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 .
<
downstream o f t h e shock a c c o r d i n g t o t h e r e l a t i o n f o r a normal shock wave ( F i g . 3.4a).
I t i s c l e a r t h a t an important q u e s t i o n w i t h
T h i s leads
t o a reversed g r a d i e n t i n y - d i r e c t i o n .
respect t o t h e p r a c t i c a l a p p l i c a t i o n o f s u p e r c r i t -
According
i c a l a i r f o i l s i s , how gradual the change from
t o Z i e r e p , t h e shock wave t h e r e f o r e has t o be
shock-free f l o w t o t h e n e i g h b o u r i n g f l o w c o n d i -
curved i n t h e sense a s i n d i c a t e d i n f i g u r e 3 . 4 b - i ,
t i o n s w i t h shock waves i s , o r , i n o t h e r words,
while,
what a r e the margins w i t h i n which t h e Mach number
i n a d d i t i o n , a readjustment of t h e p r e s s u r e
g r a d i e n t i s r e q u i r e d j u s t downstream o f t h e shock
and i n c i d e n c e may be v a r i e d around the design con-
wave, which r e m i t s i n a r a p i d expansion ( F i g . 3.4b-2).
d i t i o n w i t h o u t s e r i o u s d e t e r i o r a t i o n o f t h e fa-
O f t e n t h i s expansion can be n o t i c e d i n
v o u r a b l e low-drag p r o p e r t y ( l i t t l e o r no wave drag)
s u r f a c e p r e s s u r e d i s t r i b u t i o n s o f a i r f o i l s , where
o f the shock-free flow. For t h e a e r o e l a s t i c i a n ,
i t manifests i t s e l f as t h e s o - c a l l e d Z i e r e p cusp
t h e i m p o r t a n t q u e s t i o n becomes how t h e aerodynamic
(see, f o r i n s t a n c e , f i g u r e 3.3).
l o a d i n g w i l l change d u r i n g o s c i l l a t i o n s around the
There are, how-
ever, circumstances, depending on t h e v e l o c i t y gradients i n y-direction,
design c o n d i t i o n .
the c u r v a t u r e o f t h e
a i r f o i l c o n t o u r and t h e t h i c k n e s s o f t h e boundary layer,
3.3
i n which t h i s e f f e c t cannot be observed,
as i n the exampies o f f i g u r e 3.2.
SOME PARTICULAR FLOW PATTERNS ON AIRFOILS WITH FLAP
A p a r t from t h e f l o w p a t t e r n s discussed i n
3.2
t h e preceding s e c t i o n s , t h e r e a r e some a d d i t i o n a l
SHOCK-FREE FLOW
shock c o n f i g u r a t i o n s t h a t may occur only on a i r The f l o w p a t t e r n s discussed so f a r a r e char-
f o i l s w i t h a f l a p , s i n c e they a r e d i r e c t l y r e l a t e d
-22-
=RESIURE
COEFFICIENT
C
SHOCK - F R E E OEIIGN COHOITION
,-. _' '.
Urn
't
a: CHORDWISE PREIIURE
M,
F i g . 3.5
-
0.725
1.0
O
OIiTRIBUTIONS
--
U?
no 0.89
.-.
-
IHOCK-FREE DESIGN COHOITION
U T = 0.748 eo= 0.81'
-.
A
nQ
--
,--.
\.___' 0.748
u,
0.89
u0
-.
0.m 0.85'
& <--,
b: FLOX
*%
PATTERN1
0.748
i
.__--
01
Mx
0.7a -00 i
I n f l u e n c e of Mach number end i n c i d e n c e on pressure d i s t r i b u t i o n and flow p a t t e r n of a "shockfree" a i r f o i l i n i t s d e s i g n c o n d i t i o n .
t o t h e a b r u p t change i n downwash a t t h e l e a d i n g edge o f t h e f i a p ( F i g . 3.6).
f i a p ( F i g . 3.6b),
A l 1 a d d i t i o n a l possi-
t h e expansion has t o be
f o l l o w e d by a normal shock i n o r d e r t o re-
b i l i t i e s c o n t a i n one o r more of t h e f o l l o w i n g
duce t h e f l o w v e l o c i t y from supersonic t o
elementary f l o w p a t t e r n s , w h i c h a r e well-known from
subsonic.
t h e behaviour o f subsonic and supersonic flows around. sharp-edged c o r n e r s (Refs.
