Investigations of the Transonic Flow Around Oscillating Airfoils

DOCUMENT CONTROL SHEET O R I G I N A T O R ' S REF.

NLR TR 77090 U

i

SECURITY CLASS.

Unclassified

ORIGINATOR

National Aerospace Laboratory (NLR) Amsterdam. The Netherlands TITLE

Investigations of the transonic flow around oscillating airfoils

AUTHORS

DATE

H. Tijdeman

I

21-x-77

PP

146

ref

I88

DESCRIPTORS

Supercritical wings Ai rfoi 1 s Unsteady flow Transonic flow Pressure distribution Pressure measurements

Wind-tunnel tests Wind-tunnel walls Wing oscillations Aerodynamic loads

ABSTRACT

Exploratory wind-tunnel experiments in high-subsonic and transonic flow on a conventional airfoil with oscillating flap and a supercritical airfoil oscillating in pitch are described. I n the analysis of the experimental results, emphasis is placed upon the typical aspects of transonic flow, namely the interaction between the steady and unsteady flow fields, the periodical motion of the shock waves and their contribution to the overall unsteady airloads. Special attention is paid to the behaviour of the supercritical airfoil in its "shock-free'' design condition. Moreover, it is discussed to what extent linearization of the unsteady transonic flow problem is allowed if the unsteady field is considered as a small perturbation superimposed upon a given mean steady-flow field. Finally, the current status of unsteady transonic flow theory is reviewed and the present test data are used to evaluate some of the recently developed

NLR TR 77090 U

INVESTIEATIONS OF THE TRANSONIC FLOW AROUND OSCILLATING AIRFOILS by

H. Tijdernan

SUMMARY

E x p l o r a t o r y w i n d - t u n n e l experiments i n high-subsonic and t r a n s o n i c f l o w on a c o n v e n t i o n a l a i r f o i l w i t h o s c i l i a t i n g f l a p and a s u p e r c r i t i c a l a i r f o i l o s c i l l a t i n g i n p i t c h are described.

I n t h e a n a l y s i s o f t h e exper-

imental r e s u l t s , emphasis i s p l a c e d upon t h e t y p i c a l aspects o f t r a n s o n i c flow,

namely t h e i n t e r a c t i o n between t h e steady and unsteady f l o w f i e l d s ,

the p e r i o d i c a l m o t i o n of the shock waves and t h e i r c o n t r i b u t i o n t o t h e o v e r a l l unsteady a i r l o a d s . S p e c i a l a t t e n t i o n i s p a i d t o t h e b e h a v i o u r o f t h e s u p e r c r i t i c a l a i r f o i l i n i t s "shock-free" it

design c o n d i t i o n .

Moreover,

i s d i s c u s s e d t o what e x t e n t l i n e a r i z a t i o n o f t h e unsteady t r a n s o n i c

f l o w probiem i s a l l o w e d i f t h e unsteady f i e l d i s considered as a small p e r t u r b a t i o n superimposed upon a g i v e n mean s t e a d y - f l o w f i e l d .

Finally,

the c u r r e n t s t a t u s of unsteady t r a n s o n i c f l o w t h e o r y i s reviewed and t h e p r e s e n t t e s t d a t a a r e used t o e v a l u a t e some o f t h e r e c e n t l y developed c a l c u l a t i o n methods.

D i v i s i o n : F l u i d Dynamics Prepared: HT Approved: HB

6

Completed

: 21-X-77

Ordernumber: 524.109/101,618 TYP.

: H6

T h i s r e p o r t s e r v e d t h e a u t h o r as a t h e s i s t o o b t a i n a Ph. D . degree o f D e l f t T e c h n o l o g i c a l U n i v e r s i t y

SUMMARY

Exploratory wind-tunnel experiments in high-subsonic and transonic flow on a conventional airfoil with oscillating flap and a supercritical airfoil oscillating in pitch are described. I n the analysis of the experimental results, emphasis is placed upon the typical aspects of transonic flow, namely the interaction between the steady and unsteady flow fields, the periodical motion of the shock waves and their contribution t o the overall unsteady airloads. Special attention is paid to the behaviour of the supercritical airfoil in

it5

"shock-free" design condition. Moreover,

i t is discussed to what extent linearization o f the unsteady transonic

a smali perturbation superimposed upon a given mean steady-flow field. Finally, f l o w problem is allowed if the unsteady field is considered a s

the current status o f unsteady transonic flow theory i s reviewed and the present test data are used to evaluate some o f the recently developed calculation methods.

- 5-

CONTENTS Page I

BACKGROUND AN0 OUTLINE OF T H E S I S

9

I. I

Background

9

1.2

Outiine of thesis

11

PART i: INTRODUCTORY CHAPTERS 2

THE FLOW AROUND OSCILLATING AIRFOILS

15

2.1

D e s c r i p t i o n o f unsteady a i r i o a d s

15

Some n o t e s on t h e unsteady-flow equations

16

2.2

2.2.1 2.2.2 2.2.3 2.3

3

The unsteady-flow equations Moderately subsonic and supersonic f l o w Transonic flow

M A I N CHARACTERISTICS

3.1

19

Present s t a t u s o f t h e research on unsteady t r a n s o n i c f l o w

OF THE STEADY TRANSONIC FLOW AROUND AIRFOILS

20 20

Transonic f l o w s w i t h embedded shock waves

3.1.1 3.1.2

Development o f f l o w p a t t e r n w i t h Mach number, f l a p angle, and i n c i d e n c e C h a r a c t e r i s t i c s o f a normal shock wave

3.2

Shock-free f l o w

22

3.3

Some p a r t i c u l a r f l o w p a t t e r n s on a i r f o i l s w i t h f l a p

22

3.4

Viscous aspects

24

PART l i : SCOPE AND DESCRIPTION OF THE EXPERIMENTAL INVESTIGATIONS

4

5

SCOPE OF THE NLR INVESTIGATIONS

29

4.1

Problem d e f i n i t i o n

29

4.2

Approach

29

TECHNIQUE FOR UNSTEADY PRESSURE MEASUREMENTS

30

5.1

P r i n c i p i e . o f t h e measuring technique

5.2

T h e o r e t i c a l model f o r t h e dynamic response of tube-transducer 5.2.1 5.2.2

5.3

5.3.2

5.3.3

5.4.2

33

The dynamic response i n s t i l l a i r Influence o f the a i r f l o w V e r i f i c a t i o n i n a j o i n t ONERA-NLR i n v e s t i g a t i o n

P r a c t i c a l a p p l i c a t i o n i n wind-tunnel

5.4.1

31

The p r o p a g a t i o n o f p r e s s u r e waves through c y l i n d r i c a l tubes S o l u t i o n f o r complete tube-transducer systems

The dynamic c h a r a c t e r i s t i c s o f t u b e - t r a n s d u c e r systems

5.3.1

5.4

30 systems

35

tests

Choice and c a l i b r a t i o n o f t u b e - t r a n s d u c e r systems Measuring equipment and data r e d u c t i o n

38

6 WINO-TUNNEL MODELS AND TEST SET-UP NACA 64A006 a i r f o i l w i t h f l a p

38

6.2

NLR 7301 a i r f o i l

39

6.3

Wind tunnel

41

6.4

Optical flow studies

41

6.1

7 TEST PROGRAM

42

7.1

NACA 64A006 a i r f o i l w i t h f l a p

42

7.2

NLR 7301 a i r f o i l

42

-6-

PART I l l : ANALYSIS OF RESULTS

8

THE INTERACTION BETWEEN THE STEADY AND UNSTEADY FLOW FIELD

8.1

I n t r o d u c t o r y remarks

47

8.2

The i n f l u e n c e o f Mach number on the a i r l o a d s o f t h e NACA.64A006 a i r f o i l w i t h f l a p

47

8.2.1 8.2.2

8.2.3

G r a p h i c a l experiment

53

8.4

The I n f l u e n c e o f i n c i d e n c e and mean f l a p a n g l e

54

8.4.2

8.5

58

Unsteady p r e s s u r e d i s t r i b u t i o n s Unsteady aerodynamic c o e f f i c i e n t s 62

ON THE P E R I O D I C A L MOTION OF SHOCK WAVES 9.1

I n t r o d u c t o r y remarks

62

9.2

Shock s t r e n g t h and shock p o s i t i o n i n steady f l o w

62

9.3

Types o f shock-wave m o t i o n observed i n unsteady f l o w

64

9.4

I n t r o d u c t i o n o f an a n a l y t i c a l model

66

9.4.1 9.4.2

9.5

R e l a t i o n between shock p o s i t i o n and shock s t r e n g t h A p p l i c a t i o n o f t h e a n a l y t i c a l model

69

A d d i t i o n a i remarks

9.5.1 9.5.2

Some p r o p e r t i e s o f t h e unsteady shock r e l a t i o n s P o s s i b l e use o f t h e shock-wave model

THE UNSTEAOY AERODYNAMIC CHARACTERISTICS OF THE "SHOCK-FREE'' NLR 7301 AIRFOIL

70

10.1

I n t r o d u c t o r y remarks

70

10.2

Unsteady p r e s s u r e d i s t r i b u t i o n s

71

10.2.1 10.2.2 10.2.3

F u l l y subsonic f l o w ( c o n d i t i o n I ) Transonic f l o w w i t h shock wave ( c o n d i t i o n 1 1 ) The "shock-free'' d e s i g n c o n d i t i o n ( c o n d i t i o n I l l )

75

10.3

Unsteady aerodynamic c o e f f i c i e n t s

10.4

Remarks on the motion o f t h e shock wave

78

10.5

The i n f l u e n c e o f the t r a n s i t i o n S t r i p

81

10.6

Some a d d i t i o n a l e f f e c t s

83

10.6.1 10.6.2 10.7 I1

Unsteady p r e s s u r e d i s t r i b u t i o n s Unsteady aerodynamic c o e f f i c i e n t s

