Free-flight static stability measurements of cones in hypersonic flow

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TECHNICAL NOTE

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FREE-FLIGHT STATIC STABILITY MEASUREMENTS

OF CONES IN HYPERSONIC FLOW

by KoR. ENKENHUS S 0 CULOTTA P. KROGHANN RHODE-SAINT-GENESE, BELGIUM JANUARY 1970 '- .•• • I

TECHNICAL NOTE

66

FREE-FLIGHT STATIC STABILITY MEASUREMENTS

OF CONES IN HYPERSONIC FLOW

by

K.R. ENKENHUS S. CULOTTA P. KROGMANN

The authors are sincerely grateful to Professor B.E. Richards for his advice and assistance in both the theoretical and experimental phases of the work.

The wind tunnel tests were carried out with the help of J.L. Royen and

F. Vandebroeck.

Summary List of symbols List of figures List of tables 1. Introduction CONTENTS l. i i iv v 2. Theory

5

2.1 The displacement method

5

2.2 The curvature method

8

2.3 Test of a sphere to check the dynamic

pres-sure and obtain the initial impulse 10

2.4 Determination of the dynamic pressure, free-stream velocity and angle of attack from

experiment al dat a 11

2.5 Numerical calculations 13

3.

Model design, construction and measurement

14

4.

Results and discussion 18

4.1

Experimental values of lift and drag

coeffi-cient 18

4. 2 Comparison of results with theory

5.

Conclusions

Appendix Determination of Longshot Test Section

18 19 Cond~ tions 20 Appendix 2 References Tables Figures

The VKI Free Flight Data Reduction

Programme 23

32

34

39

SUMMARY

Free flight tests of 150 _and

₉

0 _half-angle

cones have been carried out in the VKI Longshot tree piston

hypersonic tunnel at M=15 (Re/ft 3 x 10 6 ) and M=23 (Re/ft

1.3 x 10 6 ). The motion of the models was recorded by expos1ng

successive images anto one photographic plate by use of a

multiple spark light source.

The lift and drag coefficients were

cal-culated from the analysis of the trajectory using an IBM 1130 computer. The force coefficients agreed well with values

obtained trom force balance measurements by Maddalon at M=20.3

in a helium blowdown wind tunnel, but differed somewhat from Nextonian predictions.

LIST OF SYMBOLS E g h ho I 10 11 L L c m M p Po

=

n/qlS

=

L/qlS

-= (aM/acx)/q1Sn c

én

+

CM

=

ac

_M

/a(

2

v\

)

q .

Ia

PO.P

,

I 0 q =

lpv

2 2 1 q1 =

-pv

2 2 R s S t t c - ii

-drag coefficient (D = -drag) lift coefficient (L = lift)

pitching moment coefficient slope (M= pitching moment)

pitch damping coefficient drag

base diameter of cone

constants j f proportionality

=

Cn/CD

7

CL/CL '

respectively

N

N

sum of errors squared

acce eration due to gravity enthalpy persunit mass

supply enthalpy

model moment of inertia in pitch

to

initial impulse

Ir

qàt due to tunnel starting ptocess

integrals in displacement method nozzle throat-to-exit length length of cone

mass of model Mach number

stat1c pressure supply pressure

equivalent perfect gas supply pressure

Pitot pressure

dynamic nressure of free stream dynamic press'lre of stream

relative to moving model gas constan·, per unit mass entropy per unit mass

base area of cone model time

thickness of the hollow cone frust rum

T T

_o

To P

u,

w

x,z

GREEK SYMBOLS a ö

o

y P SUBSCRIPTS r c ~ -p d e f R-N P s t o temperature supply temperature

equivalent perfect gas supply temperature velocity of mmdel relative to tunnel

in x and z directions, respectively free stream velocity

stream velocity relative to moving model

tunnel fixed co-ordinates (Fig. 1)

true angle of attack (angle between VI and model axis)

angla of relative wind VI' to x-axis

model pitch angle (angle oetween model axis and x-axis)

angle between free-stream velocity V and x-axis

ratió of specific heats density cone center-of-gravity center-of-pressure drag experimental

final value (at end of observed motion)

lift

Newtonian value plastic

sphere tip

initial value when first picture of àotion is taken (also flow stagnation value)

iv -LIST OF FIGURES

1. Model flight orientation

2. Trace of pitot pressure on oscillograph

3.

Model geometry nomenclature

4.

Design of

9

0 _{half-angle cones with hollow plastic frustrum}

5.

Design of

9

0

half-angle cones with solid plastic frustrum

6.

Nominal dimensions of the

9

0 _{and 15}0 _{half-angle cones}

7.

Completed models and their moulds

8.

Shadowgraph taken of the motion of a

9

0 _{half-angle cone} at a nominal angle of attack of 300

9.

Lift coefficient of 150 _{half-angle cones vs angle of} attack at Mach 15 and 23

10. Drag coefficient of 150 _{half-angle cones vs angle of} attack at Mach 15 and 23

lt.

Lift-to-drag ratio of 150 _{half-angle cones va angle of} attack at Mach 15 to 23

12. Lift coefficient of

9

0 _{half-angle cones vs angle of} attack at Mach 15

13. Drag coefficient of

9

0

half-angle cones vs angle of attack at Mach 15

14. Lift-to-drag ratio of

9

0

half-angle cones vs angle of attack at Mach 15

LIST OF TABLES

1. Resu1ts of 15° ha1f-ang1e cone tests at Mach 15. 2. Resu1ts of 15° ha1f-ang1e cone tests at Mach 23. 3. Resu1ts of

9°

ba1f-ang1e cone tests at Mach 15.

1. INTRODUCTION

In the past several years, the free flight teehnique had been developed to determine the l i f t , drag and dynamie stability of mmdels at hypersonie speeds in the VKI Longshot Free-Piston Hypersonie Tunnel. The models are held in the test seetion by fine threads, and when the flow starts, the threads burn away and the subsequent free-flight motion ean be photographically recorded for typieally 15 milliseconds.

In 1ynamie stability tests-, models with a positive statie margin undergo oscillatory motion during their flight, from whieh aerodynamie moment derivatives sueh as CM and

. . . a .

the damp~ng der~vat~ves sueh as CM. + CM ean be determ~ned. Work of this type has ineluded inv~stiga%ions of a pitehing 30° half angle eone by Levin at M=16.6 (Ref. 1), and by Zoet at M=15 (Ref. 2), and a study of the eoupled angular motion of an elliptic cross-section eone at M=15 by Rowe (Ref. 3). Details of these tests may be found in the referenees quoted above.

In statie stability tests, models with zero statie margin are employed. Such rnodels fly at a nearly constant

angle of attack, and the drag coeffieient, CD' and lift

coeffi-eient, CL' may be determined from the horizontal and vertieal rnotions of the model, x(t) and z(t) . The first work of this kind was carried out by Krogmann (Ref.

4)

.

who tested 15° half angle cones at M=15 and 23. Later tests of 9° half angle cones were made at M=15 by Culotta and Enkenhus. The data reduetion methods originally developed by ~rogmann

have also been subsequently mod~difed and refined, and the new data reduetion program has been used to re-evaluate the original experimental results of Krogmann.

The purpose of this report is to present the results of all the statie stability experiments whieh have been earried out to date, and to deseribe the most reeently developed data reduetion

- 2

-techniques. In addition to the work described here, drag measurements have recently been made by Six and Schlechten

of Louvain University by an optical technique (see Ref.

5).

Two basically different methods have been evolved for determining lift and drag coefficients from the observed free flight motion. In the so-called curvature method~ the exnerimental values of CL and CD are obtained by changing thëir value's 1n the model mot ion program until the predicted curvatures ~, ~ match the experimental values.The experimental values of curvature are obtained by a least squares fit to the data. The idea behind this method is obvious, sinde the curvatures of the x and z motions are a direct measure of the correspoàding accelerations, and hence of the force coefficients. In the so-called displacement method, CL and CD are obtained from the total displacements xf-x o ' zf-zo during the observation times.