(c) Lambda shock wave. When a supersonic f l o w
74, 7 5 ) :
passes a concave c o r n e r , an o b l i q u e shock wave occurs,
( a ) Subsonic expansion around a sharp-edged
through which t h e flow remains
convex c o r n e r . T h i s r e s u l I s i n a supersonic
supersonic b u t changes i t s d i r e c t i o n . For
r e g i o n and an o b l i q u e shock wave emanating
the examples o f f i g u r e 3 . 6 ~ . the o b l i q u e
from t h e corner.
shock w i I I be f o l l o w e d by a normal one t o
Since downstream o f an
o b l i q u e shock t h e f l o w remains supersonic,
a r r i v e a g a i n a t a subsonic mean flow.
a second normal shock o r a f a n o f compres-
shocks merge i n t o a s o - c a l l e d lambda shock.
s i o n waves i s r e q u i r e d t o reduce t h e v e l o c i -
D u r i n g t h e p r e s e n t wind-tunnel
t y t o subsonic speed ( F i g . 3.6a).
80th
experiments, almost
e v e r y t y p e o f shock p a t t e r n a s sketched i n f i g u r e
3.6 has been r e a l i z e d . However, i n o r d e r t o sim-
( b ) Prandtl-Heyer expansion. When a supersonic f l o w passes a convex c o r n e r , t h e f l o w ex-
p l i f y t h e a n a l y s i s for t h e a i r f o i l w i t h f l a p , t h e
pands c o n t i n u o u s l y , u n t i l t h e f l o w i s d i -
f l o w p a t t e r n s i n which o n l y normal shocks occur
r e c t e d a l o n g t h e c o n t o u r downstream of t h e
a r e emphasized.
corner.
I n t h e examples o f t h e a i r f o i l w i t h
-23-
3.4
V I S C O U S ASPECTS LAMINAR BOUNDARY
N o r m a l l y , the e f f e c t o f v i s c o s i t y i n an a t tached subsonic f l o w around an a i r f o i l
i s confined
t o a t h i n l a y e r adjacent t o the surface o f the a i r f o i l , which i s c a l l e d the boundary l a y e r , and t o the wake.
I n t h e boundary l a y e r ,
the velocity rises
from z e r o a t t h e s u r f a c e t o an almost c o n s t a n t TRANSITION POINT
v a l u e a t t h e o u t e r edge. Usually t h e boundary l a y e r s t a r t s a t t h e l e a d i n g edge as a l a m i n a r bound-
Fig. 3.7
a r y l a y e r , which a f t e r a c e r t a i n d i s t a n c e changes
Boundary layer in a subsonic attached flow.
from l a m i n a r t o t u r b u l e n t ( t r a n s i t i o n ) ( F i g . 3 . 7 ) . The presence o f t h e boundary l a y e r changes t h e e f f e c t i v e c o n t o u r o f the a i r f o i l and,
thus, has an
e f f e c t on t h e p r e s s u r e d i s t r i b u t i o n and t h e aerodynamic loading. The magnitude o f t h i s e f f e c t depends, among o t h e r s , on t h e geometry and t h e i n c i dence o f t h e a i r f o i l and t h e Reynolds number, t h e l a t t e r b e i n g an i m p o r t a n t parameter f o r t h e growth o f t h e boundary-layer t h i c k n e s s and t h e l o c a t i o n
SEPARAI;ON BUBBLE
of the t r a n s i t i o n point. The behaviour o f the boundary l a y e r i s of
Fig. 3.8
even more importance i n t r a n s o n i c f l o w than i n
subsonic f l o w , s i n c e here i t has a c o n s i d e r a b l e i n f l u e n c e on t h e p o s i t i o n and s t r e n g t h o f the shock wave. The p r e s s u r e jump t h r o u g h t h e shock wave,
Interaction Of a shock wave with a turbulent and laminar boundary layer and its effect on the pressure distribution.
o f t h e boundary l a y e r .
in
O f s p e c i a l importance i s
whether t h e shock wave i n t e r a c t s w i t h a laminar o r
i t s t u r n , has a l a r g e i n f l u e n c e on t h e development
a t u r b u l e n t boundary layer. As sketched i n f i g u r e
3.8,
t h i s may lead t o c o n s i d e r a b l e d i f f e r e n c e s i n
shock p a t t e r n s and pressure d i s t r i b u t i o n s .