The i n f l u e n c e o f frequency

8.5.1 8.5.2

10

Steady p r e s s u r e d i s t r i b u t i o n s Unsteady p r e s s u r e d i s t r i b u t i o n s Unsteady aerodynamic c o e f f i c i e n t s

fl.3

8.4.1

9

47

The e f f e c t o f Mach number The e f f e c t o f t h e a m p l i t u d e o f o s c i l l a t i o n

84

Concluding remarks

SOME CONSIDERATIONS ON A LINEARIZED TREATMENT OF UNSTEADY TRANSONIC FLOWS I1 . 1

11.2

I n t r o d u c t o r y remarks

86

Flow c o n d i t i o n s w i t h an o s c i l l a t i n g shock wave

86

11.2.1 11.2.2

11.3

Local e f f e c t s o f a shock wave C o n t r i b u t i o n o f a shock wave t o t h e o v e r a l l aerodynamic loads 89

Special flow conditions

li.3.i 11.3.2

11.3.3

11.4

86

"Shock-free" f l o w Flow w i t h a double shock Flow around an a i r f o i l w i t h f l a p 92

Concluding remarks

-7-

PART I V : THE CURRENT STATUS OF UNSTEADY FLOW T H E O R Y ANO EVALUATION OF SOME NEW METHODS

Page

FOR UNSTEADY TRANSONIC FLOW 12

REVIEW OF CALCULATION METHODS FOR TWO-DIMENSIONAL UNSTEADY FLOW 12.1

C l a s s i f i c a t i o n of the v a r i o u s methods

95

12.2

L i n e a r i z e d subsonic l i f t i n g - s u r f a c e t h e o r y

99

12.2.1 12.2.2 12.2.3 12.3

12.3.2

Local-Mach-number corrections i n l i n e a r i z e d l i f t i n g - s u r f a c e t h e o r y Methods based on the i i n e a r i z e d t r a n s o n i c s m a l l - p e r t u r b a t i o n equation

Wethods f o r n e a r - s o n i c f l o w w i t h o u t shock waves

102

12.5

Methods f o r t r a n s o n i c f i o w w i t h shock waves

102

12.6

General remarks Methods based on t h e E u l e r equations Methods based on the p o t e n t i a l e q u a t i o n

104

Role o f t h e NLR r e s u l t s

EVALUATION OF SOME NEW CALCULATION METHODS FOR UNSTEADY TRANSONIC FLOW

105

13.1

I n t r o d u c t o r y remarks

105

13.2

Comparisons between t h e o r y and experiment i n steady and quasi-steady flow

106

13.2.1 13.2.2 13.2.3

13.2.4

13.3

13.3.2

13.3.)

13.4

C o r r e c t i o n for tunnel-wal i i n t e r f e r e n c e Subsonic f i o w T r a n s o n i c f l o w w i t h shock wave "Shock-free" f l o w

Comparisons between t h e o r y and experiment i n unsteady f l o w

13.3.1

i5

1O0

12.4

12.5.1 12.5.2 12.5.3

14

The i n t e g r a l e q u a t i o n r e l a t i n g downwash and ioad d i s t r i b u t i o n The K e r n e l - f u n c t i o n method The D o u b l e t - L a t t i c e method

Hethods f o r h i g h - s u b s o n i c f i o w 12.3.1

13

95

Ill

Pressure d i s t r i b u t i o n s Aerodynamic c o e f f i c i e n t s Shock-wave motions

117

Concluding remarks

IMPACT OF THE NLR INVESTIGATIONS AND FUTURE PROSPECTS

ii8

14.1

Impact o f t h e NLR i n v e s t i g a t i o n s

118

14.2

Future prospects

118 1 19

REFERENCES

A P P E N D I X A : DEFINITION OF STEADY AND UNSTEADY AERODYNAMIC QUANTITIES APPENDIX E : THE DYNAMIC RESPONSE OF TUBE-TRANSDUCER SYSTEMS APPENDIX C :

DERIVATION OF THE QUASI-STEADY AN0 UNSTEADY SHOCK RELATIONS

APPENDIX D :

LIST OF SYMBOLS

A P P E N D I X E : SUMMARY I N DUTCH (SAMENVATTING I N HET NEDERLANDS)

-8-

1

BACKGROUND'AND OUTLINE OF THESIS

For t h e t r a n s o n i c f l i g h t regime, w i t h i t s mixed

BACKGROUND

I .I

subsonic-supersonic flow p a t t e r n s , these means a r e l e s s developed. Here the a e r o e l o s t i c i a n i s

Under c e r t a i n c o n d i t i o n s , s t r u c t u r e s l i k e a i r p i a n e wings and t a i i surfaces may experience

hampered s e r i o u s l y by the l a c k o f e f f e c t i v e c a i -

v i b r a t i o n s o f an u n s t a b l e n a t u r e .

c u l a t i o n methods t o determine t h e unsteady a i r

T h i s phenomenon, c a l i e d " f i u t t e r " ,

loads. For wing s e c t i o n s i n two-dimensional flow,

i s an aero-

e l a s t i c problem, determined by the i n t e r a c t i o n o f

c a l c u l a t i o n methods become a v a i l a b l e a t the moment,

the e i a s t i c and i n e r t i a l forces o f t h e S t r u c t u r e

b u t t h e c u r r e n t p r a c t i c e f o r wings o f general

and the unsteady aerodynamic forces generated by

planform s t i l l i s t h a t rather a r b i t r a r y interpola-

the o s c i i l a t o r y m o t i o n o f the s t r u c t u r e i t s e l f .

t i o n s and e x t r a p o l a t i o n s a r e being made on the

In g e n e r a l , two o r more v i b r a t i o n modes a r e i n -

b a s i s o f c a l c u l a t e d a i r l o a d s f o r pure subsonic and

volved

-

supersonic f l o w .

f o r i n s t a n c e bending and t o r s i o n a l v i b r a -

t i o n o f a wing

-

t o v e r y expensive wind-tunnel

which, under t h e i n f l u e n c e o f the

unsteady aerodynamic forces,

In many cases, one has t o r e s o r t

-

i n t e r a c t w i t h each

SUPERSONIC TRANSPORT WING ( R E F . 6 ! SUBSONIC SWEPT WING lREF.j! SPACE SHUTTLE WING iREF.71

d

o t h e r such t h a t the v i b r a t i n g S t r u c t u r e e x t r a c t s

I

energy from the passing a i r s t r e a m . T h i s leads t o a progressive increase i n amplitude o f v i b r a t i o n ,

YI w

u s u a l l y ending up i n a d i s i n t e g r a t i o n o f t h e

m

structure.

=IL

cc

experiments.

0.3

.

MACH CORRECTION OF 'REF.3 19465

i

As f o r a g i v e n c o n f i g u r a t i o n o f a s t r u c t u r e the unsteady aerodynamic f o r c e s i n c r e a s e r a p i d l y w i t h f l i g h t speed, w h i l e the e l a s t i c and i n e r t i a f o r c e s remain almost unchanged, n o r m a l l y t h e r e e x i s t s a c r i t i c a i f i i g h t speed ( " f l u t t e r above which f l u t t e r

OCCU~S.

speed"),

n u t t e r speed versus Ilach number curve showing the "transonic dip".

~ i g .1.1

Because o f t h e d i s -

a s t r o u s c h a r a c t e r o f the phenomenon, the a i r c r a f t

T h i s s i t u a t i o n i s v e r y u n s a t i s f a c t o r y , es-

x a n u f a c t u r e r s have t o prove t h a t t h e f l u t t e r soeeds of t h e i r ?roducts a r e w e l l o u t s i d e t h e

p e c i a l l y s i n c e experience shows (Refs. 1-4)

F l i g i t onveiope, and i n t h i s r e s p e c t they have t o

f l u t t e r problems o f t e n become most c r i t i c a i f o r

n e e t severe a i r w o r t h i n e s s requirements.

t r a n s o n i c - f i o w c o n d i t i o n s . The main reason f o r

I n many cases the demands

that

t h i s i s t h e r a t h e r p e c u l i a r behaviour o f the un-

for f l u t t e r f r e e -

dom a r e t h e d e t e r m i n i n g f a c t o r s f o r t h e c o n s t r u c -

steady aerodynamic forces i n t r a n s o n i c f l o w s ,

:ion o f *lings a r d t a i i surfaces.

p a r t i c u l a r when s t r o n g shock waves a r e i n v o l v e d .

For t h i s reason,

in

much a t t e n t i o n has been p a i d t o t h e development o f

T h i s i s r e f l e c t e d , f o r i n s t a n c e , i n the behaviour

adequate c a l c u l a t i o n methods t o p r e d i c t the f l u t -

of t h e f l u t t e r speed f o r b e n d i n g - t o r s i o n f l u t t e r

:er

as a f u n c t i o n o f Mach number ( F i g . 1 . 1 1 , which

c h a r a c t e r i s t i c s o f a i r c r a f t . The v i b r a t i o n

c h a r a c t e r i s t i c s o f t h e S t r u c t u r e a t zero a i r s p e e d

shows t h e s o - c a l l e d " t r a n s o n i c d i p " ,

c a n be determined a c c u r a t e l y by s o p h i s t i c a t e d

r e l a t i v e l y low f l u t t e r speeds i n t h e t r a n s o n i c

c a l c u l a t i o n methods o r by ground v i b r a t i o n t e s t s .

f l i g h t regime.

Therefore,

wind-tunnel

the accuracy o f the f l u t t e r p r e d i c t i o n

a region o f

I n a d d i t i o n t o t h e r e s u l t s o f some

i n v e s t i g a t i o n s (Refs.

5 - 7 ) , a l s o the

depends m a i n l y on the knowledge o f t h e unsteady

Mach-number c o r r e c t i o n as proposed i n 1946 f o r a

aerodynamic f o r c e s .

f l u t t e r c r i t e r i o n f o r wing t o r s i o n a l s t i f f n e s s (Ref. 8)

I n t h e subsonic and supersonic f l i g h t

i s given i n Tigure 1.1,

the presence o f an o l d problem.

regimes, the unsteady aerodynamic f o r c e s can be p r e d i c t e d ressanabiy w e l l by t h e o r e t i c a l means.