While the methods described above are simple in principle, there are a number of factors which must be taken into account if exc~llent results are to be obtained. These will be discussed very briefly here in order to give the reader

a general view of what is involved before presenting the

det~ils. These factors may b~ grouped under the headings

of non-uniform test section conditions, the effect of model motion, and tunnelstarting transients.

a) Non-uniform Test Section Conditions

In Longshot (Ref.

6)

the nitrogen test pas

compressed by the pmston is trapped in a reservoir supplying the nozzle ~t pressures as high as 4000 atmospheres and tem-peratures up to 250Q oK. As the gas flows out the conical nozzle, a conical flow field is produced in which the dynamic pressure decays with time as the supply pressure and temperature in the reservoir fall. (Ref.

7)

.

The decay of supply pressure (or pii,ót pressure) is of the order of 50% in 10 ms in Mach 15 tests. Effects which must

therefore be taken into account i~ data reduction include:

(&)

the influence of real gas effects in the nozzle expansion process on test section conditions;

(2) the change lbf angle of attack 'til the conical flow field

when the model is off the nozzle axis;

(3) the decay in dynamic pressure with axial distance due to'

the conical expanding flow; and

(4)

the variation of dynamic pressure with time at any given

location due to the decaying supply conditions.

b) Effects of Model Motion

(1) The free stream dynamic pressure and the angle of attack at the model location are corrected for the effect

of model motion relative ~o the wind tunnel, but these

corrections are small.

(2) The effect of the variations in angle of attack,

a,which occur during the flight are accounted for by Rssuming

that CD and CL depend on a according to the Newtonian

theory, but that the experiment al values are unknown, constant

multiplës of the Newtonian values C

DN CLN i . e • ,

C L ( a)

=

e" r; ( a )

Rl LU

}

vhere e

d and et are to be determined by comparing the

predicted moti6n with observation. This correct ion is quite significant.

..

4

-c) Tunnel Starting Transients

It might be supposed that the displacement method

of data reduction would be more accurate than the curvature

method, since the former does not in~olve differentiation

of experiment al data.

However, the tunnel starting process imparts j -lrt'!t'iïa'Ï'l-·

axial and veltical velocities u and w to the model which

o 0

influence the x-and z distances mO'trn during the observationtime.

The first photograph of the model position 1S taken at a time to which is at, or shortly af ter, the time at which peak

dynamic pressure has been achieved. The initial velócities

u _o and w can, in principle, be determined from the initial

0

sl~pes of the experimentally measured displacements x (t) e and z (t), respectively. However, this mnvolves numerical

e

differentiation of data in a region where the slipe is

changing rapidly, and the accuracy has been found to be

unsatisfactory. This problem has been overcome by employing data obtained from the simultaneous testing of a sphere, which

has a known drag coefficient in hypersonic flow. As we shall see, the sphere test not ohly permits a check on the dynamic pressure obtained fr om pitot probe readings

but also allows the tunnel starting impulse I _o

to be experimentally determined.

The initial model velocities can then calculated from u _o

=

w _o

=

ft

0

=

r

q(t)dt o be e.ccurately

When the above refinements are taken into account,

it 1S found that the displacement and curvature methods g1ve data which agree weIl with each other, and values of CL and CD obtained are within a few percent of the results of other investigations based on force balance measurements.

2. THEORY

We begin by deriving expressions for the motion of the center of gravity of a free-flight model in a conical time-varying hypersonic flow (see Fig. 1). At any time t , the model e.g. is a distance x downstream of the conical nozzle exit, and a distance z above the nozzle axis. It is assumed that the motion of the model is confined to the x-z plane. The model has a pitch angle

e,

and an angle of att ack Cl. The relat i ve wind' ~tIla.~-ë&,· an angle ö

=

e -

Cl with the x-axis. Then the equations of motion are,

( 2 )

" n,S

-Z

=

(CL(Cl)cosö + C (Cl)sinö)~ - g

n

m

where

Cn(Cl), CL(Cl) are the drag and lift coefficients,

respectively, based on model base area S; m is the model mass; q1

=

t

p

V;

is the dynamic pressure of the airstream of

density pand velocity V

1 relative to the model; and g 1S the acceleration due to gravity.

Let to be the time, af ter the flow is fully established, at which the first photograph of the model position (= x , z ) _o

0

is taken. This time is selected to be at ar shortly af ter the peak Pitot pressure has been "reaë.h-~-êf (see Fig. 2). By integrating Eqs (2) and (3) and employing Eq. (1), we bbtain

u

and w = where

I,

1 2 13 14

=

=

_

.

=

S m

6

-ft

J

CD (a )qlcose dt t N 0 rtC L (a)Cl1 sine dt J to N (t

l

C

_n

(a)ql sine dt iJ TIl to rt

I

CL (a) ql cos ö d t Jt N 0 w o - g(t - t o )

and uo' Wo are the velocity components imparted by the flow

starting process. Since the model is initially mounted very close velocities freestream to the nozzle u and w are 0 0 velocity

V,

i t

centerline, and the induced very small compared with the

follows that e

₌

0 and

_V,

=

V

during the starting process. The angle of attack remains

essentially constant at the initial value a during this period. Tt follows that

S CD (a )1₀ S

1

( 6 )

u o = e

,

w

=

e" -CL (a )T d m

N

0 0 f; m N 0 0 where

\

fto I ::: _{!let) dt}

J

0 0

1S the initial impulse, equal to the time integral of the

freestream d,rnamic pressure during the starting process. The displacements are obtained by integrating

S [e d

I 5-

e t

I 6

]+uo(t-to)+x o

( 7 )

x

=

_m S [e

1

8

+15

I

J+w (t-t )+.1g (t-t )2+ z (8 ) z

=

_{m i .d}

₇

0 0 2 0 0 where

15

=

Jt

1

₁

dt

17

=

f:

I

₃dt to 0

16

:: ft I

2

dt

18

=

ft 1

₄

dt to to

At t=tf' the model e.g. is observed to have eo-ordinates (xf' Zf)' By inserting Eqs (6) into (7) and (8) and solving for e

d and ei' we obtain

= et = m S m S (x - x )

[I

+ C

I

(t - t

)J

+

I

[z -

z

_.1.

g (t - t ) 2

J

f 0

8

L o f 0

6

f 0 2 f 0

[r

+C I t ] -~(x

-x

)1 D o f S f 0 ( 10 )

Eqs (9) and (10) are used to deter1 in.~~e,.;t1l;f}:: drag

an~ l ift eoefficients by the displaeement methode We shall

i

r

di~ate presently how the initial impulse l a n d the

o

qU~Jtities a(t) , ql (t) required for the evaluation of integral

1

,

-1

(x. ,z.)

~ ~

8

-2.2 The curvature method

In the curvature method the model c.g. position at the time t. of the ith data point is calculated

~

from Eqs

(7)

and

(8)

and compared with the observed position x. , z . . We shall illustrate the curvature tecpnique by

trea-1e ~e

ting only the equations for the x-motion, since the equations for the z-motion are similar. Since the initial velocity u

o

is not known

à

priori, the motion x(t) is first predicted assuming an arbitrary value, (u =0). Letting öx.=x. -x.,

o ~ ~ ~

the difference between the measured and predicted xeposition, and 6u = u _oe - u the sum of the errors squared in the fit

0 '

o

is

N

E

=

i~l [6x.-6u (t.-t ~ , ~ 0

'J2

The best fit value of 6u i ' then that for which E is a

oaE

minimum, i.e. for which =

o.

This leads to the result

6u

=

o a6u _o

N

N

E 6x.(t.- t ) / E i=l ~ ~ 0 i=1 (t.-t )2 ~ 0 ( 12 )

The displacement x(t) is now recalculated on the computer with the previous value of u increased by 6u •

o 0

The difference !x. between the observed and predicted

~

motion is now assumed to be given by the parabolic variation

ts.x.=a +alt+alt2

~ 0 . T

The predicted curvature will then match experiment at the values of e

d and et for which ~~i=O, i.e. a2=O. The sum of the errors squared in fitting the parabola is

The conditions ~or E2 to be a minimum are

=

=

0

The resuiting three equ~tions (15) may be soived for

the three unknown a o ' al and a2' The condition ~or zero

difference in curvature between the e ~erimentaiiy observed

and theoreticaiiy predicted motions is

N P = constant x a2 = t . l. where

t~

_l.

