT Al=‘,F16.7,’ EXTRAPOLATED OMEGA WT~II)=APPwIl l=‘,t/O.l)

5:O SlJMWTTlIl=SUMWTTII-lI+INCRT*~hT~III+WT~II-lII/2 C c IF I<‘.! IS ! *TEl STtD IN ALL Ttli. LALCULATtD SLUPtS AT EACH UF i PUIIJTS FOR KEFcKANCE KtCIJVE THE C FROM FIRST COLUMN c I >. Ttit FC:L LilW Ihf; L A R U \

b

650 cc, 4T IY’lE hT( I I )=l.U~tm~~ti~LrGP SU~hl~II~=SU~~T~II-1I+~tLIMtGA-~II-Z~~IN~~T~~~wT~II~+~TIII-l~I/Z

IF (DARS I (E!‘CtGA-OMtGA)/IJMtbA).Lt.O.Ol) GO TO 750 ST3KtAI 1 )=Cl)ti;l STuRtEI 1 )=cuMcGA IF (KKKI.;t.Zl GU TO 690

IF (KKIK.cO.ZI GO TO 690 CGtAL=CGtA1/2.

(,;1 Ii) 200

660 Cc:dT I NLE GMXl=hT(II-1)

STI;R,A(~)=C.~EA~ STgRtt(2I=l.U+OKLGA-UMXl CKIK=i NRITi (6,670)STO~EA(Zl,OMX1

670 t-O&MAT ( ’ CUti-FICIENT Al=‘,F15.7, ’ EXTKAPOLATtD WTIOMEGA 1 )=‘,,C15.7)

if- (!~A~S(IL~MX~-~~.O+CIVCGA)I/I~.O+OMEGA)).LE.O.O~~ GO TO 750 IF (KKnI.GE.2) GLI TO h90 IF (KKKK.bE.2) GIJ TO 680 CGEAL=Z*LUEA~

GO 10 ZOO 680 kKKI=KKKI+l

IF (KKKKK.GT.6) GO TO 700 hKKKK=KKKKK+l

COE,~l=ST~htAI2)+ISTUl~EA(1)-STORtA(2EI2))/ (STOI~EE(Z)+ 1 (OME!;A-STt,REt ( 1) 1 1

GO TU 2slJ 650 IF (KKKKC.GT.6) 6cI TO 70U

KKKKK=KKC KK+I COE4l=STGREA~2~+~STOKtn(l)-STORtA(Zf~*ISTORtE~2I~/ ISTOdEE(Z)+

l(lj~lEGA-ST~~EEIlII) Gi, ru 200

700 COtA1=IS:OREA(lI+STOKEAo)/2. CO TO 2ciU

710 kdllt (6,720) CGtAl 720 FG&MAT I’ Al LONVERGtS TO SAME VALUE BUT OMtGA NOT SATISFIED Al=

1 ‘,F15.71 ~LJ T., 731)

730 kHITE (6,740) 740 FLJRCAT I’ ITEKATIUN DID NOT CONVtRGE, NEXT DATA TAKEN’)

GO TU 1U 150 kl( l+ILIMITl=1.O+UMEGA

S”M~T~1+ILIMIT~=SUM~T~1I-l~+(1+ILIMIT-~II-l~~~INCRT~~l.~+OME’GA- 1 hT(II-I 1)

ISTEP=Il IMIT/ZO JK=l KK<=l kRITE (6, 760)

160 FORMATI ’ Z k(L) INTEGKAL 0 - 2 k(S) OS P* 1 CPIZI SHGCK SHAPE OY/UZICALI DY/DZIBOOY)‘I

Ld?=iJ.o XKB=l.O/ICOtiAl-1.0)

PRESUR=-III3.0*XKR+4.5)*XKB+3.OI*XK6+0.5) + IIII~.O*XKI~+~.O)*XK~ 1 +5.0)*XKD+Z.O)*XK6)*IALCGICOEAl~l +2.*TUOU-L.*OMEGA*CGNIC

ILLL=l

SHOCKS=DtLTA+TOOCl CPPP=2~~SNALP~~~2~*~1.O+[EPS*(Z.+CUNIC+CMEGA+~R~SU~~l~ IF INUCLCD.EO.2) GU TO 765 IF (ANALTC.EU.~.I GO TO 770

AZL=ZBP CALL LGR IZ,Y,ALL,NP,AK,DERFD) DYBYUX=lt1MOR/XBARI-ItPS*TALP)*II3.~AKI4I+ZBP~Z.+AKI31)oZBP+AKI.!))