-9-

which i l l u s t r a t e s

The f i r s t t r a n s o n i c - f l u t t e r problems were encountered d u r i n g w o r l d w a r li

p i a c e w i t h s t r o n g shock w a v e s , a s an the conven-

by a i r c r a f t o f

advanced d e s i g n a t t h a t time (Typhoon,

t i o n a i - t y p e wings, b u t w i t h o n i y v e r y weak snock waves o r even w i t h o u t them.

Fury),

which were a b l e t o p e n e t r a t e t h e t r a n s o n i c regime

The advantages o f i o c a l supersonic regions

d u r i n g a d i v i n g f l i g h t . A number o f a i r c r a f t i o s t

On

a i l e r o n s and t a i l s , sometimes ending up i n f a t a l

u t i l i z e d i n s e v e r a l ways. For i n s t a n c e , i n compar-

t h e wing w i t h o u t n o t i c e a b l e wave drag can be

a c c i d e n t s . These e a r i y experiences gave t h e t r a n -

i s o n w i t h c o n v e n t i o n a l wings, one may use t h i c k e r

s o n i c regime i t s v e i l o f m y s t i c i s m and c o n t r i b u t -

wings o r reduced sweep angles a t che same c r u i s e

ed t o t h e many myths t h a t came i n t o b e i n g about

Mach number. Another p o s s i b i l i t y i s t o increase

the d i f f i c u l t i e s a s s o c i a t e d w i t h c r o s s i n g t h e “sound b a r r i e r “ .

t h e c r u i s e Mach number a t c o n s t a n t wing t h i c k n e s s

A t t h a t t i m e i t was i m p o s s i b l e

and sweep angle. The f i r s t a p p l i c a t i o n w i l l lead

t o g e t aerodynamic d a t a for t h e t r a n s o n i c range, because t h e r e were no t r a n s o n i c wind t u n n e l s a v a i l -

t o a more e f f i c i e n t wing c o n s t r u c t i o n , the second w i l l l e a d t o an increased c r u i s i n g speed, b o t h

a b l e and t h e r e was l i t t l e o r no s u p p o r t by theo-

w i t h o u t e x t r a drag p e n a l t y .

r e t i c a l means.

i n c o n t r a s t w i t h the f i r s t phase i n t h e h i s -

D u r i n g t h e f i r s t f i f t e e n years a f t e r t h e war, t h e knowledge o f t r a n s o n i c flows

t o r y o f t r a n s o n i c f l i g h t , t h e r e c e n t developments

i s improved

a r e s u p p o r t e d - a t l e a s t as f a r a s steady f l o w i s

-

c o n s i d e r a b l y by the experience g a i n e d w i t h a num-

concerned

b e r o f f l y i n g models and research a i r c r a f t ,

o r i e n t e d c a l c u l a t i o n methods, which enable t h e

t h e Bel1 X - i ,

like

and the development o f t r a n s o n i c

by a number o f h i g h l y computer-

p r e d i c t i o n o f t r a n s o n i c f l o w p a t t e r n s around a i r -

wind t u n n e l s w i t h s l o t t e d and porous w a i l s , which

f o i l s and wings, w i t h and w i t h o u t shock waves.

g r e a r i y enlarged the p o s s i b i l i t i e s f o r obtaining

I m p o r t a n t c o n t r i b u t i o n s i n t h i s respect a r e the

aerodynamic data under c o n t r o l l e d c o n d i t i o n s . The

methods developed by Nieuwland (Ref. 9 ) and

main i n t e r e s t i n t h i s p e r i o d , however, was n o t

B o e r s t o e i (Ref. 1 0 ) f o r the design o f SuDerCrit-

t h e t r a n s o n i c regime i t s e i f . The a t t e n t i o n was

ical airfoiis.

focussed p r i m a r i l y on the development o f m i l i t a r y

methods f o r steady t r a n s o n i c f l o w ,

a i r c r a f t f o r supersonic operations,and

made t o references 11-16,)

the tran-

s o n i c speed range was o n l y a t r a n s i e n t phase t h a t

(For a review

of comoutational reference i s

The e x t e n s i v e compu-

t a t i o n s r e q u i r e d f o r t h i s purpose became p o s s i b l e

had t o be passed s a f e l y w i t h o u t e x c e s s i v e drag

by the enormous development i n computer technolo-

r i s e and w i t h o u t severe v i b r a t i o n and s t a b i l i t y

gy and numerical mathematics d u r i n g the i a s t two

problems.

decades.

T h i s s i t u a t i o n has changed s i n c e t h e l a t e

I n a d d i t i o n , the knowledge about the

p h y s i c a l behaviour o f steady t r a n s o n i c f l a w s has

s i x t i e s , when a renewed i n t e r e s t f o r t r a n s o n i c

been i n c r e a s e d c o n s i d e r a b l y by t h e fundamental

f l i g h t s t a r t e d . A s f a r a5 m i l i t a r y a v i a t i o n i s

experimental

concerned,

Pearcy e t a l .

t h i s increased i n t e r e s t stems from t h e

demand f o r a new g e n e r a t i o n of “air-combat fighters”,

(Refs.

171,

18, 19) and Spee (Ref. 20).

As a l r e a d y mentioned, t h e s i t u a t i o n w i t h

l i k e the F-16 and F-17, which r e q u i r e

respect t o unsteady f l o w i s s t i l l v e r y u n s a t i s -

an o p t i m a l m a n o e u v r a b i l i t y under t r a n s o n i c - f l o h conditions.

i n v e s t i g a t i o n s o f Holder (Ref.

f a c t o r y . S t i m u l a t e d , however, by t h e renewed i n -

In c i v i l aviation, the interest i s

t e r e s t i n t r a n s o n i c f l i g h t and the encouraging

s t i m u l a t e d by the new concept o f t h e s o - c a l l e d

p r o g r e s s i n s t e a d y - f l o w computations, t h e e f f o r t s

“supercritical

t o s o l v e t h e unsteady-flow problem have become

wing8‘, which should make i t pos-

s i b l e t o t r u i S e a t t r a n s o n i c speeds w i t h o u t the

s t r o n g e r than ever b e f o r e .

usual drag p e n a l t y a s s o c i a t e d w i t h t h e presence

I t i s c l e a r t h a t t i i e 5ucce55 o f new c a l c u -

o f shock waves. T h i s can be achieved by shaping

l a t i o n methods f o r unsteady t r a n s o n i c f l o w w i l l

the wing geometry i n such a way t h a t t h e t r a n s i -

depend l a r g e l y on t h e r e l i a b i l i t y t h a t can be

t i o n from i o c a i f l o w r e g i o n s w i t h supersonic f l o w

achieved i n d e s c r i b i n g phenomena t y p i c a l f o r un-

t o the a d j a c e n t subsonic r e g i o n s does not take

steady t r a n s o n i c f l o w . However, i n c o n t r a s t w i t h

-10-

P a r t I I d e s c r i b e s t h e wind-tunnei

the s t e a d y - f l o w case, e x p e r i m e n t a l data t h a t a r e

investiga-

s u f f i c i e n t l y d e t a i l e d t o v e r i f y fundamental theo-

t i o n s o f b o t h t h e NACA 64A006 a i r f o i l w i t h o s c i l -

r e t i c a l assumptions or t o c o n f i r m the v a i i d i t y o f

i a t i n g f l a p and the NLR 7301 a i r f o i l p e r f o r m i n g

c a l c u l a t e d r e s u l t s were v e r y scarce, and i t i s i n

o s c i l l a t i o n s i n p i t c h . D e t a i l s a r e discussed o f the

t h i s r e s p e c t t h a t t h e p r e s e n t work aims t o con-

technique f o r unsteady p r e s s u r e measurements de-

tribute.

veloped a t NLR ( c h a p t e r set-up ( c h a p t e r ter

1.2

53,

t h e models and t e s t

6 ) , and t h e t e s t program (chap-

7). I n p a r t l i l , an a n a l y s i s i s g i v e n o f the

OUTLINE O F THESIS

r e s u l t s o f t h e experiments.

I n t h i s a n a l y s i s , em-

phasis i s placed upon t h e e f f e c t s t h a t a r e t y p i -

I n an a t t e m p t t o improve t h e p h y s i c a l i n -

c a l f o r h i g h subsonic and t r a n s o n i c flow.

s i g h t i n t o t h e n a t u r e o f unsteady t r a n s o n i c f l o w s

8,

i n chap-

r e s u l t s f o r t h e NACA 64A006 a i r f o i l a r e

and t o f u r n i s h experimental evidence t h a t c o u l d

ter

s u p p o r t t h e development o f t h e o r e t i c a l o r semi-

used t o i l l u s t r a t e t h e mechanism o f t h e i n t e r a c -

e m p i r i c a l methods, i t was decided a t NLR t o per-

t i o n between t h e steady and unsteady f l o w f i e l d s .

form a program o f e x p l o r a t o r y wind-tunnel

F u r t h e r i t i s demonstrated t h a t the presence o f

inves-

t i g a t i o n s on o s c i l l a t i n g a i r f o i l s i n two-dimen-

o s c i l l a t i n g shock waves may c o n t r i b u t e s i g n i f i -

s i o n a l f l o w , i l i t h t h e a i d o f a s p e c i a l technique

c a n t l y t o the o v e r a l l unsteady a i r l o a d s . I n chao-

developed by 8ergh (Refs. 21 and 2 2 ) , d e t a i l e d

t e r 9 , t h e e x i s t e n c e i s shown o f t h r e e d i f f e r e n t

steady and unsteady p r e s s u r e d i s t r i b u t i o n s were

types o f p e r i o d i c a l shock-wave motion. These types

determined on an a i r f o i l o f c o n v e n t i o n a l type

a r e e x p l a i n e d by means o f a simple a n a l y t i c a l

(NACA 64A0063

model. Next, i n c h a p t e r IO, some aspects o f t h e

w i t h an o s c i l l a t i n g t r a i l i n g edge

f l a p and on an advanced-type s u p e r c r i t i c a l a i r f o i l

unsteady f l o w around a s u p e r c r i t i c a l a i r f o i l a r e

(NLR 73013 performing o s c i l l a t i o n s i n p i t c h .