=

N

r

i=l (t.- t )n l. 0 N t. l. n l!.x.t. l. l.

=

l . -.1;,1 l!.xl. . (tl. .-t )n 0 t. l.

t1

l. f::.x. l. l!.x.t. l. l. l!.x.t~ l. l.

=

0 ( 16) The vaiues o~ e

d and eQ, must now be varied in the computer

program untii P=O. The caicuiation are first performed with e

d=eQ,=l, i.e. with Newtonian vaiues of CD and CL to obtain

P=P 1 • The corresponding vaiue of u is obtained by repeated

o

use o~ Eq.(12) untii convergence. Next, the process is

repeated for en=1 but with e_d =0.9 and e =1.1 to obtain P 2 ,

Yv 2 d 3

P3' A parabola is numericaiiy fitted through those three vaiues of Pand the vaiue of e

d ~or which p=o i s found.

Caicuiations are now made in a simiiar way using the

z-motion data to obtain Wo and eQ, ' The x-caiculation are

now repeated with this corrected vaiue o~ eQ, to obtain a

corrected e

d; then the z-caicuiations are repeated with the

corrected ed' By successive iterations, the finai vaiues

o~ the initiai veiocities U_{o and} V_o and of e

- 10

-thus obtained.

2.3

~~~~_9f_~_~E~~!~_~9_2~~2~_~~~_~z~~~i2_E!~~~~!~

~~~_9È~~i~_~~~_i~i~i~1_i~E~1~~ _

Experiments have shown (Ref.

8)

that the drag coeffi-cient of a sphere is CD = 0.917 at hypersonic Mach numbers.

s

The testing of a sphere (a ping-pong balI filled with wax or plasticine) along with the model under investigation is

there-fore extremely useful in flow calibration, in the following way. The experimental value of sphere drag coefficient (CD)

s e is first calculated by the curvature method. If the value differs from C

n

=0.917, then it is assumed that the overall level

of the

~easured

dynamic pressure q(t) was in error, and should e

be changed to the value (C

n

s e

)

q(t)=q ( t ) . -e C Ds ( 17)

Eqs

(6)

and (7) then yield the following value for the initial impulse

s

_s

x -x _{f o m}

--Is

s

( 18)

where C

n

_s =C

n

_N

is employed in calculating the integral

Is.

In practice, Eq.(17) has led to a correct ion of dynamic pressure w.hich is less than about

5%.