1 *ZBP*ZdP GO TO 780

770 AZOAR=ZBP+SOEPS*TALP DYHYDX=HMOR-III3.*COFPo*AZBAR)+Z.aCOfPI3~~~AZ~AKtCOFPI2~~*AZl~A~

1 *AZbAR 780 dT=ATANIDYBYDXI

dTl=ALPl+BT PSTARL(l)=CPPP*COS(BTlI PSTARO(l)=CPPP*SINIBTl) .’

785 ~RITtI6,H90ILBP,WTI1 ),SUMWTIl ),PRESUR,CPPP,SHGCKS 790 JK=JK+ISTEP

ILLL=ILLL+l IF (JK.L,T.Il+ILIMIT)) GO TO 910

ZBP=LJK-l)*INCRT IF IJK.GE.ILIMITI ZBP=OMEGA

h = 1 . ‘LO TU 800

800 M=M+l IFlWT(Ml.6T.LHPl GO TO 810

IF Ih’T(M).EO.ZBP) GO TO 820 ti0 TO HO0

Ml0 ZYIP=IM-2)9I~CKT+INCRl~II ZBP -WTIM-lI)/(kTIM)-hTIM-l))) GU TO &30

820 LYIP=(M-I)*IkCRT 830 IF IZYIP.GT.ZtiP) GU TO 840

IF (ZYIP.CQ.ZBP) ~0 TO 790 CALL ClLh ( ZYIP,ZHP,FCTZB,YINT) CALL c~G6 I ZYIP,ZBP,SINT,SLOPtI b0 TO 850

840 CALL b]Gb (ZUP,LYIP,~CTZYI,YI~Tl CALL i?G6 (Z~P,ZYIP,SI~T,SLOPt) SLOPt=-SLOPE

650 PKtStiK=-l~O-WTIJK)~~2-2~S~M~TIJKI+2~ZBP~hTIJKIt2~OELTA+YIi~I~ 1 I~~~T~J~~-WT~JK-1~I/I~CRT~~I-l.Otl.O/IWlIJK~-Z~P~~42) t2.aTOUC 2 -Z,*COhIC*UMtGA

CPPP=2~fSNALP~~*2)+(l.O~IEPS*I2.*CONICoCMEGAtPRESURl)) DYBYDL=-hT(JK)-1.0/(WTIJKI-ZBP)+SLGPE SHOC&S=DCLTA+TUOU-SUMhTIJK) IF (tiUCLCD.tQ.2) GO TU 8bO IF (AYALTC.tti.1.) GO TO 860

AZL=ZHP LALL LGR IZ,Y,AZL,NP,AK,DERFD) DYaYDX=(HMOK/XBAR)-(tPS4TALP)*((3.~AK(4)~Z~Pt2.*AK(3))*Z~PtAK(2))

1 *ZHP*ZtlP bt) Tu 470

860 AZHAR=ZBP*SQtPS*TALP UYHYDX=hMUR-~l(3.*CO~P(4)~A~0AR)tZ.*COfP~3))~AZ~ARtCUFP(~)!~Af~A~

1 *AZb&R 870 bT=ATANIDYBYDXI

BTl=ALPltBT PSTARLIILLL)=CPPP*COS(BTlI

C c

C C C

C

. . c

PSTARDlILLLI=CPPP*SIN(BTl~ 8tiO w3ITt(6,tiJOlZHP,kT(JKl,SUMWT(JK~,PRESUR,CPPP,SHOCKS,DYBYDZ,D~Kl(JK

1) B&IO

YOO

910

320

333

FURMAT If-lU.517E15.6) IF (vUCLCD.tO.2) GO TC 790

WKITt (6,YOO)OYBYDX,bT f-OKMAT ( ’ UYBYDX=‘,E16.7,’ SLOPE IN RADIANS=‘,E15.6) bU TU 790

IF (NOCLCD.tC.2) GO TO 10 CL=PSTAKL(lI-PSTAdL(21)