discussed w i t h t h e a i d o f r e s u l t s f o r the NLR 7301

In

a d d i t i o n , t i m e h i s t o r i e s o f t h e shock-wave motions

airfoil.

were recorded. The i n v e s t i g a t i o n s were l i m i t e d t o

around t h e "shock-free"

a t t a c h e d f l o w , so u n s t e a d y - f l o w phenomena l i k e

a i r f o i l , as wel1 as r e s u l t s f o r some t y p i c a l o f f -

"buffet"

o r "buzz",

design condition of t h i s

design c o n d i t i o n s , a r e considered. F i n a l l y , i n

occurring i n situations with

s e v e r e l y separated f l o w s ,

I n t h i s chapter, r e s u l t s f o r o s c i l l a t i o n s

c h a p t e r 1 1 , t h e q u e s t i o n i s r a i s e d t o what ex-

were i e f t o u t o f con-

t e n t l i n e a r i z a t i o n o f t h e unsteady-flow problem i s

s i d e r a t ion.

possible,

i f t h e unsteady f i e l d i s considered as a

small p e r t u r b a t i o n o f a g i v e n mean steady c o n d i t i o n . The t h e s i s i s s u b d i v i d e d i n t o f o u r p a r t s .

P a r t I V s t a r t s w i t h a review of t h e c u r r e n t

P a r t I s t a r t s w i t h a general d e s c r i p t i o n

s t a t u s of two-dimensional unsteady-flow t h e o r y , i n

o f the f l o w around o s c i l l a t i n g a i r f o i l s , which i s

which emphasis i s p l a c e d upon t h e recent develop-

f o l l o w e d by a c o n s i d e r a t i o n o f s p e c i f i c problems

ments i n t h i s f i e l d ( c h a p t e r 1 2 ) .

a s s o c i a t e d w i t h t r a n s o n i c flows.

I t i s shown t h a t

I t i s shown t h a t

i n t h e p a s t few years, e s p e c i a i l y i n the USA,

i n t h e t r a n s o n i c speed range an e s s e n t i a l c o u p l i n g

con-

s i d e r a b l e progress has been achieved i n s o l v i n g

e x i s t s between t h e steady and unsteady f l o w f i e l d ,

t h e complicated u n s t e a d y - f l o w equations numeri-

which does n o t o c c u r f o r m o d e r a t e l y subsonic and

c a l l y . The e x p l o r a t o r y i n v e s t i g a t i o n s a t NLR have

supersonic speeds. T h i s c o u p l i n g causes one o f t h e

been made j u s t i n t i m e t o support these deveiop-

main d i f f i c u l t i e s i n t r a n s o n i c unsteady aerodynam-

ments, and a t p r e s e n t t h e NLR data a r e being used

ics,

e x t e n s i v e l y f o r t h e purpose o f comparison i n the

s i n c e i t i m p l i e s t h a t t h e unsteady-flow

problem

m a j o r i t y o f the recent t h e o r e t i c a l studies. A t

can be t r e a t e d no l o n g e r independently

o f t h e s t e a d y - f l o w problem. For t h i s reason, p a r t I

t h e same time, t h i s S i t u a t i o n enables the a u t h o r

i s concluded w i t h a b r i e f d e s c r i p t i o n o f t h e main

t o conclude t h i s work w i t h a f i r s t e v a l u a t i o n o f

c h a r a c t e r i s t i c s o f steady t r a n s o n i c flows (chap-

a number o f new computational methods f o r unsteady

t e r 3).

t r a n s o n i c f l o w ( c h a p t e r 13).

-11-

P A R T I INTRODUCTORY CHAPTERS

-Ij-

2

2.1

THE FLOW AROUND OSCILLATING AIRFOILS

OESCRIPTION OF UNSTEADY AIRLOADS

When an a i r f o i l

i 5

performing s i n u s o i d a l

o s c i l l a t i o n s around a g i v e n mean c o n d i t i o n , the c i r c u l a t i o n and, hence, t h e l i f t f o r c e and l o c a l pressures show p e r i o d i c a l v a r i a t i o n s .

In o r d e r t o

keep t h e t o t a l v o r t i c i t y c o n s t a n t ( a c c o r d i n g t o H e l m h o l t z ' theorem), each time-dependent change i n c i r c u l a t i o n around t h e a i r f o i l

i s compensated by

t h e shedding o f f r e e v o r t i c i t y from t h e t r a i l i n g edge. T h i s v o r t i c i t y , which has t h e same s t r e n g t h a s t h e change i n c i r c u l a t i o n b u t i s o f o p p o s i t e

Fig. 2.2

s i g n , i s c a r r i e d downstream by the f l o w ( F i g . 2 . 1 ) . Due t o t h e v e l o c i t i e s a t t h e a i r f o i l

induced by

Example of unsteady pressure sigoals and overall loads o n B sinusoidally oscillating airfoil at subsonic'speed.

the f r e e v o r t i c e s , t h e instantaneous i n c i d e n c e o f

around t h e i r mean values. To d e s c r i b e such har-

the a i r f o i l

monic v a r i a t i o n s ,

i s changed i n such a way t h a t the

two q u a n t i t i e s a r e needed, name-

o s c i l l a t o r y p a r t o f t h e l i f t l a g s behind the

l y magnitude and phase s h i f t w i t h respect t o t h e

motion o f t h e a i r f o i l .

motion o f the a i r f o i l (Fig. 2.3).

An e q u i v a l e n t

way o f d e s c r i p t i o n i s i n terms o f a complex number.

The main parameter g o v e r n i n g t h e unsteady f l o w i s the s o - c a l l e d reduced frequency, k, de-

I n t h e l a t t e r n o t a t i o n , the r e a l p a r t o f a pres-

f i n e d as k = wP,/U_,

sure p e r t u r b a t i o n (or load) is i n phase w i t h the

which i s p r o p o r t i o n a l t o the

r a t i o o f t h e chord l e n g t h 2 2 (Fig. 2.1).

and t h e wave l e n g t h L

m o t i o n o f t h e a i r f o i l , and t h e imaginary p a r t i s i n q u a d r a t u r e w i t h i t . In o t h e r words,

T h i s parameter i s a measure f o r t h e

unsteadiness o f t h e flow.

the r e a l

p a r t i s t h e a c t u a l p r e s s u r e p e r t u r b a t i o n a t the

For s i m i l a r i t y o f t h e

f l o w around an o s c i l l a t i n g f u l l - s c a l e a i r f o i l and

i n s t a n t t h e o s c i l l a t i n g a i r f o i l reaches i t s maximum

i t s wind-tunnel model r e p r e s e n t a t i o n i t i s re-

p o s i t i v e d e f l e c t i o n , whereas t h e imaginary p a r t

q u i r e d t h a t , besides the i m p o r t a n t parameters f o r

r e p r e s e n t s t h e pressure p e r t u r b a t i o n a t the i n s t a n t

steady f l o w ( a i r f o i l shape, i n c i d e n c e , Mach number

t h e a i r f o i l passes i t s mid p o s i t i o n i n p o s i t i v e

and Reynolds number), a l s o the reduced frequency

direction. Of

f o r the model t e s t s i s t h e same as i n r e a l i t y .

t h e d i s t r i b u t i o n o f t h e unsteady pressures along

i n t e r e s t t o the a e r o e l a s t i c i a n

i 5

t h e chord o r o v e r t h e wing and t h e i r i n t e g r a t e d

As a t y p i c a l example, f i g u r e 2.2 g i v e s some t i m e h i s t o r i e s o f t h e l o c a l p r e s s u r e s and t h e r e -

values, which represent the o v e r a l l unsteady l i f t

s u l t i n g l i f t and moment on an a i r f o i l performing

and moments.

i t i s usual t o p r e s e n t steady and un-

o s ~ i l l a t i o n si n p i t c h . Both t h e p r e s s u r e s and t h e

steady pressures i n t h e form o f dimensionless

overall loads show almost s i n u s o i d a l v a r i a t i o n s

coefficients,

as d e f i n e d i n Appendix A.

This Ap-

p e n d i x a l s o c o n t a i n s t h e d e f i n i t i o n s o f the overa l 1 steady and unsteady aerodynamic c o e f f i c i e n t s ; t h e s i g n conventions a r e a c c o r d i n g t o t h e AGAR0 ( A d v i s o r y Group f o r Aerospace Research and Development o f t h e NATO) Hanual on A e r o e l a s t i c i t y , VARIATION IN INCIDENCE

JW casu*

VARIATION IN LIFT

J L co, iw,-u1

MAIN PARAMETER

REDUCED FREQUENCY k='$=

Volume V i

(Ref. 2 3 ) .

T h i s way o f d e s c r i b i n g unsteady pressures o r 7

2

loads i s only v a l i d i f t h e aerodynamic q u a n t i t i e s v a r y s i n u s o i d a l l y i n time, or, i n o t h e r words, as

Fig. 2.1

l o n g as a l i n e a r r e l a t i o n s h i p e x i s t s between the

Flow around an Oscillating airfoil

-15-

DESCRIPTION IN TERMS OF MAGNITUOE AND PHASE ANGLE PRESSURE I N POINT A

P

*P~*~~==P,*P~C.IIW>*"~I =P,'Ipl

WITH

DESCRIPTION I N TERMS OF A COMPLEX NUMBER

cos

wi

COIWf

-

= P%

:Plli"U>lIi"W,

p1 =MAGNITUDE OF PRESSURE PERTURBATION

WITH A p '

+

RI

= o, c m m

Alp"=

o i P H A S E ANGLE

Fig. 2.3

1 ~IA P ' +A P " i!P"

P = P . + A P = P , + P ~ R ~ e"Wwl+wl!