2.4.

~~~~!~i~~~i~~_~f_~~~_~l~~~i~_E~~~~~~~

,

!~~~~~~~~~_!~!~~i~l_~~~_~~6!~_~f_~~~~~~_f~~~

~~E~!i~~~~~!_~~~~ (a) Dynamic pressure

The only measurement required to obtain the free stream dynamic pressure in a hypersonic flow is that of Pitot pressure, p~, since perfect gas table for a y=1.4 gas show that

( 19)

to a high degree of accuracy for M>10. The nitrogen gas in the test section of Longshot does behave as a thermally

and calo~ically perfect gas to a high degree of accuracy.

During a run, the pitot pressure is, however, only measured at the nozzle exit, so i t is necessary to make an assumption about how q varies with x. For a hypersonic flow the velocity in the test section is near the limiting velocity VI ' =/~

~m 0

and thus its variation with axial distance may be neglected, while by the continuity equation pVA=constant. Therefore,

1 2 ' 1 , , ,

q=2PV var~es as A ~.e. ~nversely as the cross sect~onal area

, L

of the conical jet, or as (L+x)2, where L is the throat-to-exit length of the nozzle. We thus obtain

(20) where

( 2 1 )

1S found to be an adequate representation of the exponential decay with time.

- 12

-(b) Freestream velocity

I ;, a hypersonic flow the freestream velocity V is

near the lim~ G ing value

l2h

and hence does not increase

o

significantly with distance as the flow continues to expand downstream of the nozzle exit. It does vary with time,

however due to the decay of the supply er.thalpy with time.

An analysis made by Enkenhus (see appendix 1) shows that, 1n

a real nitrogen flow, the decay of velocity is related to the decay of dynamic pressure by

(22)

This relation is exact under the assumption that h«h

o

(hypersonic flow) and that the flow entropy and test section Mach number remain constant during the run. Integration of

Eq. (22) yields for y=1.4,

V(x,t)=V(O,O) [ 9(0.0) _q(x,t)

J

1/7

(23)

The initial velocity at the nozzle exit, V(O,O), is calculated

from nozzle calibration data as explained in Appendix 1.

The dynamic pressure ql acting on the model differs from the

freestream value q due to the mot ion of the model relative to the wind tunnel. Let the freestream velocity vector V make an angle ~ with the x-axis.

Then

(24)

where

(c) Angle of attack

The angle of attack of the model is (see Fig. 1)

a=0+ê (27)

where 0 is the pitch angle and ö is the angle that relative wind (of velocity V ) makes with the x-axis :

~ . -l(Vcoscl>-u)

u=s~n =

Vi

t

1

+1 (

V cos

p -

u ) 2J (V cos ~ - u ) (2

8 )

6 Vi Vi .

The matching of the observed mudel motion to

theoretical predictions using the displacement and curvature methods described above was done on an IBM 1130 computer. A description of the Free Flight Program and a complete listing of all the subroutines used is presented in Appendix 2.

IJ

!

r

- .. l~~ -{

3. MODEL DESIGN. CONSTRUCTION,AND MEASUREMENT

In order to produce cone models of suitable weight and zero static margin, the construct ion method adopted was to employ a molded t locel plastic body of either hollow or solid section, with a metal nose tip. The geometry of the

models is shown in Fig.

3.

The geometrical parameters which are sufficient to define the configuration are the cone semi-vertex angle

e ,

the base diameter D , the thickness t of the hollow

c c c

frustrum, and the length Lt of the metal tip.

The design problem is the following : a cone of the chosen semi-vertex angle

e

must be constructed with x .

=

x .

e e g cp

in order to have zero statie margin, and with ballistic

coeffi-.

/11'

2C

/11'

2 . •

c~ents mg

4

De D and mg

4

DcC

L wh~ch ~~sure that the model w~ll

not fly out of the viewing area in the available testing time, but giving as large displacemen~as otherwise possible. The latter are governed by the choice of the base diameter DC and total model mass m • The paramete~which can be adjusted to

c

achieve these aims are D , t and Lt' and the densities p a n d

c c p

P

t of the plastic frustrum and metal tip, respectively.

Expressions for the mass and center of gravity of the model are easily obtained by using the concept that it ~s built up by the superposition of a system of simple solid cones. The model can be thought of as being formed of a solid plastic cone of length LC and density P

p ' less a cone of length Lo and den-sity P

p ' plus a metal cone of length Lt and density Pt , less a plastic cone of length Lt and density P

p • The center-of-gra-vity of each elementary cone 1S located 3/4 of the distance from its tip .to the base.

The mass is then m

=

P V + P (V - Va - V )

c t t P t

and the center-of-gravity location is

x _cg

=

~.75(Pt - pp)VtL_t + _{(O.75 Vc} - Vo)ppLc + _{O.25p p VoLn}

m c

These are two equations ~n the two unknowns Pp and Ptt which may be solved to give

(0.75V

c

- Vo )L

c

m (xc _c - 0.75 Lt )

g

An IBM 1130 computer program was written, which

gene-rated design data in the following way :

Xc

IL

and m , it calculated : g c L = D 12tan6 c c c gi ven 6 , _c D , _c t _c , Lt'

DO

=

D -

2t

Icose

c c c

=

0 for solid cones)

LO

m p = LtD _c

IL

_c

=

L

Do/D

c c

( 4 )

=

p (V - V t - V 0 ) p c

=

m - m _p

From the ca1culations it was concluded that base diameters

D _c

=

5 cm. and 7 cm. would be suitable for cones of 90 _{and 15}0

- 16

-resultsi some useful design charts for 9° hollow cones (t =0.5cm)

c

and solid cones are shown in Figs. ~ and 5, respectively. These

graphs give the mass of the metal tip necessary to provide a

specified center-of-gravity location, x , /L 9 for a model

cg c

of 10 gms total mass. A tip length Lt = 2 cm was assumed for

the hollow cones o For the solid cones i data are presented for

values of Lt from 2 to 6 cmo For any model mass differing from

10 gms, the mass of tip required is proportional to the total

mass.

As an example of the use of these curves, consider the

model designed for testing at 15° angle of attack o Since this

model has an

LID

> l~ the upward~ rather than the rearward

motion is the critical one, ioeo, the design is governed by mg/CLA. A model mass of about 35 gms was calculated to give

an appropriate displacement during the testo From the data of

Madda~on (Ref o 9) the center-of-gravity location for neutral

stability was estimated to be Xc

IL

= 00680 Figo 5 then showed

g c

that for a tip length Lt = 4 cmi mt = 1039 gms for a 10 gm

modelo The corresponding mass of tip for a 35 gm model is

i6

x 1039 = 409 gmso A

~

cm long tip was machined of aluminium

and found to weigh 4.73 gms o The desired model weight was

4.73 x 4

therefore 10 = 3 gms o An appropriate amount of Clocel

1039

beads was placed in the mold and bakedo Upon final assembly,

i t was foünd that the mass of the model was m = 34.248 gms.

c

The actual center of gravity was now calculated with the aid

of Fig. 5 as follows o For m = 10 gms _i the corresponding tip

10 c .

mass would be

34

0248 x 4073 = 10380 F1go 5 then shows

x

IL

= 0.6807, which is satisfactorily close to the design

cg c

value. Finally, the center-of-gravity was determined