CD=PSTARD(l)-PSTARD(d1) DO 920 IMM=2,20,2 CL=CL+4.U*PSTARLIIMM)+2.0+PSTARL(l+IMM) CC=CD+4.U*PSTAKD(IMM)+2.U*PSTARD(l+IMM)

CiJNlihlJE CL=CL/60.0 cD=cD/bo. CLdYCD=CL /CD WRITE (6,Y30)CL,CD,CLBYCD kORMAT( ’ LIFT COEFF. CL=‘,E15.6,’ DRAG COtFF. CU=‘,E15.7, ’ LIFT/D

IKAG KATIU L/O=‘,E15.7) GO Tu 10

1000 CUNTIhUE RETURN

LND

FUNClION FCTZB(XI &tAL INCRT DIMENSIGii WT(405),SUMWT(405),tlTAf405I CLUBLE PRECISION WT,SUMhT,EITA COUbLE PRECISION WS CilMMC?N WT,E~TA,SIJ~WT,II,INCKT,T,TT,ZBP,ZYIP XI:~T=INCRT Irs=x CALL ;u~v(L (h,wS,O.O,xINT,II,WT,IER) FCTZB=(WS-ZBp)**3/(WS-X)**2 HETUPN END

FUNCTIldJ fCTZYI(X) REAL INCkT LtIMENSIUY wT~~U~),SUKWT(~U~I,EITA(~O~) DGU@LE PRECISION wT,SUMkT,EITA DbUELE PdECISION hS C OMMOLl WT,tITA,SUMWT,II,INCRT,T,TT,ZBPtZVIP

i INT=INCKT ws=x ‘,.;;‘zy;,VIL (6,kS,O.O,XIkT.IIvWTvIERI

=-(WS-ZBP)**3/(WS-Xi**2

RETURN END

.

C C C

10 20 30

40 50

60 70

80.

SUGHtiUTINt NtVIL IN,XvXl,RINT,H,W,IEH) DIMENSIOk F( l&I,W(MI OOUOLE PRECISION W,X I t-R=0 IF (N-2)20,40,40 WRITE (6.30) FORMAT ( ‘NEVIL ERK0R.N 2 OR 18’) IER=l

GO TU 180 IF (N-18)50,50,20 U=(X-Xl)/RINT J=IFIX~LJ+O.OOOO1) I=J+N/Z.+O.l

K=O IF (M-N)160,60,60 IF (I-M+lIB0,80,70 K=M-N LO TO 100 KK=J-N/2.+1.1 IF (KK)lOG,90,90

90 K=KK 100 UU=U-K

LJU 110 L=l,N Ll=K+L F(L)=WlLl)

110 LUNTINUE LL=l J=N-l-LL

120 JJ=J+l u=uu DO 130 L=l,JJ L2=L+l F(L)=((U+l-Ll*F(L2I-(u+l-L-LLI*F(L))/LL

130 CONTINUE LL=LL+l If- (J)150,150,140

140 J=J-1 GO TU 120

150 X=F(l) GU TO 180

160 WHITE (6,170) M 170 FORMAT ( ‘NEVIL ERR0R.M N.CONTINUE WITH N=M=‘,IZ)

N=M I ER=2

GO TO 10 180 CONTINUE

HE TURN ENO

2: L

~UBHUIJTIhE LtiK (A,B,C, IP,D,E) DIk!tNSIUN A(IP),b(IP),D(r),t(l),AZi6),AY(5) OC!UBLE PRtCISION h,B,D,E,AZ,AY,UI~,DIMl,AN~Ml,A~U~2~ANUM3 Ul l)=O.O ul2)=0.0 1;(5l=c.o Di4)=0.0 rJLI 10 I=l,IP K=I IF (C -A(K)~20,100,1U

LO CUNT IIVUE 20 IF f(K+Z).GT.IP) SD TL 80

IF (K.Lt.3)GO TO 60 UI l=A(K )-C DIZ=C -A(K-1) IF (DIZ.GT.DIl) GO TO 40 LO 30 L=1,5 M=K+L-4 AZIL)=A(M)