PI m n

I

, REAL PART OF PRESSURE PERTURBATION

m , IMAGINARY

PART

Description O f unsteady pressures.

displacement o f t h e a i r f o i l and the unsteady a i r -

2.2

SOME NOTES ON THE UNSTEADY-FLOW EQUATIONS

loads. T h i s i s , however, n o t always t r u e , e s p e c i a l l y n o t i n separated flows o r i n r e g i o n s near o s c i l -

The j o i n t i n f l u e n c e o f a i r f o i l t h i c k n e s s ,

l a t i n g shock waves (see, f o r i n s t a n c e , t h e p r e s s u r e

i n c i d e n c e and a m p l i t u d e o f v i b r a t i o n i s d i f f e r e n t

v a r i a t i o n a t x/c = 0 . 4 6

f o r moderately subsonic and supersonic f l o w and

i n f i g u r e 2.4).

I n such

cases, emphasis w i l l be p l a c e d upon t h e f i r s t

f o r t r a n s o n i c flow. T h i s w i l l be demonstrated by

F o u r i e r component a f the s i g n a l s , because i n f l u t -

c o n s i d e r i n g t h e b a s i c f l o w equations f o r a l l three

t e r i n v e s t i g a t i o n s t h i s i s t h e o n l y component t h a t

speed regimes.

can g e n e r a t e n e t energy a t t h e frequency o f the

2.2.1

a i r f o i l motion.

The unsteady-flow equations

The b a s i c equations far an i d e a l two-dimens i o n a l i n v i s c i d flow, which express c o n s e r v a t i o n o f mass, momentum i n x- and y - d i r e c t i o n , and energy, can be w r i t t e n a s : .Ic=.o1 . i P

.I1

.10

.i6 .64

.eo

i'

where e r e p r e s e n t s t h e t o t a l energy per u n i t volume, g i v e n by INCIDENCE

Fig. 2.4

"NITEAD"

PRESSURE5

LIFT

UOHENI

e = (y

Example O f unsteady pressure signals and overall loads on an oscillating airfoil in transonic flow with a shock

-

I ) - 1 p + + P ( u ~+

v'),

(2.2)

and where p and p a r e t h e d e n s i t y and pressure,

wave.

-16-

w h i l e U and V represent t h e v e l o c i t y components i n x-

and y - d i r e c t i o n ,

f o r flows w i t h weak shock wave5 (Mach number j u s t

r e s p e c t i v e l y . The q u a n t i t y y

upstream o f the shock wave les5 than about 1 . 3 ) .

denotes t h e r a t i o o f s p e c i f i c heats.

The terms t h a t a r e l i n e a r i n O a r e placed on t h e

The boundary c o n d i t i o n a t the m v i n g a i r f o i l s u r f a c e , S(x,y,t)

= O,

l e f t . The terms on t h e r i g h t - h a n d s i d e a r e o f t h e

r e q u i r e s the v e l o c i t y com-

second and t h i r d degree.

oonent normal t o the s u r f a c e t o be zero:

as

as

as

- + u - + v - = o a t ax ay

2.2.2

(2.3)

Moderately subsonic and supersonic f l o w

When i t i s t r i e d t o f u r t h e r s i m p l i f y t h e The s o l u t i o n s a t i s f y i n g ( 2 . 1 )

t o (2.3)

i s made

p o t e n t i a l e q u a t i o n (2.61,

u n i q u e by t h e K u t t a c o n d i t i o n , which r e q u i r e s t h a t

t h e commonly used ap-

proach i s t o assume t h e p e r t u r b a t i o n s t o be s m a l l ,

a t the t r a i l i n g edge and across t h e t r a i l i n g v o r t e x

so t h a t terms o f second and h i g h e r o r d e r can be

sheet t h e pressure and f l o w d i r e c t i o n a r e c o n t i n -

neglected. The r e s u l t f o r moderately subsonic and

uous.

supersonic flow, where 1 1 Various degrees o f a p p r o x i m a t i o n can be made

-

M-1

i s o f the o r d e r I ,

i s the l i n e a r equation

t o s a t i s f y b o t h equations (2.1) and t h e accompan y i n g boundary c o n d i t i o n s . A g r e a t s i m p l i f i c a t i o n i s a t t a i n e d when i t i s assumed t h a t t h e f l o w i s i s e n t r o o i c and i r r o t a t i o n a l : The boundary o f the a i r f o i l can be expressed a s pp-'

= constant S(x,y,t)

and _av- - =au

ax

ay

= y

-

[f'(x)

+

a(x)

+

g(x,t)] =

c

O

o

o,

x $ 2 i (2.8)

i s t h e two-valued f u n c t i o n denoting

i n which f'(x) The l a t t e r r e l a t i o n a l l o w s t h e i n t r o d u c t i o n o f a

t h e t h i c k n e s s d i s t r i b u t i o n , a ( x ) the incidence p l u s

d i s t u r b a n c e v e l o c i t y p o t e n t i a l O, d e f i n e d by

t h e camber d i s t r i b u t i o n , and g ( x , t )

the time-

dependent d e f o r m a t i o n o f t h e a i r f o i l .

u=u,+m

v=m

Y

.

Then e q u a t i o n ( 2 . i ) , t o g e t h e r w i t h (2.4)

'

By i n t r o d u c t i o n o f ( 2 . 8 ) ,

(2.5)

the boundary con-

d i t i o n (2.3) reduces t o and

(2.5),

can be combined i n t o a s i n g l e e q u a t i o n f o r (Refs. 2 4 , 2 5 ) :

Both t h e d i f f e r e n t i a l e q u a t i o n (2.7) and the boundary condition (2.9) are l i n e a r i n

@.They

form

t h e b a s i s o f numerous c a l c u l a t i o n methods t o det e r m i n e aerodynamic loads on t h i n steady and osc i l l a t i n g a i r f o i l s . The l i n e a r i t y i m p l i e s t h a t sol u t i o n s s a t i s f y i n g t h e unsteady p a r t o f boundary c o n d i t i o n (2.9)

can be considered s e p a r a t e l y from

s o l u t i o n s s a t i s f y i n g t h e steady p a r t s . I n t h i s decomposition, Although e q u a t i o n i s e n t r o p i c flow,

i l l u s t r a t e d i n figure 2.5,

lating airfoil

(2.6) i s o n l y v a l i d f o r

the o s c i l -

i s r e p l a c e d by an i n f i n i t e l y t h i n

a i r f o i l o s c i l l a t i n g i n a u n i f o r m p a r a l l e l flow.

i t remains a good approximation

-17-

For p r a c t i c a l a p p i i c a t i o n s i n f l u t t e r c a l c u l a tions,

Solutions o f the t h i n - a i r f o i l

i t i s usual t o c o n s i d e r only t h i s unsteady

two- and three-dimensional

p a r t o f the s o i u t i o n .

equations for both

subsonic and supersonic

flows a r e documented v e r y well i n the l i t e r a t u r e and have been v e r i f i e d e x t e n s i v e l y by experiments

1"

I i E l ü l PROBLEY

UNSTEADY PROBLEM

(Refs.

P

23, 26-40). I n g e n e r a l , i t appears t h a t

s a t i s f a c t o r y p r e d i c t i o n s can be o b t a i n e d f o r a t tached moderately subsonic and supersonic flows. OsciLuriw

iH8 C I:NES I

CMBER

AIRFOIL

f

INCIDENCE

Fig. 2 . 5

OSCILLAIIHG PLATE

An e x c e p t i o n i s formed by o s c i l l a t i n g t r a i l i n g edge f l a p s , f o r which t h e I n f l u e n c e o f v i s c o s i t y

Decomposition i n t o a symmetrical nonl i f t i n g airfoil, an i n f i n i t e l y t h i n c u r v e d p l a t e , and a0 i n f i n i t e l y t h i n oscillating plate.

i s o b v i o u s l y dominant.

I n such cases. one has t o

r e i v on wind-tunnel t e s t s .

When a p p l i e d t o s i n u s o i d a l motions, t h e time-dependent deformation i s d e s c r i b e d by

g(x,t)

= g í x ) eiwt

.

2.2.3

(2.IO)

Transonic f l o w

In the t r a n s o n i c regime, w h e r e ' t h e Mach numb e r i s c l o s e t o one,

the p o t e n t i a l equation ( 2 . 6 )

By s p l i t t i n g up t h e d i s t u r b a n c e p o t e n t i a l '4 i n a

can no longer be l i n e a r i z e d completely. üy assuming

steady and unsteady p a r t

5111.11

p e r t u r b a t i o n s , most o f the n o n l i n e a r terms i n

(2.6)

can be e l i m i n a t e d b u t , a s d e r i v e d by Landahi

(Ref.

241, t h e f o l l o w i n g e q u a t i o n , which i s essen-

t i a l l y n o n l i n e a r , has t o be r e t a i n e d : the unsteady p a r t o f e q u a t i o n (2.7)

and t h e boundary c o n d i t i o n (2.9)

becomes:

The corresponding boundary c o n d i t i o n remains the

yields:

same as t h a t g i v e n i n (2.9). The n o n l i n e a r i t y o f ( 2 . 1 5 ) r a t e solution of

m,

prevents a sepa.

due t o a i r f o i l thickness and

i n c i d e n c e and t o O s c i l l a t i o n S . T h i s means t h a t a i r where k = m!./U_

denotes t h e reduced frequency based

loads on s i n u s o i d a l l y o s c i l l a t i n g a i r f o i l s a l s o

on a r e f e r e n c e l e n g t h L equal t o I .

depend on a i r f o i l t h i c k n e s s and incidence. Conse-

i n accordance w i t h t h e l i n e a r i z a t i o n ,

the

q u e n t l y , the study o f o s c i l l a t i n g a i r f o i l s i n

unsteady p r e s s u r e jump across t h e a i r f o i l surface

t r a n s o n i c f l o w i s much more complicated than i n

can be found from

AC

P

= AC

P+

-

AC

P-

moderately subsonic and supersonic f l o w .