COMBINED FLOWPATTERNS
A t t h e r e l a t i v e l y low Reynolds numbers t h a t
u s u a l l y can be reached i n wind-tunnel
the
boundary l a y e r o f t e n remains l a m i n a r up t o t h e
a:sM*Li
SUPERSONIC REGION A T THE CONVEX CORNER BETWEEN W I G *NO FLAP
shock wave, w h i l e ,
&
t h e boundary l a y e r most p r o b a b l y w i l l be t u r b u l e n t
t o c o n s i d e r a b l e scale e f f e c t s . For c o n v e n t i o n a l
.--
b:PRANOTL .MEIER EXPANSION AROUND CONVEX CORNER BETWEEN WING AN0 FLAP
C:.I-SHOCK WAVE IN THE CONCAVE CORNER BETWEEN WING AND FLAP
f o r f u l l - s c a l e Reynolds numbers,
when i n t e r a c t i n g w i t h t h e shock. T h i s g i v e s r i s e
\*
4 cj '.-
4
Some particular flow patterns on
a i r f o i l s , a u s e f u l technique f o r reducing these scale effects i s t o a r t i f i c i a l l y force transition o f t h e boundary l a y e r upstream o f t h e shock by means o f a t r a n s i t i o n s t r i p . As w i l l be explained below, t h i s technique becomes q u e s t i o n a b l e f o r the
+\ _ _ I
modern-type a i r f o i l s .
I n t h e i n t e r a c t i o n between a shock wave and
t h e boundary l a y e r , d i f f e r e n t stages can be d i s -
.-
Fig. 3.6
tests,
t i n g u i s h e d ( F i g . 3.9).
"weak"
811
when t h e boundary l a y e r remains attached,
and "strong"
airfoil with flap.
-24-
The i n t e r a c t i o n i s c a l l e d
when s e p a r a t i o n occurs.
IHOCK
number. a t f i r s t the bubble remains l o c a l i z e d , b u t then i t spreads o u t r a p i d l y towards t h e t r a i l i n g edge, which leads t o a sudden f a l l i n p r e s s u r e a t
WEAK INTERACTION
t h e t r a i l i n g edge ( p r e s s u r e divergence). T h i s i s accompanied by a s i g n i f i c a n t drop i n c i r c u l a t i o n , and o f t e n unsteady phenomena l i k e "buffet"
4
"buzz"
SEPARATION BUBBLE
:1-:
For modern a i r f o i l s , MOOERATELI STRONG INTERACTION
and
s t a r t t o occur.
N L R 7301
l i k e the
(Fig.
3.5a). w i t h t h e i r steep p r e s s u r e g r a d i e n t s towards t h e t r a i l i n g edge,
a second t y p e o f s e p a r a t i o n may
occur, which s t a r t s from t h e t r a i l i n g edge ( r e a r s e p a r a t i o n ) . Some p o s s i b l e developments o f t h i s SHOCK INDUCE0 SEPARATION
s o - c a l l e d 8 - t y p e s e p a r a t i o n a r e sketched i n f i g u r e
STRONG INTERACTION
F i g . 3.9
D i f f e r e n t stages of shock-wave/boundaryl a y e r i n t e r a c t i o n ( t u r b u l e n t boundary layer).
3.10.
The r e a r s e p a r a t i o n , which
sicai
subsonic type, depends s t r o n g l y on t h e t h i c k -
approaching t h e t r a i l i n g edge and on the p r e s s u r e
(Ref.