experimen-tally by weighing the tip of the model while the base was

supported and the model was horizontalo This gave Xc

IL

=0

₀₆₇₃₀

g c

The difference between prediction and measurement can be

attributed to variations in the density of the plastic (sinee

the clocel beads were not of uniform siz~, to minor variations

from the ideal geometry, and to experimental errors 1n weighing

cones which were constructed and flown are shown in Fig.

6.

Only the

9

0 _{cone flovn at zero angle of attack was hollow; the} rest were solid. Photographs of the molds and two completed models are shown in Fig.

7.

18

-4.

RESULTS AND DISCUSSION

4.1

Experimental values of l i r t and drag coerricient

Eighteen tests wera carried out on two model shapes at ditferent angles or incidence and flow conditions in three test series. i) Uin. tests were carried out on a

15°

halt-angle cone at Mnom

=

15,

ii) four on a

15°

ha~f-angle _{cone at Mnom}

=23.

and iii) tive on a

9°

half angle cone at Mnom=l~o A typical

shadowfraph of the motion of a 9~ cone and the c4libration sphere is shówn in -Figo

8

0 The raw data obtained trom these tests was

reduced using the computer programme outlined in Appendix 2, and the results trom the 3 series are tabmlated in Tables I, 2 and

3

respectivelyo

Spher~ tests were carried out at the same time as the latter

9°

ha1t-angle cone tests, and the results from ~he computer programme are presented in Table

4.

402 Comparison of resultB with theory

The values ot the coetficients determined by the

displacement method and the cu~ature method trom the Tables

1-3

tor the tests on the two modeIs. the

15°

angle-cone and the

9°

halt-angle cone, have been plotted in FigaQ

9-10

and Figso

12-L4

respectively. Figso

9

and

12

show the l i t t coetticient, CL'

Figso

10

and

13

the drag coeffi cient. CD. and Figs o

11

and

14

the l i t t to drag ratio, LID. plotted agains't:-the Anean angle of incidence ot the model during its measured mo~iono On each tigure is plotted the values determined by Newtonian impact theory as calculated using the formulas fr om RefD

8,

and alBo

the experimental results obtained by Maddalo. in Ref.

9

using a force balance in a helium test gaso

Figso

9

and

12

show the experimental results ot CL to be somewhat below Nevtonian theory over the vhole range ot

angles of incidence examinedo The drag coefficient_a Cn' aB seen

in FigBQIO and

13

alBO deviates slightly away from Newtonian theoryo In this case the experrmental results are slightly above theory ror angle s of' inc iden ce below

40°

and :,.sl.ight~ below theQ':-t for

of the coefficients are the same as shown by Maddalon. and in fact agreement to within a few per cent is seen between these datao This agreement with the N.A.S.A. data is further

empha-sized in the graphs of l i f t to drag ratio given in Figs.

11

and 14, where again both these experiments show the same devi-ation from theory.

CONCLUSIONS

Free-flight statie stability tests on

15°

and

9°

half angle cones have been carried out in the Longshot free-piston wind tunnel at Mach numbers of

15

and 23. The motion of th~

models was analyzed using an IBM 1130 computer to calculate the lift and drag coefficients. The following conclusions were

reached :

1. Suitable model motions were generated on all model configu-rations and at all test conditions ₉ allowing high precision measurements of model displacements to be made using a multiple spark souree shadowgraph techniqueo

20 The data reduction was achieved by integrating the equations of motion, using measured test section conditions as input data, and comparing the results with the experimental data. Two data reduction techniques~ the displ acement method and the curvature

method₉ were employed in the data evaluationo Both methods agree with each other within

5%

,

and resulted 1n values of C

_n

and CL differing from Newtonian theorY9 but very close to results obtained by Maddalon using a force balance in a helium tunnel.

3. The displacement method requires the values of the initial velocities

Uo

and

Wo

which are caused by tunnel starting tran-sients. These were found signifi cant enough not to be neglected from calibration tests on spheres. The curvature method does not require such measurements.

REFERENCES

10 Levin, G.M. : "Development of the Free-Flight Teehnique for Determination of Statie and Dynamie Coeffieients using Longshot"

VoK.I., PR 68-212, 1968

2. Zoet, V. : "Aerodynamie Damping of a 300 _{Half Angle Cone}

at M

=

15"

V.K.I. PR 70-259, June 1970

3. Rowe, M. : "The Coupled Angular Motion of an Elliptie Cone in Hypersonie Flow"

V.Kolo PR 70-270, 1970

4. Krogmann, Po : "Free Flight Statie Stability Measurements on Cones at Mach 15 and 24"

V.K. I. PR69 ",

5. Six, L.. Slecht en ~ Jo "Det ermina t ion 0 f Aerodynamie Forc e s entering on a 90 _{eone at zero Angle of Attaeh in}

Hypersonie Flow M

=

15 by an Optical Method"

U-V. K. I •• May l.97"0 "+i'

6. Riehards, BoEo, Enkenhus~ KoRo "Hypersonie Testing in the VoK.lo Longshot Free Piston Tunnel"

AIAA Journal~ Volo 8 No 6 j June 1970, pg. 1020

70 Enkenhus ~ KoRo : "On the Pressure Deeay Rate in the Longshot Reservoi r"

VoKoI. TN 40~ 1967

80 Wells~ WoRq Armstrongj WoOo : "Tables of Aerodynamic

Coeffi-eients obtained from Developed Newtonian Expressions for Compl ete and Partial Conie and Spherie Bodies at Combined Angles of Attaek and Side Slip with Some Comparisons with Hypersonic Experimental Data" NASA TR R-12_{7 i} 1962

90 Maddalon i Do V 0 : iI Aerodynamic Charaet erist ie s o f the Sharp

Right Cireular Cone at Mach 200 3 and Angles of Attaek

to 1100 _{in Helium"}

NASA TN D-3201 ~ 1966

10 0 Culottai So p Riehards ~ BoE. : "Methods for Determining

Conditions in Real Nitrogen Expanding Flows"

VoKolo TN 58i Febro 1970

110 "Equations, Tables and Charts for Compressible Flow"

NACA REPORT H35 p Ames Aeronautieal Laboratory, 1953

120 Culotta~ SOi Enkenhus p KoRo : "Analytical Expressions for the Thermodynamie Properties of Dense Nitrogen" VoKolo TN30i Deco 1968

- 22

-13. Grabow, M., Brahinsky, HoSo : "Thermodynamic Properties of Nitrogen from 300 to 50000 _{K and from 1 to 1000 Amagats"} AEDC-TR-66-69, Aug. 1966

140 IBM Application Program, 1130 Scientific Subroutine Package (1130-CM-02X ), Programmer' s Manua1

Determination of Longshot Test Section Conditions

The test section conditions ~n Longshot vary with

location and time as the gas trapped in the reservoir exhausts through the conical hypersonic nozzle. The flow is calibrated fr om readings of the supply pressure PO. supply temperature To.

and pitot pressure Po vs time~ It has been shown that the

isentropic expansion of real nitrogen from the supply conditions

(PO. To) produces the same test section conditions as a y

=

1.4

perfect gas expanding from equivalent perfect gas conditions (To • Po _{p p} ) (Ref. 10), where

TO

=

hol305R

P

POp

=

eXP{3051nTop -

siR

+ Cl}

(A-l)

(A-2)

In Eq. (A-2 ), the constant Cl has the value 3.138 for Po in

P

atm and To in

oK.

Once the equivalent perfect gas supply

con-p