= -(z/u_)(aUax

+

Formally, the n o n l i n e a r i t y o f the unsteady ik@),

(2.14)

f l o w f i e l d may be circumvented by assuming t h e unsteady e f f e c t s t o be v a n i s h i n g l y m a l l d i s t u r b a n c e s

where AC

P+

and A C

P-

denote t h e unsteady p r e s s u r e

o f t h e steady f l o w around t h e a i r f o i l

c o e f f i c i e n t s f o r t h e upper and lower s u r f a c e , res-

When i t i s assumed t h a t

pectively. From (2.12) and (2.13)

i n a given

mean p o s i t i o n .

i t follows that the

main parameters g o v e r n i n g t h e unsteady problem a r e the reduced frequency k, t h e f r e e - s t r e a m Mach numb e r Mm, and t h e v i b r a t i o n mode g ( x ) .

e q u a t i o n ( 2 1 5 ) can be s p l i t i n t o two p a r t s , one

-18-

f o r the steady p o t e n t i a l 0 imposed unsteady p o t e n t i a l For t h e p o t e n t i a l

eo,

O

and one f o r t h e super-

t i o n w i l l be p a i d t o t h i s s u b j e c t .

a.

When the experimental evidence t o support

the r e s u l t i n g e q u a t i o n reads:

t h i s f a s t development i s considered,

i t can be con-

cluded t h a t v e r y l i t t l e m a t e r i a l i s a v a i l a b l e . With only a few e x c e p t i o n s , the numerous experiment a l i n v e s t i g a t i o n s i n the p a s t a r e n o t s u i t a b l e for which i s t h e well-known t r a n s o n i c s m a l l - p e r t u r b a -

t h i s purpose, s i n c e they were l i m i t e d t o t h e d e t e r -

t i o n e q u a t i o n f o r steady t r a n s o n i c flow.

m i n a t i o n of o n l y one o r two o v e r a l l c o e f f i c i e n t s ,

The e q u a t i o n g o v e r n i n g t h e unsteady f l o w f i e l d

l i k e t h e h i n g e moment (Refs. 41-63)

becomes:

moment (Refs. 64-68).

-

+xx [ I

M2

-

M~[(Y+I)/U_]@~~] + + y y

-

2ikM2+x +

-

(y+l)(H~/U,)00xx4x

= O

.

I n t h e m a j o r i t y o f these i n -

vestigations,

t h e f r e e - o s c i l l a t i o n technique has

been a p p l i e d ,

i n which the model o r c o n t r o l sur-

face

+ k2M2+

o r the pitching

i s suspended i n s p r i n g s ,

i n such a way t h a t

f r e e o s c i l l a t i o n s can be performed. The aerodynamic

(2.18)

moment Is then determined from the change i n

frequency and damping w i t h airspeed. A s e r i o u s The boundary c o n d i t i o n on t h e a i r f o i l i s t h e same as t h a t g i v e n i n

drawback o f t h i s r e l a t i v e l y simple method i s t h a t

(2.13), and a l s o the formula for

t h e unsteady p r e s s u r e c o e f f i c i e n t (2.14)

no f u r t h e r i n f o r m a t i o n i s o b t a i n e d about t h e de-

remains

t a i l s o f t h e unsteady pressure d i s t r i b u t i o n and

val id.

the corresponding mean steady f l o w f i e l d . The unsteady-flow e q u a t i o n (2.18)

+. The

in

is linear

T h i s t y p e o f i n f o r m a t i o n can be o b t a i n e d

equation, however, has nonconstant c o e f -

only by measuring d e t a i l e d steady and unsteady

f i c i e n t s t h a t depend on t h e mean steady f l o w f i e l d @

O

. This

demonstrates a g a i n t h a t ,

pressure d i s t r i b u t i o n s on a model t h a t i s forced

in contrast with

moderately subsonic and supersonic flow.

i n t o a n o s c i l l a t o r y motion. For t e s t s o f t h i s type,

t h e un-

however, t h e t e s t set-up,

steady f l o w f i e l d i n the t r a n s o n i c range can be

no longer t r e a t e d independently o f t h e steady f l o w field.

i n s t r u m e n t a t i o n and

d a t a - r e d u c t i o n procedures a r e much more c o m p l i cated than f o r t h e f r e e - o s c i l l a t i o n method. T h i s

Because o f t h e importance of t h i s i n t e r -

might e x p l a i n t h a t i n the p a s t unsteady p r e s s u r e

a c t i o n , the main c h a r a c t e r i s t i c s o f t h e steady

d i s t r i b u t i o n s v e r y seldom have been determined.

t r a n s o n i c f l o w around a i r f o i l s a r e b r i e f l y r e -

A f i r s t a t t e m p t t o measure l o c a l unsteady

viewed i n chapter 3 .

pressures on an o s c i l l a t i n g wind-tunnel model i n t r a n s o n i c f l o w was made by E r i c k s o n and Robinson (Ref. 41).

2.3

T h e i r method, i n which e i e c t r i c a l

PRESENT STATUS O F THE RESEARCH ON UNSTEADY

pressure c e l l s i n s t a l l e d f l u s h w i t h the model sur-

TRANSON I C FLOW

face a r e used, has been a p p l i e d s u c c e s s f u l l y by Wyss, Sorenson,and t h e i r c o l l e a g u e s a t NASA. A l -

The mathematical c o m p l e x i t y o f t h e problem o f

though they a c t u a l l y measured the pressures on

unsteady t r a n s o n i c f l o w has prevented t h e develop-

o s c i l l a t i n g c o n t r o l surfaces on two-dimensional

ment o f e f f i c i e n t

and three-dimensional wings (Refs. 43, 53,

computation methods f o r many

years.

56, 57)

and a i r f o i l s o s c i l l a t i n g i n p i t c h (Refs. 65-67), S t i m u l a t e d , however,

by t h e renewed i n t e r e s t

only o v e r a l l aerodynamic c o e f f i c i e n t s have been

i n t r a n s o n i c f l i g h t and t h e enormous developments

p u b l i s h e d , except f o r some t y p i c a l o s c i l l o g r a p h

i n steady t r a n s o n i c f l o w computations, a number o f

records o f l o c a l p r e s s u r e f l u c t u a t i o n s .

a t t e m p t s has been made i n the l a s t few years t o

The f i r s t d e t a i l e d unsteady pressure d i s -

s o l v e the c o m p l i c a t e d unsteady f l o w equations ( a

t r i b u t i o n s i n t h e t r a n s o n i c regime have been re-

review i s g i v e n i n c h a p t e r 12), and i t may be ex-

p o r t e d by Lessing, Troutman and Meness (Ref.

pected t h a t i n the coming years c o n s i d e r a b l e a t t e n -

and by L e a d b e t t e r , Clevenson and Igoe (Ref. 7 0 ) .

-19-

69)

These two s t u d i e s deal w i t h three-dimensional flow.

Probably the r e s u l t s g i v e n i n reference

Pressure tubes i s d i s c a r d e d (see chapter 5 ) . To

69

the a u t h o r ' s knowledge, s i m i l a r d a t e f o r two-

a r e n o t c o r r e c t because o f a fundamental e r r o r i n

dimensional o s c i l l a t i n g a i r f o i l s d i d n o t e x i s t a t

the d a t a - r e d u c t i o n procedure, where the e f f e c t o f

the time NLR s t a r t e d i t s e x p l o r a t o r y program on

t h e main f l o w on the dynamic response o f the

unsteady t r a n s o n i c flows.

3 MAIN CHARACTERISTICS OF THE STEADY TRANSONIC FLOW AROUND AIRFOILS As an i n t r o d u c t i o n t o t h e di'scusslon o f t h e

l o c a l Mach number j u s t upstream o f t h e shock wave

t r a n s o n i c f l o w around o s c i l l a t i n g a i r f o i l s , a

i s about 1.25 t o 1.3.

b r i e f survey w i l l be g i v e n o f the behaviour o f

stream o f t h e shock wave separates c o m p l e t e l y , t h e

t r a n s o n i c flows around a i r f o i l s a t r e s t . For m r e

f l o w around t h e a i r f o i l

When t h e boundary l a y e r down-

i s changed c o n s i d e r a b l y ,

d e t a i l e d c o n s i d e r a t i o n s o f p l a n e steady t r a n s o n i c

and o f t e n unsteady-flow phenomena l i k e " b u f f e t "

f l o w w i t h embedded shock waves, t h e reader i s r e -

and "buzz"

f e r r e d t o t h e p u b l i c a t i o n s o f Holder (Ref. Shapiro (Ref.

17),

For a e r o e l a s t i c i n v e s t i g a t i o n s , one i s in-

74) and Sears (Ref. 7 5 ) , w h i l e , f o r

t h e v a r i o u s aspects o f shock-free flow,

t e r e s t e d p r i m a r i l y i n t h e changes

reference

13,

i n aerodynamic

l o a d i n g due t o v a r i a t i o n s i n downwash along t h e

i s made t o t h e c o n t r i b u t i o n s o f Nieuwland, Spee and Boerstoel (Refs.

s t a r t t o occur.

chord. Therefore,

16, 20, 7 6 ) .

some examples w i l l be g i v e n o f

t h e development o f t h e f l o w p a t t e r n when the downwash i s changed a t c o n s t a n t free-stream Mach number.