19) as
t h e type-8 s e p a r a t i o n
i s v e r y s e n s i t i v e t o t h e Reynolds number and t o the
n o r m a l l y proceeds as f o l l o w s ( F i g . 3.10):
t y p e "A"
i n c o n t r a s t w i t h t h e preceding
type of separation (type A),
the s e p a r a t i o n
process i n d i c a t e d by Pearcy e t a l .
o f the cias-
ness and t h e v e l o c i t y p r o f i l e o f t h e boundary l a y e r g r a d i e n t . Therefore,
For c o n v e n t i o n a l a i r f o i l s ,
i 5
l o c a t i o n o f t h e p o i n t where t h e t r a n s i t i o n o f the boundary l a y e r from laminar t o t u r b u l e n t t a k e s
t h e p r e s s u r e r i s e through t h e shock causes a t h i c k e n i n g of t h e boundary l a y e r a t t h e f o o t .
place. T h i s means t h a t t h e advanced-type a i r f o i l s
If
a r e v e r y s e n s i t i v e t o t h e Reynolds number, w h i l e
t h e p r e s s u r e r i s e i s l a r g e enough, a l o c a l separat i o n bubble develops j u s t downstream o f t h e shock.
s i m u l a t i o n o f h i g h Reynolds numbers w i t h t h e h e l p
W i t h f u r t h e r i n c r e a s e i n t h e free-stream Mach
o f a r t i f i c i a l t r a n s i t i o n becomes v e r y q u e s t i o n a b l e .
'TYPE A ' '
VARIANTS OF "TYPE E"
1 REAR SEPARATION PROVOKED BY BUBBLE
--
3 REAR SEPARATION
2 REAR SEPARATION PROVOKED BI SHOCK ITSELF
i - i INTERACTION WITH BUBBLE
ALREADY PREIEHI
1 b i INTERACTION WITH SHOCK
CL +* >
I o i INTERACTION
1
WITH BUBBLE
-*
b , INTERACTION WITH SHOCK
*
P
-
U
<--! Y
C L
Y rr
2 * I
I
r FIRST E F F E O ON CIRCULATION f.l MAJOR
F i g . 3.10
EFFECTS OH CIRCULATION
Flow models f o r t h e i n i t i a l development of shock-induced s e p a r a t i o n w i t h Mach number or i n c i d e n c e f o r a t u r b u l e n t boundary l a y e r (Ref. 1 9 ) .
-25-
Pearcy e t a l . (Ref.
19) developed a method
t o determine t h e t y p e o f s e p a r a t i o n f o r a s p e c i f i c a i r f o i l and t h e c o n d i t i o n s (Mach number and i n c i dence) a t which t h i s o c c u r s by p r e s e n t i n g t h e p r e s s u r e s a t f i x e d chord-wise s t a t i o n s as a funct i o n o f t h e f r e e - s t r e a m Mach number. investigation,
I n the present
t h e i r method has been a p p l i e d t o
d e t e r m i n e t h e boundaries o f s e p a r a t i o n f o r b o t h t h e NACA 64A006 a i r f o i l and t h e NLR 7301 a i r f o i l , i n o r d e r t o assure t h a t o n l y f l o w c o n d i t i o n s a r e s t u d i e d w i t h o u t severe s e p a r a t i o n e f f e c t s . The r e s u l t f o r t h e NLR 7301 a i r f o i l ,
as obtained i n a
d e t a i l e d i n v e s t i g a t i o n o f Zwaaneveld (Ref. i s reproduced i n f i g u r e 3.11.
73).
Together w i t h t h e
boundaries f o r maximum l i f t and l i f t divergence, t h i s f i g u r e g i v e s a good i l l u s t r a t i o n o f t h e c h a r a c t e r i s t i c s and l i m i t a t i o n s o f a t r a n s o n i c a i r f o i l .
Fig. 3.11
Main characteristics and boundaries for the onset of separation of the NLR 7301 airfoil (Ref. 73).
-26-
PART II SCOPE AND DESCRIPTION OF THE EXP ER IM ENTAL INV EST IG AT1ON S
-27-
4
SCOPE OF THE NLR INVESTIGATIONS
4.1
PROBLEM DEFINITION
design,
i s the q u e s t i o n whether o r n o t the new gen-
e r a t i o n o f s u p e r c r i t i c a i a i r f o i l s possesses adThe aim af the i n v e s t i g a t i o n s a t N l R was t o
verse unsteady aerodynamic c h a r a c t e r i s t i c s , espe-
improve t h e p h y s i c a l i n s i g h t i n t o t h e n a t u r e o f
c i a l l y around t h e i r design c o n d i t i o n .