ditions have been found i the test section Mach number is

ob-tained from the ratio of pitot pressure to equivalent supply pressure by the Rayleigh supersonic pitot tube formula, with y

=

1.4

~ p'

.

..!..

= f(M) pOp

=

(

6M 2 ) 7

I

2 ( 6 ) 5/ 2 (A-3)

M2

+ 5

7M2

- I

x Measured using a KistIer type 6201 piezo-electric transducer.

a tungsten-rheni um termo-couple, and a Hydyne Type W variabIe

reluctance diaphragm transducer, respectively. with supporting equipment (see Refo 6 for fuller details).

- 2~

-The other test s ect ion flow paramet ers may now be obt a:j.ned ;from POp' Top and M using the well-known perfect gas isentropic

rela-tions (Ref. 11)0 In particular, the test section flow veiocity ~s

Y = M)yRT

where T = TO (1 .. + Y2-1 M2 )-1

P

.

}

(A-4)

Since the only test-section flow parameter which is

monitored during a run is the pitot pressure p~, it is of

inte-rest ~o enquire wh ether the other flow quantities can ~e derived

from it in a simple way in the situation where flow conditions

are varying with time.

If we assume that the flow is adiabatic, th en the test

section velocity Y and statie enthalpy hare reläted to the

supply enthalpy ho by the energy equation

h + ; y2

=

ho (A-5)

For hypersonic flow h « hO, so that the rate of decay ~f

velo-city in the test section ~s

d

dt (.Q,nY) =

(A-6)

by Eqs. (A-4) and (A-5).

If we now make the further reasonable approximations that the ~t

entropy of the flow and the test section Mach number rem~in

constant, Eqs. (A-I) , (A-2 ) and (A-3 ) yield

1 =

-3

.

5

1 d

= -

₃

_.

₅

(R.npo') dt

(A-7)

Combining Eqso (A-6) and (A-7 ) ~ we have

d

dt (R.nV)

or, upon integrationi

V(x,t)

V(o,o)

[

,

]1/ 7

_ po(x,t)

,

p~(o,o)

(A-8)

(A-9)

The decay in freestream velocity is thus determinable in a simple way from the measured pitot pressure. Since the dynamic pressure is directly proportional t o the pitot pressure (see Eqo (19)),Eq. (23 ) of Section (2.~ follows immediately.

26

-Appendix 2

The VKI Free Flight Data Reduction Program

The free-flight data reduction program is written

~n Fortran IV and is used on an IBM 1130 computer. The program consists of a MAIN routine and 17 subroutines, the listings of which are given below. A brief description of the program follows. The reader should also note the explanations given by the

COMMENT statements inserted into the various routines. Card 1. READ(2,300l) OPT

3001 FORMAT (Fl.O)

If OPT = 1 calculation of lift and drag by both displacement and curvature methods OPT = 2 calculation by displacement method

only

OPT = 3 calculation by curvature method only CALL DAT2(AO~ TF_îI DT_a ERR)

The purpose of subroutine DAT2 is to read computing

options~ model and flow constants~ according to the following

statements ~

Card 2 {READ(2 s 3000) LRUN p PO§ TO}- the run

number~

and the

3000 FORMAT (I3~ 2E12i

6)

supply pressure Po (psi) and temperature TO (OK) at the start of the runo

Card 3 READ(2, 10) NXOZj TIMO. XPO, ZPO, CC1? CC2 a CC3

lOF 0 RMAT (I 1 9 3 F 5 0 1 ij 3 E 12 0 6 )

NXOZ determines a further computing option if NXOZ

=

1, experimental data reduction

=

29 calculation of theoretical motion onlyo

TIMO

=

tO (ms )i the time of the first spark picture af ter peak P~ i s reached

XPO

=

xo (mm) ~ ZPO

=

zO(mm)9 the model coga position ~n

=

2 •

=

fit vs time (Eq. 21). The constants are chosen to give

p~ in psi when t is in milliseconds.

Card

4

_{READ(2,202)AO, CMA. CMD a CDa CL} 202 FORMAT (6E12 0

6)

AO

=

60. the initial pitch angle of the model (deg)

CMA

=

CM ,the pitching moment coefficient slope based on

a

model base area and base diameter De 0

CMD

=

CM. + CM i the damping in pitch derivative

=

a q

CD

=

CD~ CL

=

CL the drag and l i f t coefficients based on model base area o

The above aerodynamic derivative values are used to theoretically predict the free flight motion of pitching cones (NXOZ

=

2 tOPT

=

1)0 When determining CD and CL from experimental datai the values read in are ignoredo

Card 5 READ(2g202) Di Wa CIs THET a P

D

=

cone base diameter (mm)

W

=

model weight ~ms)

Cl

=

moment of inertia in pitch (Newton-meters2 )

THET :::

e

c i the cone half angle (degrees). This value

used later by subrout ine FORCE in calculating the Newtonian values of CD and CL" The options

e

₌

9°~ 15°, or 30°0 6

=

0 corresponds to a

c c

sphere test.

P

=

lOt the initial impulse ~n Newton-seconds/meter2

is

are

(Eq. This value is used only in cone data reduction by the displacement method o It is obtained as a

printout from a previous run for a sphere test.

28

-Card

6

READ(2,202) TF₈ DTB ERR

TF

=

t

f , the time of the last data point (seconds)

DT

=

6t, the integration time step (seconds) (typically

10- 4 )

ERR

=

allowable relative error ~n any quantity determined

in the integration process (typically 10-5 )

The subroutines prints the run number and option, and terminates.

CALL FLOW (CC1. PO! TO, VO, RVO! EM, OPT)

This subroutine is given the constant CCl

=

Cl ~n Eq. (21)

which enable the initial pitot pressure

P~~.

0)

=

eCI to be

found. (Note that Cl is such that p~ is in psi.). It is also

given the initial values of supply pressure PO

=

po(lb/ft2 ) and

supply temperature TO

=

TO(OK).

CALL RHO (PO, TO! RO)

CALL TROPY (RO, TO, SO)

CALL ENTAL (PO,RO? TO. HO)

The density RO

=

Po (slugs/ft 3)9 the dimension less

entropy SO

=

so

IR,

and the enthalpy HO

=

hO ft-lbs/slug are

obtained by calling real nitrogen subroutines. These subroutines employ the equations derived by Culotta and Enkenhus (Ref. 12)

which match the AEDC tables of Grabow and Brahinsky (Ref. 13)

to better than 1% for densities up to 1000 amagats 9 and

tempe-ratures from 300 to 5000oKo

Next, the equivalent perfect gas supply conditions TOP

=

TO and POP = PO are calculated by Eqso (A-l) and (A-2),

res-p p

pectively. The initial free stream velocity VO

=

V(O,O) (meters/

sec), the mass flux per unit area RVO

=

PoVo (kg/m2-sec), and

the Mach number EM

=

M(09 0 ) are th en calculated using perfect

gas relationso Since the Rayleigh supersonic Pitot formula,

Eq. (A-3) is an implicit equation in M, it is solved by an

CALL DATl (AO, DT i TF, ERR)

This subroutine prints model and flow constants. It

also prints heading for experimental data when NXOZ

= 1.

CALL CUBE

This subroutine determines the coefficients XDBI. XDB2. XDB3 of the cu~ic {X

=

XDBI + XDB2 0T + XDB3;T2 + XPO}

loeo x

=

al + a2 t + a3 t + Xo

to x(t) and ZDBl, ZDB2, ZDB3 of the cubic

{

z

=

ZDBl + ZDB20T + ZDB3oT 3 + ZPO} to zet).

i.e. z = bI + b2t + _{b 2}t 2 + Zo

The coefficients are such that x and z are in meters when

T

=

t is in seconds .

The subroutine also obtains the coefficients DAO, DAl,

DA2 in the quadratic {ALC

=

DAO + DAl.T + DA20i}

for eet) ioeo 8

=

Co + clt + C2 t2

(8 in degrees i T

=

t in seconds) by least squareso

Card 7 00 0 7+N READ (2,109 ) TIMS(I). SPM(I)i ZPM(I), ALPR(I)

109 FORMAT (F 703~ F 803~ F 803, F

6

02)

These N data cards contain experiment~l data obtained