$0 AY(L)=tl(M) GO TU 120

‘tD !I0 50 L=l,5 M=K+L-3 AZ(L)=A(M)

f,G AY(L)=U(M) CO TO 120

60 uu 70 L=1,5 AZ(L)=AlL)

70 AY(L)=RfL) bJ TU 120

to 110 90 L=l,S pi= IP+L-5 AZlL)=AfM)

90 AY(L)=blh) GO Tir 12(J

100 IF (X.LE.3) GO TLI 60 If((K+Z).GE.IP) GO TO 60 uo 110 L=1,5 h=K+L-3 AZ(L)=A(h)

110 AY(L)=b(M) 125 AZ(6)=AZl 1)

DC1 130 1=1,5 DI~=~AZ~i)-AZ~2))*lAZ(l)-AZ(3))~(AZ(l)-AZ~4))~(AZ(l)-AZ(5)) bIMl=AY( I )/DIM AtiUMl=AZ(L)+AZ(3)+AZi4)+AZ(5) ANUM~=(AZ(2)4AL(3))~.(AZ(L)*AZo)+(AZ(2)~AZ(5))+(AZ(3)~AZ(4))

l+(AZ(3)*AZ(S))+(AZ(4)*AZ(5)) AN~M~=(AZ~2)~AZt~)*AZ(4))+(A2o*AZ~3)*AZ(S))+(AZ(2)~AZ~4)~AZ(5))+

1(AZl3)*AZl4)*AZl5)) G(l)=D(l)-(AhUM3*DIMl) 0(2)=D(Z)+(Ar~LM2+DIMl) 0(3)=D(3)-(ANUMl*DIMl) D(4)=D14)+DIMl

AZ(l)=AZ(Z) AL(2)=AZ(5) AZ(3)=AZ(4) AZ/4)=AZ(5) AZ(5)=AZ(6) AZ(6)=AZ(l)

130 CONTIhUt t(l~=((4.0*D~4)*C+3.0*D~3)~*C+2.0*0(2))*C+D~l~

G(Z)=L.O*D(Z) 0(3)=3.0*D(3) D(4)=4.0*D(4) Kt TUdN tNU

SUBROUTINE OG6 ( XL,X”:FCTT,Y) A=O.5*(XL+XU) B=XIJ-XL C= .4662348*8 Y=.OB566225*lFCTT(A+C)+FCTT(A-C)) C= .3306047*B Y=Y+.lB03808+(FCTT(A+CL+FCTT(A-C))

* C=.1193096*B Y=B*(Y+.233Y57O*(FCTTlA+C)+FCTTlA-C))) KE TURN END

r

FUNClION SINT (X) REAL INCHT DIMEI~SIO~ WT(405),SUMWT(405),EITA(405) uUUBLE PRtCISION kT,SUMiiT,EITA DOUBLE PdtCISIDIU k+K COMMON WT~tITA,SUMWT,II,INCRT,T,TT,ZBP,ZYIP XINT=INCKT WK=X CALL NtVIL ~6vWK,0.0,XINT,II,~T,lER~ SINT=l.O/((WK-X)4*2) KETUKFJ END

SUHKCJUTINE ANSLUP lA,B,C,D,E,F,G) DI#EhSIw R(4).C(l),D(4) UUUBLE PKECISi& B,C

t1=1.o/t EZ=E*F*G

B(l)=El*D(l) B(Z)=L.b*El*G(2)*E2

C(l)=((B(4)*A+B(3))*A+B(2))aA+B(l) RETURN EhD

* * * ,

FIG. I Notation

ti %+l

FIG.2. Variation of w(t) Vs t (not to scale)

(a) (b)

F16.3. Streamline Patterns

FIG.4 Parameter a Vs C2 for flat delta wing

I

.