unsteady t r a n s o n i c flows and t o s u p p o r t the d e v e l opment o f t h e o r e t i c a l o r semi-empirical
A f i n a l aspect t h a t i n e v i t a b l y has t o be
prediction
methods. From t h e c o n s i d e r a t i o n s i n c h a p t e r 2,
c o n s i d e r e d i s t h e i n f l u e n c e o f t h e boundary l a y e r . As d e s c r i b e d i n c h a p t e r 3,
it
t h e boundary l a y e r
i5
i s c l e a r t h a t such an e x p l o r a t o r y i n v e s t i g a t i o n
o f c r u c i a l importance f o r steady t r a n s o n i c flow
s h o u l d be focussed on t h e f o l i o w i n g aspects.
and,
F i r s t l y , the d i s c u s s i o n s about t h e unsteady-
t h e r e f o r e , may have a s i g n i f i c a n t i n f l u e n c e
a l s o on t h e unsteady a i r l o a d s .
f l o w e q u a t i o n s have shown t h a t i n t r a n s o n i c flows 4.2
an e s s e n t i a l c o u p l i n g e x i s t s between t h e steady and
APPROACH
unsteady f l o w f i e l d s . T h i s i m p l i e s t h a t , besides
In view o f t h e afore-mentioned aspects,
the usuai parameters f o r unsteady f l o w ( f r e e - s t r e a m
it
Mach number, reduced-frequency and v i b r a t i o n mode),
was decided a t NLR t o p e r f o r m steady and unsteady
a l s o t h e a i r f o i l shape, t h e mean i n c i d e n c e and the
p r e s s u r e measurements on Some c h a r a c t e r i s t i c a i r -
mean f l a p angle have t o be considered.
f o i l s i n two-dimensional f l o w . P r i o r i t y has been
I n an inves-
t i g a t i o n o f t h e i n f l u e n c e o f these a d d i t i o n a l param-
g i v e n t o the c o n c e n t r a t i o n on i n v e s t i g a t i o n s i n
eters,
two-dimensional flow, s i n c e they a r e n o t only s i m -
d e t a i l e d i n f o r m a t i o n should be o b t a i n e d
p l e r than f o r three-dimensional
about b o t h the unsteady and t h e mean steady flow. Secondly, the unsteady t r a n s o n i c - f l o w
flow, b u t a l s o
o f f e r t h e o p p o r t u n i t y t o supplement the pressure
problem i s e s s e n t i a l l y n o n l i n e a r (see eq. 2.15).
measurements by o p t i c a l f l o w s t u d i e s o f the p e r i -
A s shown i n s e c t i o n 2.2.3,
o d i c a l motion o f t h e shock waves. F u r t h e r i t was
the problem can be
reduced t o a l i n e a r one, be i t t h a t the r e s u i t i n g
a n t i c i p a t e d t h a t t h e development a f t h e o r e t i c a l
e q u a t i o n (2.18)
methods
possesses s p a t i a l l y v a r y i n g coef-
-
i n analogy w i t h the devclaoment of
-
f i c i e n t s . T h i s l i n e a r i z a t i o n was achieved by
steady t r a n s o n i c computationai methods
assuming t h e unsteady f l o w t o be a small p e r t u r -
s t a r t w i t h t h e two-dimensional fiaw problem.
b a t i o n o f the n o n l i n e a r steady f l o w f i e l d . course,
would a l s o
An e x c e l l e n t t o o l t o reach t h i s goal was the
Of
experimental technique developed a t NLR, which en-
t h i s l i n e a r i z a t i o n can always be e n f o r c e d
by t a k i n g 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 Small
a b i e s t h e measurement
enough, b u t the important q u e s t i o n a r i s e s whether
steady and unsteady pressures on a s t i l l a t i n g wind-
t h i s can be achieved w i t h o u t a r e t u r n t o imprac-
tunnel modeis ( d e t a i l s o f t h i s technique a r e g i v e n
t i c a l l y small values. Therefore,
i n c h a p t e r 5).