from spark photographs of the model motion, giving the

time t(ms), and experimental values of x and z (mm)

and the pitch angle e (degrees)o x and z are taken as

zero at the first data pointo The values are corrected

later by adding xo~ zo~ respectivelyo The program accepts

an arbitrary number of data cardso

Card 7+N+IA blank card indicates end of experimental datao

WRITE (la 110) TIMS(I)a XPM( I). XCa DEXg ZPM(I), ZC i DEZ,

ALPR(I)p ALCp DEA

Subroutine DATI now prints t~ xe'

8, ~e (time in ms, lengths ln mm,

Xg óx, Z , z, AZ,

e

,

c e

angles in degrees).

The experimental data and the accuracy of the least square

30

-CALL DISP

This subroutine calculates CD and CL from the cubic fits of the x and z motionsi using the equations

C x

=

'qïS

xm

C z CD

=

C

_x

cosö + C

_z

sinö C

=

C cosö - C sinö L z x

zm

= qlS

The Newtonian values CD

N, CLN are calculated by calling subroutine FORCEo

WRITE (1 ~ 604) AT, CD ~ CDN. CL, CLN Values of CD

=

CD, CD

N

=

CDN, CL

=

CL, and CLN

=

CLN

are printed out each millisecond for times t

=

AT < t f (millisec)

The cubic functions for x and z imply a linearly varying

curvature, which do~not adequately represent the actu.l vari-ation. For this reason, the results printed by subroutine DISP only give an indication of the drag and lift coefficient to be expected fr om the experimental datao The results are useful

~n indicating the consistency of the data at various times

during the run o

CALL START (Y, DERY. PRMT. TIME. NDIM, N)

This subroutine first sets the initial values of the

integrals 11 - Is

=

Y(I) (I

=

1,8) required in the

displace-ment method, equal to zero (See Eqso

(

5

)

and

(8))0

In the Runge-Kutta integration package used, the initial values of DERY{I)

(I

=

1,8) are inversely proportional to the anticipated average

magnitudes of 1

1 and must sum exactly to unityo The computer

uses this information to ensure that each 1

1 wil 1 have less than

the specified related error af ter integrationo

Other quantities which must be specified are

NDIM

=

8

=

number of differential equations

PRMT(l)

=

0

=

the initial time (time is measured from to) PRMT(2)

=

TIME{N)

=

tf - to

PRMT(3) = 0.05 )( TIME (N) = _Á t , the integration step S1ze

1

(t f t 0 )

=

₂₀

-PRMT(4)

₌

.0001 = a110wable fractional error in the

integrat ion process

CALL RKGS (PRMT. Y! DERY. NDIM. IHLF. FCT7. OUTP7 J AUX)

This statement ca11 the standard IBM Runge-Kutta inte-gration routine RKGS which integrates an arbitrary number of ordinary differential equations. The operation of RKGS is fu11y described in Ref. 14.

CALL FCT7 (T. Y, DERY)

This subroutine supplies the differential equations found 1n section 2.1 to subroutine RKGS. For the displacement method these are : dIl DERY(l)

=

_dt"

= _CDN (a)qlcoso dI₂ DERY(2) =

-

= _{CL N} (a)qlsino dt dI3 DERY(3) =

_dt"

= _CDN (a)qlsino dI4 DERY(4) =

-

= _{CLN (a)qlcoso} dt dIs DERY(S) =

_dt"

= 1₁ = Y(l) DI6 DERY(6)

₌

-

= 1₂ = Y(2) dt DI7 DERY(7)

=

_dt"

= 13= Y ( 3 ) DIa DERY(8)

₌

_~ = 14 = Y(4)

32

-CALL OUTP7 (Tt Y, DERY! IHLF, NDIM, PRMT)

Subroutine OUTP7 is entered each time an integration step has been calculated by RKGS. It only contains a dummy sta-tement. When the integration is finished control reverts to the ma1n programme.

CALL DISP2 (Y, IHLF)

Subroutine DI SP presents the results of calculations by the displacement methode (If IHLF ~ 11, there is an

integra-tion error). For a sphere test, it calculates and writes the

initial impulse, 10, and velocity

Uo

from Eqs. (18) and

(6)

respectively. For a cone test, the subroutine prints 10 and the

values of the integrals I(i), i

=

1,8. It finishes by printing the calculated values of the fraction of the Newtonican values of the drag and l i f t coefficients, e

d and et' respectively (Eqs.

(9)

and (10» and the initial values of velocity

Uo

and

If we have OPT

=

1 or

3,

the programme computes the motion using the curvature method.

CALL CURV (AO, FT, DT, ERR)

Subroutine CURV may be used to calculate theoretically only the pitching oscillations of a cone (if the control

variabIe NXOZ

=

2) or to calculate by the curvature method e

d

and et of free-flying non-oscillating models by comparing predicted motion with experimental data (if NXOZ

=

1).

The case for

NXOZ

=

2 15 dealt with in Ref. 2.

Preparations are first made to begin the integration process. The initial values of the integrals are :

Y(l)

₌

x eo Y(2)

₌

z eo Y(3)

₌

_uo Y(4)

₌

_Wo

Again as in subroutine START, the initial values of DERY(I)i (I

=

li4) are inversely proportional to the anticip~ted

magnitudes of 11 and must sum to unity.

Other quantities that must be specified are

NDIM

=

4 number of differential equations PRMT(l)

=

0

PRMT(2)

=

TIME(N)

=

tfinal - tinitial PRMT(3)

=

Ät. the integration step size

PRMT(4)

=

00001 allowable error in the integration process

See the comment statements ~n CURV which further explain the operation.

CALL RKGS(PRMT, Y, DERY. NDIM. IHLF, FCT6, OUTP6. AUX)

As before

CALL FCT6(T. Y, DERY )

This subroutine supplies the differential equations of the model motion to RKGS. When NXOZ

=

1, the differential equations to be integrated are

DERY(l)

=

u

=

Y(3) DERY(2)

=

w

=

Y(4)

DERY(3)

₌

x

(see Eqn. (2

»

DERY(4)

=

Z

(see Eqn. (3

»

- 34

-CALL OUTP6 (T, Y, DERY, IHLF, NDIM, PRMT)

When NXOZ

=

1, the subroutine calculates the errors DX and DZ and the quantity P (from Eq. (16)) required to find the curvature. The control then reverts to CURV, which carries out the iterative calculations of e

d and ei which as outlined in section 2.2, correspond to P

=

O. The final values of e

d

and ei match the data closest when convergence occurs. CURV writes each successive value of e

d and ei to give P

=

0 and indicates when convergence occurs. OUTP6 then writes T, X, 6X, Z 6Z and

Q.

Finally CURV writes the initial model velocity components, UO and WO.

The listing of the maln programme and the 17 sub-routines are given on the next pages, and several typical data runs are shown afterwards to illustrate the computer programme.

COHI.IOIl Q,ll;(OZ, IS, IT,IIP,lIQ,II,P, Tl, T2, T3,OX1,OXTl,0;:T2,DXB, T111E,XP

C01~1.IOU ZP , ALP, EI'I:I, UO, \10, OPT , Cl1, V 1, V, VO, RV, RVO, 1'0, TO, Eli COt,U'1011 TIIET, S, 0, 11, Cl, r. L, CD, Ct,IA, C:1D, Cll, 13ETA, QO, PI , TI, CG, EO, EL

co/mou X0131,XO~2,ÄDG3,ZOOl,ZOD2,ZOD3,OAO,DA1,DA2,CC1,CC2,CC3,TltlO,

1XPO, Z PO

C FREE-FLlGIIT I'ROGRA,I-- CO:1l3INEO VER510tl

REAO (2, 300110PT 3001 FORtlA TC F 1. 0) 999 CALL OAT2(AO,TF,OT,ERR) REAO(2,5555)EOF 5555 FORtIAT(E12.G) CC1-CC1-EOF

CALL FLOI'IC CC1, PO, TO, VO, RVO, EiI, OI'Tl S-3.11l1S9*0*0/4. IF(OPT-2)500,501,501 500 CALL DAT1(AO,DT,TF,ERf1) 501 A02AO*3.1 1l 1S9/180. QO=.SIl3*10132 1l .6/1 1l .G959*EXP(CC1) IF(UXOZ-2)S02,G18,S02 S02 CALL CUIlE IF(OPT-2)S03,503,G18 S03 CALL DISP

CALL STARTCY,DERY, PR:n, TI.I[,llDIII,lll

CALL RKGS (PRIIT, Y, OERY, IJO I :1, IIILF, FCT7, OUTP7, AUX) CALL DISP2(Y,IIlLF) G13 IF(OPT-2)G19,999,G19 G19 CALL CURV(AO,TF,OT,ERR) GO TO 999 WO /I XEQ 02