Present calculation

0 Experiment (Ref.5)

M= 597

a = 23.8

O-5-

0 O 0 0 O-4 -

t o-3 -

u” 0.2 -

0.1 -

L I I 02 O-4 06 0.8 I-0

FIG.5 Spanwise pressure distribution on circular- arc

cross-section delta wing R = 2/3

O-8

O-6

o-4

I

0

-08 I

-0-2

-0-3

-0-4

-0-S

o-5

0.4

o-3

0.2

0. I

-@6t I I I I I I I I I

0.2 O-4 0.6 0.8 I-0

I -9 l-8 I.7 I*6

l-4

l-3

I*2

1-I

I*0

o-9

0~8 o-7 O-6 o-51

FIG.6 Spanwise P* distribution on flat -delta winq

O-8

r

I------- 7

-

FIG.7 Theoritical non-dimensional shock shapes on flat delta wings

R 72567 I<54 425 12 73 F’ Produced m England by Her Majesty’s Stationery Office, Reprographlc Centre, Basddon

ARC CP No.1277 June, 1973 Shanbhag, V. V.

NUMERICAL STUDIES ON HYPERSONIC DELTA WINGS WITH DETACHED SHOCK WAVES

I In this report a numerical procedure is described for In this report a numerical procedure is described for calculating the inviscid hypersonic flow about the lower calculating the inviscid hypersonic flow about the lower surface of a conical wing of general cross-section. The surface of a conical wing of general cross-section. The method is based on thin-shock-layer theory and the method is based on thin-shock-layer theory and the cross-section of the wing may be either described by a cross- polynomial (up to fourth degree) or given as tabulated

section of the wing may be either described by a polynomial (up to fourth degree) or given as tabulated

data. The actual numerical scheme is an improvement on data. The actual numerical scheme is an improvement on that used by earlier workers and the computation time is that used by earlier workers and the computation time is much shorter. This reduction in computation time has much shorter. This reduction in computation time has

bgzen/ been/

ARC CP No.1277 June, 1973 Shanbhag, V. V.

NUMERICAL STUDIES ON HYPERSONIC DELTA WINGS WITH DETACHED SHOCK WAVES

ARC CP No.1277 June, 1973 Shanbhag, V. V.

NUMrlRICAL STUDIES ON HYP%SONIC DLLTA WINGS WITH DL'TACHED SHOCK WAVES

In this report a numerical procedure is described for calculating the inviscid hypersonic flow about the lower surface of a conical wing of general cross-section. The method is based on thin-shock-layer theory and the cross-section of the wing may be either described by a polynomial (up to fourth degree) or given as tabulated data. The actual numerical scheme is an improvement on that used by earlier workers and the computation time is much shorter. This reduction in computation time has

been/

~beer exploited to produce a complete Iterative procedure for the calculation of the pressure distribution and shock shape on a given wing at given flight conditions. (In earlier work graphical interpolation was used.)

The report includes a complete set of tabulated noi:-dimensional pressures and shock shapes for flat l+:ings with detached shocks for reduced aspect ratios from 0.1 to 1.93, and some sample results for wings with caret and bi-convex cross-sections.

been exploited to produce a complete iterative procedure for the calculation of the pressure distribution and shock shape on a given wing at given flight conditions. (In earlier work graphical interpolation was used.)

'I'he report includes a complete set of tabulated non-dimensional pressures and shock shapes for flat wings with detached shocks for reduced aspect ratios from 0.1 to 1.99, and some sample results for wings with caret

lcen exploited to produce a complete iterative procedure for the calculation of the pressure distribution and shock shape on a given wing at given flight conditions. (In earlier work graphical interpolation was used.)

'The report includes a complete set of tabulated non-dimensional pressures and shock shapes for flat wings with detached shocks for reduced aspect ratios from 3.1 to 1.99, and some sample results for wings with caret and bi-convex cross-sections.

R 72567/R54 425 12/73 I=’

C.P. No. 1277

0 Crown copyrrght 19 74

HER MAJESTY’S STATIONERY OFFICE Government Bookshops

49 Hugh Holborn, London WCIV 6HB 13a Castle Street, Edmburgh EH2 3AR

41 The Hayes, Cardlff CFl ISW Brazennose Street, Manchester M60 8AS

Southey House, Wine Street, Brrstol BSI 2BQ 258 Broad Street, Brrmingham Bl 2HE 80 ChIchester Street, Belfast BT1 4JY

Government pubhcations are also avadable through booksellers

C.P. No. 1277 ISBN 011 470861 4