t h e q u e s t i o n how
o f a l a r g e number o f mean
f a r l i n e a r i z a t i o n about a g i v e n mean s t e a d y - f l o w
A f t e r some p r e l i m i n a r y measurements on a c i r c u l a r
c o n d i t i o n i s j u s t i f i e d should be a p o i n t o f inves-
a r c a i r f o i l w i t h o s c i i i a t i n g f l a p , which were n o t
t i g a t i o n . This i s important, since i n aeroeiastic
v e r y successful because o f severe f l o w - s e p a r a t i o n
i n v e s t i g a t i o n s , where the unsteady aerodynamic
e f f e c t s , a f i r s t s e r i e s o f e x t e n s i v e t e s t s was con-
f o r c e s have t o be combined w i t h i n e r t i a l , s t i f f -
ducted i n 1966 on an NACA 64A006 a i r f o i l p r o v i d e d
n e s s , and damping f o r c e s o f t h e a i r c r a f t s t r u c t u r e ,
w i t h a 2 5 per cent
a l l r o u t i n e methods a r e l i n e a r . From t h a t s i d e
c o u l d be forced i n t o s i n u s o i d a l motions. These
t h e r e c l e a r l y e x i s t s a s t r o n g i n t e r e s t i n some
t e s t s , which were l i m i t e d t o z ero mean l i f t condi-
s o r t o f l i n e a r i z a t i o n o f t h e unsteady aerodynamic
t i o n s , r e v e a l e d a number o f i n t e r e s t i n g e f f e c t s i n
p rob iem.
t h e high-subsonic and t r a n s o n i c speed regime (Ref.
chord t r a i l i n g - e d g e f l a p t h a t
71). U n f o r t u n a t e l y , a few years l a t e r i t was d i s -
A third point o f interest, related to
covered t h a t an important e f f e c t had been over-
t h e c u r r e n t developments i n s u p e r c r i t i c a l - w i n g
-29-
looked i n t h e measuring technique, which had a
s t i l l of current interest, not only f o r f l u t t e r
c o n s i d e r a b l e i n f l u e n c e on t h e u n s t e a d y - f l o w data
purposes, b u t a l s o i n connection w i t h t h e recent
a t h i g h speed i n a q u a n t i t a t i v e sense (Refs. 39,
development o f s o - c a l l e d “ a c t i v e - c o n t r o i ”
systems.
72; see a l s o s e c t i o n 5 . 3 . 2 ) . The f o l l o w i n g s t e p i n the e x p l o r a t i o n o f un-
Because o f t h e increased i n t e r e s t i n unsteady t r a n s o n i c Flow,
steady t r a n s o n i c f l o w s was an i n v e s t i g a t i o n on a
i t was decided t o repeat t h e ear-
l i e r t e s t s and t o extend t h e program t o f l o w con-
s u p e r c r i t i c a l a i r f o i l t h a t c o u l d be o s c i l l a t e d i n
d i t i o n s w i t h non-zero mean l i f t , t o i n v e s t i g a t e t h e
pitch.
e f f e c t s o f mean i n c i d e n c e and mean f l a p a n g l e on
s e l e c t e d , which was designed f o r shock-free f l o w
t h e unsteady a i r l o a d s . These measurements t o o k
w i t h t h e hodograph method o f ö o e r s t o e l (Refs. 10.
p l a c e i n 1972. The main reasons t o c o n c e n t r a t e t h e
78). T h i s a i r f o i l was chosen,because
i n v e s t i g a t i o n a g a i n on t h i s s p e c i f i c a i r f o i l w i t h
s e n t a t i v e f o r t h e new g e n e r a t i o n o f a i r f o i l s t h a t
i t i s repre-
i s o f interest for application i n supercritical-
f l a p were:
- o s c i l l a t o r y changes
wing design. Moreover,
o f t h e downwash a r e c o n f i n e d
i t s steady-flow character-
t o t h e r e a r p a r t o f t h e a i r f o i l , which c o n s i d e r -
i s t i c s were known a l r e a d y from an e x t e n s i v e wind-
ably f a c i l i t a t e s the i n t e r p r e t a t i o n o f the a i r -
tunnel i n v e s t i g a t i o n (Ref. 73). The u n s t e a d y - f l o w
loads
-
For t h i s purpose t h e “NLR 7301” a i r f o i l was
t e s t s on t h e NLR 7301 a i r f o i l were performed i n
on t h e whole a i r f o i l ;
1976.
by v a r i a t i o n o f mean incidence, mean f l a p angle. and f r e e - s t r e a m Mach number, a g r e a t v a r i e t y o f
The experimental i n v e s t i g a t i o n s w i l l be
steady f l o w f i e l d s can be generated, about which
described i n t h i s p a r t o f the thesis, w h i l e i n the
o s c i l i a t o r y p e r t u r b a t i o n s can be i n v e s t i g a t e d
n e x t p a r t an a n a l y s i s o f t h e r e s u l t s w i l l be g i v e n ,
(see a l s o c h a p t e r 3 ) ;
i n which t h e v a r i o u s aspects mentioned i n s e c t i o n
t h e unsteady a i r l o a d s on c o n t r o l surfaces a r e
4.1 a r e emphasized.