*LOCAL, OAT2, RIIO, TROPY, EIITA L, OA Tl, CU:lE, DIS P, START, FCT7 , OUTP7, DI:> P2, *LOCALFCTG,OUTPG C C C C C C C C C C 3000 C 10 C C 205 210 C C C C 200 202 C C C C C C SUBROUTINE DAT2 SUOROUTIIIE DAT2(AO,TF,OT,ERR)

DItlENS I 011 TI :IE(20), XP (20), ZP(20) ,ALP( 20)

COllMOti Q, I~XOZ, IS, t T , liP, IIQ, 11, P, Tl, T2, T3, 0;0, OXTl, Dn2, 0:<T3, T I :IE, XP COlli 10 ti ZP, ALP, Elm, UO, \'10, OPT, Ql, VI, V, VO, RV, RVO, PO, TO, Eli

COI'II.IOII THET, $, 0, 11, Cl, CL, CD, CilA, CiIJ, CU, BETA,QO, PI, TI, C::O, EO, E L C01.1t40N XDB 1, XOB2, :<O1l3, zoal, ZD[l2, ZOB3, DAO, OA1, OA2, CC 1, CC2, CC3, T I ~IO,

1XPO,ZPO

READ OATA-- IIXOZ= A COIITROL VARIAGLE, =1 l'IlIEII REDUCt,IG EXP. OATA,

AllD=2 111 mOUT EXP DATA, \/llEII :IOTl otl IS TO BE FOullD US I liG 11 E\1TO 111 All VALUES OF TIIE LIFT Alm ORAG COEFFICIEIITS.

TlI·IO= TIl·IE (l·11 LLl5EC) OF FIR5T SPARK PICTURE AFTE!': STARTlliG PRESSURE PEAK

XPO= IlllTiAL IlISTAlICE OF t40DEL CG AFT OF NOZZLE EXIT (14:1)

ZPO= ItllTIAL DI5T OF 1400EL CG Af}OVE tlOZZLE CEIITERLINE (1411)

CCl,CC2,CC3 ARE COEFFtCIEIIT'> lil QUADRATIC FIT OF PITOT PRE:iSUnE

DECAY I~ITII TIIIE. SEE PITOT DECAY PitOGRAil LISTlWl.

REAO OATA CARD 1 REAO(2,3000)IRUIl,PO,TO FORMAT(13,2E12.G) READ DATA CARD 2

READ (2,10) HXOZ, TI 1·10, X PO, Z PO, CC!, CC2, CC3 FORMAT(ll,3FS.I,3E12.G)

IF(NXOZ)205,20S,200

TBE COl4PUTER IDElnlFIES Tl-IE EtlD OF A SET OF EXP. DATA BY A [lLANK CARD, FOR HitlCII NXOZ=O.

IJRITECl,210)

FORMAT(/'EUO OF DATA') CALL EXIT

READ DATA CMD 3

AO= HIITIAL AfIGLE OF ATTACK (DEG) CIM= PI TClIlllG l~o;IENT COEFF I Cl EIIT CMO- OYNArIIC PITCII OAiU'llIG COEFFICIEln READ (2,202 )AO, CIIA, CliO

FORIIAT( GEI2. G) REAO DATA CARD Il

0= r40DEL [lASE DIA:IETER (1,11 LLlIIETERS) 11= I~ODEL \~E I GliT (GRAliS)

CI= HODEL PITCH 140ilEIlT OF IIIERTIA A[lOUT CG (KILOGRAIH~ETEI1:; SQ.)

TIIET= CONE SEIlI-VERTEX A~GLE (DEGREES)

P IS Il1iTiAL 114PULSE FRO/I SPIlERE TEST

REAO (2,202) O,I~,CI, TIlET,P

D=D/1000.

W=\'I*9.80665/1000. S=3.11l159*0*0/ k.

C CU-LEI/GTH OF COlliCAL NOZZLE FflOl1 TIlROAT TO EXIT (J4ETERS)

C CG- TEST GAS CONSTAIIT 111 MKS UrllTS

SUBROUTI NE DIIT2 PIlIlE 02

CG=2.96780E+02 C READ Dil TA CIIRD 5

C TF=FINAL TII·1E OF IrlTEGIlATION CSECONDS). \/IIEI'I rtEDUCIIIG EXP. DIlTIl,

C TF SIlOULO BE LONGER TIlAll TIIE TII1E OF TIlE LAST DIlTIl POlllT

C DTalilTEGRATION TlllE STEPCSECOlIDS). IIIIEil REDUCIIIG EXpErtl~IEt1TIIL

CDATA, TF flUST BE 11 SUI3-i·1ULTI pLE OF TIIE TII1E BETI/EEil OIITII POlIlTS. C [RR= ALLOI/IIOLE FRACTIOIJIIL ER!10R Itl II/TEGrtATION. O.f)O~l IS SUl TAB C PROGRA~1 AUTOI1ATI CALLY ADJUSTS IIITEGRIITI ON STEP TO KEEP ~II TIII N

C SPECIFIED ERROR. READC2,202) TF,DT,ERR

NN=OPT

WRITECl,100)IRUN,rJN

100 FORI1ATC' FREE-FLlGIIT Pf10GRl\tl RUil '13,' OPTION '11/1>

900 RETURrl END

SUBROUTINE FLOU

SUBROUT I NE F LOl/ C CCl, PO, TO, VO, RVO, Eil, 0 PTl C PO = INITIIIL SUPPLY PRESSURE CPSI) C TO - INITIAL SUPPLY TEMP. CDEG K)

PA-PO TA=TO

PO=PO*2116.22/14.6959 TO=TO*1.8

CALL RIIO CPO, TO, RO)

CALL TROPYCRO,TO,50) CALL EIITIIL(PO,RO, TO,1I0)

C EQUIVIILANT PERFECT GilS SUPPLY COIIOITIOIIS

TO P=llO/ 3.5/1776. POP=EXPC3.5*ALOGCTOP/l.8)+3.1379-S0)*2116.22 PITOT-EXPCCCl)*211G.22/1 4 .6959 FT-ALOGCPITOT/POP) EIU-I0. Ef.12-12. K-O G-EI1l 10 G2-G*G Ill-G. *G2/(02+5.) 112-6./0.*G2-1. ) 11l2-Hl*Hl 11 13=tI12*111 IU4-11l2*1112 11l7-H13*H14 H22-112*1I2 1i23 a 1l22*112 1125-H22*1123 II=SQRTCH17*H25 ) F=ALOGCf() DF=F-FT IFCABSCDF/FT)-.OOOlll,I,2 2 IFCK-1l3,4,4 3 Fl=F G=EI\2 K=1 GO TO 10 4 F2=F G= EI·\2- C EI·12-EIU l * C F 2 -FT> / C F2-Fl l EMl=EI12 Fl=F2 E/.12 =G GO TO 10 1 EI~=G RI·I=I. +0. 2*E;1*EI~ RI·12 = Rl1 * RI·\

SUB ROUTI NE F LOI'I

RII3=R~12 *R/·l RI14=RI·12 *RI·12 RI-17=RI·14*RlU RI·172=SQRTCRII7 ) TINF=TOP/RM PINF=POP/RIH2 RINF=PINF/CI77G.*TINFl V I NF=SQRTC7. *1776. * CTOP-TIIIF» VO-VI NF*.3048 RO=RINF*16.02*32.17 RVO-RO*VO IFCOPT-2)50,51,51 50 I/RITECl,1000)PA, TA,VO,RVO,EI"I

1000 FORMATC/'PO- 'EI2.5,' PSI TO- 'E12.5,' K VO= 'EI2.5,' M/SEC RVO= , lEI2.5,' KG/CM-S)**2 M= 'FG.2/)

51 RETURN WO

C CALCULATES OEIISITV R (SLUGS/FT**3l PC=70912.8 VC=1>6.3641 TC=22 6.98 1~=28. 016 RIIOC=H/VC ZC-0.291 AI>=1.3131> AAl-2001>.919 AA2=1611.22S M3=S44.0S3 AAI>=1>3.SI> 1l1l1-21. 696 IlB2=73.S89 IlB3-37.161 1l1l1>=3.919 CC1=2G71>.2S6 CC2-1960.I169 CC3=47S.934 001-0.4826 002=.8107 003-.31>21 001>=.05396 A=S.98 BE=6.26 CO-S.5 XC=l.+IlE C1=XC*XC*XC C2=IlE*C3.*IlE-1. l C3-3.*BE**2-G.*BE-1. C4-IlE* (BE-3. ) CFC-0.188607801>1 AS=2.*(CO-A4)+IlE AG-AS-BE OB=(1.-CO-A+2.*BE)/2. B1-C3/C2 TR=T/TC TR2-TR*TR TR5-SQRT CTR) TR2 5-TR2 *TRS

C F I RST GUESS FOR RIIO IS BASED OtJ All EL-1I01l LE GAS RIl-RIlOC/ (TR*PC/ (PO*ZC) +B 1)

502 J=O

SUil ROUT IIJE RIIO 500 J=J+1 RR=RIl/RHOC RR2=RR*RR Rfl3=RR2*RR lll= (M* O. -l./TR )+AS/T:l2-'\6/ H2 5) *Rfl2 U2=OB*(1.-1./TR2)*RR3 P=- PC*TR* Wl+ln)

OP=- (2. *IH+3. *ln) *TR/ Rfl*PC/RIiOC IF(RR-l. )2,2,3 2 0=C2-C3*RR+CI>*RR2 P=P+TR*PC*Cl*RR/O E02=0*D OP=OP+C1*(C2-CI>*Rfl2)*TR/ED2*PC/flIlOC GO TO 4 3 AA=AA1-AA2*RR+AA3*RR2-AAI1*RR3 BO=IlIlI-BB2*RR+IlB3*RR2-BIl4*RR3 CC=CC1-CC2*RR+CC3*RR2 00=001-DD2*RR+003*RR2-001>*RR3 TRE-EXP(DD*TR) PP=AA+BB*TR-CC*TRE RLOG=ALDG(RR) RLOG2=RLOG*RLOG P=P+PC*CFC*RLOG2*PP P=P+PC*TR*(1.+IlE*RR2l OP-OP+2. *BE*RR*TR *PC/ RIIOC AAO=-AA2+2.*AA3*RR-3.*AA4*RR2 BBO--BB2+2.*BIl3*RR-3.*BBI>*RR2 CCO=-CC2+2.*CC3*RR 000--D02+2.*DD3*RR-3.*ODI1*RR2 OW-2.*RLOG/RR*PP+RLOG2*(AAD+BBO*TR-CCO*TREl OP-CFC*PC/RHOC*(OU-RLOG2*CC*DOO*TR*TRE)+OP I> R-RII-(P-PO)/DP RT-ABS«R-RH)/RHl IF(RT-.00001l30,30,5 5 RIl-R IFCJ-50)SOO,500,501 501 WRITEC1,260)R

260 FORHATC/I1X181IRHO UNCOIJVERGEO -Ell. 11 l GO TO 502

30 RETURN EUO

PAGE 04 SUBROUTINE TROPY(R,T,SR)

C USES ENKEI~IIUS-CULOTTA EllUATIOil OF STilTE

C GIVEN DENSITY fl (SLUGS/FT**31 11110 TEilPEilllTUflE T (DEG. :l) C FINDS OlllENSIOllLESS EllTROPY Sf1=S/R

PC~70912.8 VC=I16.36111 TC=226.98 ZC=0.291 \1=28.016 RC=\I/VC AI1=1.31311 AA1=2004.919 1\1\2=1611.225 1\A3=5411.053 MI1=I13.311 IlB1=21. 696 IHl2=73.589 BB3=37.161 0011=3.919 CC1=26711.25G CC2=1960.I169 CC3=I175.9311 DD1=0.I1826 002=.8107 003=.31121 0011=.05396 A=5.98 OE=6.26 CO=5.5 XC= 1. +BE Cl=XC*XC*XC C2=BE*(3.*BE-1.1 C3=3.*BE**2-G.*BE-1. C4=BE*(BE-3. ) CFC=0.188G0780111 A5=2.*(CO-III1)+BE A6=A5-BE DO-(1.-CO-II+2.*BEI/2. SOR-13. TV-26.88311 CflT-SQflT(4.*C2*C4-C3*C3) C CALCULIITlO!1 OF S/fl Tfl~T /TC RR~R/ RC TLOG-ALOGCTR) TVT=TV/TR ETVT-EXPCTVTl SUBROUTINE TROPY RLOG=,\LOG(flR) Rfl2=flR*Rfl RR3=RR2*RR TR2=TR*Tfl TR5=SQf1T(TR) X=TVT/(ETVT-1. ) Y=ETVT/(ETVT-1.1 r.LOG2=RLOG*RLOG RLOG3=RLOG2*RLOG SR=SOR+2.5*TLOG+X+ALOG(Y) SR=SR+ZC*RR*(AI1*Tfl2-A5+1.5*IIG/Tr.S+0.5*DB*(T:l2+1.)*RR)/Tfl2 IF(RR-1. )2,2,3

C IlELO\l CfllTICI\L DEt/SITY

2 Zl=(2.*C2-C3*RR)/(RR*CRT) Z2=1./Rfl2-C3/(C2*flR)+CI1/C2

SR =SR+ (C3/CRTl *AT 1\1[( Zl) +0.5 *11 LOG (Z2) GO TO 11

C IIBOVE CRITICAL DEIISITY

3 Yl= (C3/CRT)*I\TAII( (2. *C2-C3 )/CflT)+O. 5*IILOG (1. + (CI1-C3 )/C2) Y2=-ZC*(1.-1./Rfl)*(1.+BE*RR) Y3=1l1l1*(RLOG2+2.*RLOG+2.1/flfl Y4=OB2*flLOG3/3. Y5=1l1l3*RR*(2.*RLOG-RLOG2-2.) Y6=0.5*BB4*RR2*(RLOG2-RLOG+0.5) Y7=2.*(BIl3-B81)-0.25*IlBI1 Y8=EXP(2.I109+1.*(Rfl-1.)/TR)-E=P(2.I10~1 Sfl=Sfl+Y1+Y2+ZC*CFC*(Y3+YI1+Y5+Y6+Y7-Y8) 11 RETURN ENO