3.1

TRANSONIC FLOWS WITH EMBEDDED SHOCK WAVES

The f i r s t example d e a l s w i t h a symmetrical a i r f o i l w i t h f l a p a t zero incidence (Fig. 3.2). When t h e f l a p i s d e f l e c t e d downwards over an

3 . 1 . 1 Development o f f l o w p a t t e r n w i t h Mach number,

a n g l e So, t h e speed over t h e upper surface i s

f l a p angle,and i n c i d e n c e

g r a d u a l l y increased, and t h e supersonic r e g i o n and the shock wave develop i n the same way as

When t h e f r e e - s t r e a m Mach number o f a p u r e l y

described above f o r i n c r e a s i n g free-stream Mach

subsonic f l o w around a symmetrical a i r f o i l i s i n creased, the f l o w p a t t e r n u s u a l l y develops i n a way as sketched i n f i g u r e 3.1. i c a l Mach number, M * ,

Mcl

The s o - c a l l e d c r i t -

M , r M

c

SUBCRITICAL FLOW

.s

i s reached when somewhere

i n the flow t h e l o c a l Mach number becomes u n i t y . !A=

Beyond the c r i t i c a l Mach number, a supersonic re-

i SONIC L I N E

M

g i o n appears a t t h e a i r f o i l , which i n general i s

M,r

i

SUPERCRITICAL FLOW

t e r m i n a t e d by a normal shock wave as soon as the

@

maximum l o c a l Mach number exceeds a v a l u e o f about

1.05. Through t h i s shock wave, t h e f l o w v e l o c i t y

,*-.

M.1

i s reduced from s u p e r s o n i c t o subsonic ( F i g . 3 . l b ) .

number,

q-=

the shock moves backwards, w h i l e b o t h t h e

shock s t r e n g t h and t h e s i z e o f the supersonic r e -

SUPERCRITICAL FLOW (WITH SEPARATION)

g i o n increase. A f t e r t h e p r e s s u r e jump through the shock wave has become s u f f i c i e n t l y

'I

/'

W i t h a f u r t h e r i n c r e a s e o f t h e free-stream Mach

large, so-called

@

\

shock-induced s e p a r a t i o n o f t h e boundary l a y e r

\

'

occurs ( F i g . 3 . 1 ~ ) . For a t u r b u l e n t boundary l a y e r ,

I I _ . '

t h i s shock-induced s e p a r a t i o n s t a r t s when t h e

Fig. 3.1

-20-

Influence of Mach n u b e ? on flow p a t t e r n .

3.1.2

VACA 64AOM AIRFOIL M, = 3.a75 PRESSURE COEFFICIENI

I ,to

Characteristics o f a normal shock wave

=a‘

Through a normal shock wave, a s occurring

in one-dimensional flow, the velocity is reduced from supersonic to subsonic. The pressure jump across the shock wave is determined completely by the free-stream Mach number upstream of the shock (Fig, 3.4a).

For an inviscid two-dimensional flow

around an airfoil, the foot of the shock i s normal to the contour of the airfoil, but the remaining

part is curved forward. This can be explained as follows: On a convex contour, the velocity in ydirection has to decrease upstream as well as downstream o f the shock wave. As shown by Zierep (Ref. 7 7 ) , this requirement is not compatible with a completely straight shock, because,’if the shock

PRESSURE COEFFICIENT

Fig. 3.2 Influence of flap deflection on Pressure distribution and shock pattern in transonic flow. I

.

-*r

number. At the lower surface the flow speed decreases and the development o f the flow pattern i5

I

dLOWER SURFACE

i-

UPPER

reversed: the size o f the supersonic region

decreases, the shock becomes weaker, and finally, beyond a certain flap deflection, the shock 1.0 “I,

vanishes. A similar development

in flow pattern can be

observed when the incidence o f the airfoil is varied. Ah example is shown in figure 3 . 3 , which deals with an airfoil that carries a well-devel-

ci;--

-2 r

oped supersonic region on its upper surface, terminated by a relatively strong shock wave. This

r.

example shows also that already small varlations in incidence may lead to considerable changes in pressure distribution, shock position, and shock strength. Fig. 3 . 3

ILOWER

Influence of incidence on pressure distribution and shock pattern in transonic flow.

1

-21-

a c t e r i z e d by the presence o f a normai shock wave a t e i t h e r the upper o r the iower s u r f a c e o f the I

a i r f o i l , or a t b o t h surfaces a t the same t i m e . O c c a s i o n a l i y , even two normal shock waves behind each o t h e r occur. An e x c e p t i o n t o t h i s r u l e forms

the f l o w around a I

a

so-called s u p e r c r i t i c a i a i r f o i i

i n i t s d e s i g n c o n d i t i o n . T h i s type o f a i r f o i l

O N E . DIMENSIONAL FLOW

is

shaped i n such a way t h a t , f o r a s p e c i f i c combinat i o n o f i n c i d e n c e and free-stream Mach number ( t h e design c o n d i t i o n ) , t h e t r a n s i t i o n o f t h e supers o n i c r e g i o n t o t h e a d j a c e n t subsonic r e g i o n takes p l a c e w i t h o u t n o t i c e a b l e shock wave. Away from t h i s

PRESSURE COEFFICIENT

design c o n d i t i o n , the f l o w remains no l o n g e r shockf r e e , and t h e a i r f o i l behaves more o r i e s s l i k e a conventional-type a i r f o i l . An example o f t h e f l o w around a s u p e r c r i t i c a l airfoil

b. I

SHOCK WAVE ON CONVEX SURFACE

b -2

3.5,

R A P I D EXPANSION DIRECTLY OOWNSTREAHOF SHOCK WAVE I Z I E R E P CUSP)

i n i t s design c o n d i t i o n i s shown i n f i g u r e

t o g e t h e r w i t h t h e f l o w p a t t e r n s f o r some

n e i g h b o u r i n g v a l u e s o f incidence and Mach number. The f i g u r e r e v e a l s t h a t a l r e a d y small changes i n

b. T W O . DIMENSIONAL FLOW

i n c i d e n c e and Mach number a r e s u f f i c i e n t t o d i s t u r b

Fig. 3 . 4

Normal shock waves in one- and in twodimensional flow.

t h e shock-free f l o w c o n d i t i o n . 80th t h e l o w e r i n g and the i n c r e a s i n g o f t h e Mach number r e s u l t i n

were s t r a i g h t , a decrease i n v e i o c i t y upstream o f

a f l o w p a t t e r n w i t h a normai shock wave, w h i l e the

t h e shock would r e s u i t i n an i n c r e a s e i n v e l o c i t y

same h o l d s when t h e incidence i s v a r i e d .




I o i INTERACTION

1

WITH BUBBLE

-*

b , INTERACTION WITH SHOCK

*

P

-

U

~0.5*

i s t i c f o r the present type o f " s h o c k - f r e e a i r f o i l , w i t h i t s r e l a t i v e l y b l u n t nose and i t s e x t e n s i v e

f. O.QUASI-STEADY o i=10 Hz. i =,024 n /i80 Hz. I

m

.

r

L O

0

Y

r

3

Y)

a

O L

Y

a a

I1

.-c

m

E Y

.c

m

u c a o

c

O ._ Y

-m

?

m

Y

r L LL O

I-}

where

Pi

}:I

and

U=O

should be o b t a i n e d from ( B .

47).

By p u t t i n g j = I and C = O,

( F i g . 5.3a)

u=o

1

the formula f o r the dynamic response of a s i n g l e tube-transducer system

i n s t i l l a i r i s obtained:

13} Pi

USO

:I

+ -'y-

=[coshl:Ï}

ut

O

"v

r}

sinh

o

1;

-1

Ï}I

.

(6.52)

O

T h i s e x p r e s s i o n can be shown t o be i d e n t i c a l t o t h e s o l u t i o n p u b l i s h e d by I b e r a l l (Ref. 8 6 ) . D e t a i l e d experimental v e r i f i c a t i o n s o f t h i s s o l u t i o n were p u b l i s h e d independently by Bergh and Tijdeman (Ref. 22),

who c o v e r a shear-wave-number

range O 5 s 5 8.5,

and by Watts (Ref. 9 2 ) f o r O 5 s 5 100. L a t e r ,

a d d i t i o n a l comparisons between t h e o r y and experiment were g i v e n by Karam and Franke (Ref. 88) f o r

O 5 s 5 83, by T r i e b s t e i n (Ref. 93) for O 5 s 5 9 , and by Goldschmiedc (Ref. 94) f o r O 5 s 5 2 0 .

in a l l

these cases, a s a t i s f a c t o r y agreement between t h e t h e o r e t i c a l p r e d i c t i o n and t h e experimental r e s u l t s was o b t a i n e d f o r tubes t h a t i n t h e v a r i o u s i n v e s t i g a t i o n s v a r i e d i n l e n g t h between i 5 and 600 cm. For a s i n g l e tube-transducer

system w i t h a main f l o w across t h e tube entrance, the f o l l o w i n g Formula

h o l d s ( j = I , C # O):

-i

U

+C-Pi

u=o

nÏ

a

n

iy (sinh{:r}-;

O

WL

-I

WL

.

{,P}>COSh(_r})] Y

O

O

t

(0.53)

O

T h i s formula has been v e r i f i e d i n t h e course o f t h e present i n v e s t i g a t i o n (Ref. 7 2 ) .

APPENDIX C

:

DERIVATION OF THE QUASI - STEADY AND UNSTEADY SHOCK RELATIONS

To o b t a i n t h e r e q u i r e d r e l a t i o n s between

Under t h e assumption of

i s e n t r o p i c flow,

the f l o w q u a n t i t i e s j u s t upstream and downstream

the l o c a l f l o w q u a n t i t i e s ahead o f t h e shock wave

o f a normal shock wave f o r quasi-steady and un-

a r e r e l a t e d t o t h e corresponding q u a n t i t i e s under

steady flow,

stagnation conditions,

t h e f o l l o w i n g well-known r e l a t i o n s

designated by index O,

as:

f o r steady f l o w have been a p p l i e d :

po/pl

= (i

+

&(y-l)M:

,

a O/ai = ~ T ~ / T , =I ~i i ++íy-~)i

m

E

0

Y

u

-- .. - ._ O

Y

._ - m

Y ?

-u "1 O

-

C Y )

m

-DIL

N

-

.

U .o

O

a

c .-

3

L

- m U

o

L

-

3 '

Y

O

.-2 :

m

N

.-a

+I+ .

* 1

Y

.-3m O

X

x

u

-m

.

. u

f

Y

u

c

m



a 3

a . .

m >

e > m

L

1

m

3

3

a a

Y

L

o a L

E

o m

8

c

.

Z

.-5>

Q

-

L 5 a 0 c Loa

3

2 % C

m

Y

c

m " L

C .-

a

m

Y

C

>

c

Y

L Cm m

L Y

,ám O u

C

U

o

m m c N 3

ia

a ? , a

a l m a 7 L 3 .

L

m

m

U

n c

V

m Y ' )

0"

semi span

v a r i a t i o n i n incidence, a m p l i t u d e o f pitch oscillation

tempera t u r e

Bm = ( i - M g ) '

mean temperature i n s i d e a p r e s s u r e

tube

c o n s t a n t f o r the propagation o f pressure waves through c y l i n d r i c a l tubes

time

Y

v e l o c i t y component i n x - d i r e c t i o n

6

f ree-stream v e l o c i t y velocity perturbation i n axial direct i o n ( p r e s s u r e tube) U

v e l o c i t y i n the entrance o f a pressure tube

V

v e l o c i t y component i n y - d i r e c t i o n

O

vY

Vt

il

coordinate i n radial d i r e c t i o n ( p r e s s u r e tube) (11 = r/R) t h e m a I conduct iv i t y absolute f l u i d v i s c o s i t y c o o r d i n a t e i n a x i a l d i r e c t i o n (press ur e tube) (€, = wx/ao)

t r a n s duce r vo I ume

c o o r d i n a t e i n streamwise d i r e c t i o n

volume o f p r e s s u r e tube (Vt = nR2L)

a i r dens i t y

velocity perturbation i n radial d i r e c t i o n ( p r e s s u r e tube)

V

r a t i o o f s p e c i f i c heats ( y = C /C ) P " v a r i a t i o n i n f l a p angle, a m p i i t u d e of f l a p o s c i l l a t i o n geometric (mean) f l a p a n g i e

v e l o c i t y o f a shock wave r e l a t i v e t o the a i r f o i I

V

geometr i t (mean) incidence

mean a i r d e n s i t y i n s i d e a p r e s s u r e tube

W

norma I wash

X

coordinate i n a x i a l d i r e c t i o n ( p r e s s u r e tube)

P r a n d t l number ( o 2 = UC /i) P disturbance v e l o c i t y potential

X

c o o r d i n a t e i n the d i r e c t i o n o f t h e free-stream f l o w

steady p a r t o f d i s t u r b a n c e v e l o c i t y potent ia I

l o c a t i o n o f hinge axis

unsteady p a r t o f d i s t u r b a n c e v e l o c i t y potent i aI

'h X

X

O

O

a m p l i t u d e o f shock-wave o x i l l a t i o n

phase a n g i e

d i s t a n c e between sending and r e c e i v i n g p o i n t ( x = x-5)

frequency (rad/sec)

O

x

s

(w = 2nf)

shock p o s i t i o n Subscripts:

yn

Neumann f u n c t i o n o f f i r s t k i n d o f order n

Y

c o o r d i n a t e normal t o t h e free-stream d i rect ion

1oc

G

incidence

1

v a l u e j u s t upstream o f a shock wave

UC

incidence corrected f o r w a l l i n t e r ference

2

v a l u e j u s t downstream o f a shock wave

m

- 146-

free-stream v a l u e local value

APPENDIX E : SAMENVATTING IN HET NEDERLANDS

r i e k e methoden om de gecompliceerde v e r g e l i j k i n g e n

Onder bepaalde omstandigheden kunnen de

voor de transsone s t r o m i n g op t e lossen.

v l e u g e l o f de s t a a r t v l a k k e n van een v l i e g t u i g i n

Het succes van d e r g e l i j k e methoden hangt

een i n s t a b i e l e t r i l l i n g geraken, waardoor de cons t r u c t i e meestal i n k o r t e t i j d b e z w i j k t . D i t v e r -

s t e r k a f van de betrouwbaarheid

s c h i j n s e l , d a t v e r o o r z a a k t wordt door h e t samen-

pisch

spel van de e l a s t i s c h e en massatraagheidskrachten

den beschreven. Om d i t t e kunnen beoordelen i s h e t

van de c o n s t r u c t i e en de l u c h t k r a c h t e n op h e t be-

n o d i g t e beschikken o v e r experimentele r e s u l t a t e n

t r e f f e n d e deel, wordt " f l u t t e r " genoemd. Vanwege

d i e voldoende g e d e t a i l l e e r d z i j n om basisveronder-

h e t g e v a a r l i j k e k a r a k t e r van f l u t t e r i s een v l i e g -

s t e l l i n g e n i n de rekenmethoden t e v e r i f i ë r e n en

t u i g f a b r i k a n t v e r p l i c h t om aan t e tonen d a t de

berekende r e s u l t a t e n t e toetsen.

snelheid waarbij d i t verschijnsel optreedt vol-

gegevens voor d i t doel waren e c h t e r n i e t o f nauwe-

transsone k a r a k t e r van de stroming kan wor-

Experimentele

l i j k s aanwezig op h e t moment d a t h e t i n d i t

doende v e r b u i t e n h e t s n e l h e i d s b e r e i k van h e t desbetreffende v l i e g t u i g l i g t .

waarmee h e t t y -

In d i t opzicht dient

p r o e f s c h r i f t beschreven onderzoek werd g e s t a r t .

een v l i e g t u i g aan s t r i n g e n t e luchtwaardigheids-

Om h e t i n z i c h t i n h e t gedrag van transsone

e i s e n t e voldoen.

Om genoemde reden w o r d t i n de l u c h t v a a r t i n -

stromingen om t r i l l e n d e p r o f i e l e n t e v e r g r o t e n en

d u s t r i e veel aandacht besteed aan de o n t w i k k e l i n g

experimentele r e s u l t a t e n t e v e r k r i j g e n voor de

van betrouwbare methoden voor h e t v o o r s p e l l e n van

ondersteuning van de genoemde o n t w i k k e l i n g op

de f l u t t e r e i g e n s c h a p p e n van v l i e g t u i g e n . De be-

t h e o r e t i s c h gebied,

trouwbaarheid van deze methoden hangt v o o r n a m e l i j k

R u i m t e v a a r t l a b o r a t o r i u m (NLR) een verkennend wind-

a f van de nauwkeurigheid

i s op h e t Nationaal Lucht- en

tunnelonderzoek u i t g e v o e r d aan t r i l l e n d e modellen

waarmee de l u c h t k r a c h t e n

op t r i l i e n d e draagvlakken (de zogenaamde " i n s t a -

van twee v e r s c h i l l e n d e v l e u g e l p r o f i e l e n . Met be-

t i o n a i r e luchtkrachten")

h u l p van een s p e c i a l e m e e t t e c h n i e k , z i j n g e d e t a i l -

kunnen worden bepaald.

Voor subsone en supersone snelheden z i j n h i e r v o o r

l e e r d e s t a t i o n a i r e en i n s t a t i o n a i r e drukken geme-

inmiddels goede rekenmethoden beschikbaar, maar

ten op een c o n v e n t i o n e e l p r o f i e l met een t r i l l e n d

voor h e t transsone snelheidsgebied,

r o e r en op een door h e t NLR ontworpen " s c h o k v r i j "

w a a r i n gecom-

p l i c e e r d e stromingspatronen met zowel subsone a l s

transsoon p r o f i e l

supersone snelheden, vaak gescheiden door schok-

voeren.

golven, optreden,

i s d i t nog n i e t h e t g e v a l . Deze

d a t d r a a i t r i l l i n g e n kon u i t -

Het p r o e f s c h r i f t , d a t d i t onderzoek be-

s i t u a t i e i s w e i n i g bevredigend, v o o r a l omdat de

s c h r i j f t en w a a r i n de nadruk i s gelegd op de i n -

ervaring heeft geleerd dat f I utterproblemen rela-

t e r p r e t a t i e van de r e s u l t a t e n , is onderverdeeld

t i e f vaak optreden i n h e t transsone snelheidsge-

i n v i e r delen, waarvan de inhoud hieronder i s

bied.

samengevat. Deel I b e g i n t met een b e s c h r i j v i n g van de

Door de i n de l a a t s t e j a r e n s t e r k toegenomen b e l a n g s t e l i i n g voor h e t v l i e g e n b i j transsone

stroming om t r i l l e n d e v l e u g e l p r o f i e l e n , gevolgd

s n e l h e i d (met name door de o n t w i k k e l i n g van zoge-

door een beschouwing o v e r de s p e c i f i e k e problemen

naamde " s u p e r k r i t i e k e l ' o f " s c h o k v r i j e "

d i e b i j transsone snelheden optreden. Het b l i j k t

vleugels,

w a a r b i j door een s p e c i a l e vormgeving een g e d e e l t e -

dat, anders dan b i j gematigd subsone en supersone

i i j k supersone s t r o m i n g zonder schokgolven wordt

stromingen,

i n een transsone stroming een essen-

verkregen), i s e r ook een s t e r k toegenomen b e h o e f t e

t i ë l e k o p p e l i n g b e s t a a t tussen h e t s t a t i o n a i r e en

o n t s t a a n aan methoden

h e t i n s t a t iona i re

waarmee de l u c h t k r a c h t e n Op

s t r o m i ngsvel d. H i erdoor kunnen

t r i i l e n d e draagvlakken i n transsone s t r o m i n g voor-

b e i d e stromingsproblemen n i e t l a n g e r o n a f h a n k e l i j k

s p e l d kunnen worden. Op v e r s c h i l l e n d e p l a a t s e n i s

van e l k a a r worden behandeld.

men dan ook begonnen met h e t o n t w i k k e l e n van nume-

k o p p e l i n g , w o r d t deel I a f g e s l o t e n met een samen-

-147-

I n verband met deze