-
S TECHNIQUE FOR UNSTEADY PRESSURE MEASUREMENTS
5.i
P R I N C I P L E OF THE MEASURING TECHNIQUE
The method employed a t NLR f o r t h e determinat i o n o f unsteady p r e s s u r e d i s t r i b u t i o n s on harmoni c a l i y o s c i l l a t i n g wind-tunnel models was developed about two decades ago by Bergh (Ref.
21).
The q u i n SCANIIIHGVALYE
tessence o f t h i s method i s t h a t t h e p r e s s u r e d i s -
@
t r i b u t i o n o v e r t h e model surface i s measured through a l a r g e number o f p r e s s u r e tubes, which a r e
MOD.
The use o f tubes and scanning v a l v e s
makes t h e method r e l a t i v e l y inexpensive as compared
MODEL OISPLACEUENT
AMPLIFICATION : Y
FREQUENCY w
w i t h methods i n which each tube i s r e p l a c e d by an
RI-AXIS
i n s i t u p r e s s u r e sensor. For steady p r e s s u r e measurements,
Irn.*XIP
..........
connected i n groups to only a few scanning v a l v e s (Fig. 5.la).
TEST SET-UP
PHASELAG
:a
the re-
q u i r e d v a l u e s can be determined d i r e c t l y w i t h t h e t r a n s d u c e r s i n t h e scanning valves, because t h e
-ARC L! Pi
presence o f t h e tubes does n o t a f f e c t t h e r e s u l t s .
@
More c o m p l i c a t e d i s t h e measurement o f t h e s i n u s o i d a l l y v a r y i n g pressures, because then t h e t u b i n g
TRANSFER FUNCTION OF TUBET R I N S O K E R SYSTEM
Fig. 5 . S
-30-
@
DATA REDUCTION PROCEDURE
P r i n c i p l e ai measuring technique.
system ieads t o a c o n s i d e r a b l e d i f f e r e n c e between
VOLUMETUBE) Y '.I
t h e a c t u a l pressures a t t h e model s u r f a c e and t h e
i7
~
z
~
1 ,
MODEL SURF4CE
pressures as recorded by t h e transducers. Therefore, an e s s e n t i a l s t e p i n t h e d a t a - r e d u c t i o n procedure i s the t r a n s l a t i o n o f the unsteady pressures pu, measured w i t h the t r a n s d u c e r s i n the scanning vaives, t o t h e a c t u a l pressures a t t h e model surface pi w i t h
Series connection of tubes and transducers.
Fig. 5.2
t h e use o f t h e t r a n s f e r f u n c t i o n o f t h e t u b i n g system. T h i s procedure i s i n d i c a t e d s c h e m a t i c a l l y i n f i g u r e 5 . 1 ~ : t h e v e c t o r p i i s o b t a i n e d from t h e vector p
ZR
by a c o u n t e r - c l o c k w i s e r o t a t i o n 6 and a
+U
r e d u c t i o n i n magnitude w i t h a f a c t o r CL. The values o f the a m p l i f i c a t i o n a and t h e phase l a g 6 o f t h e
i L(
I
o, I I. p,:w'i
t u b i n g system f o l i o w from t h e t r a n s f e r f u n c t i o n , as
i s indicated i n f i g u r e 5.lb. 5.1c),
As a n e x t s t e p ( F i g .
L
Fig. 5.38
Single pressure-measuring system.
Fig. 5.3b
Tube with discontinuity in tube radius.
t h e v e c t o r p i is decomposed i n a r e a l and an
imaginary component r e l a t i v e t o t h e m o t i o n o f t h e model, which u s u a l l y i s measured by means o f a c c e l erometers or displacement pick-ups.
In t h i s procedure, t h e a c c u r a t e d e t e r m i n a t i o n o f t h e t r a n s f e r f u n c t i o n s o f tube-transducer systems under the a c t u a l c o n d i t i o n s i n t h e wind tunnel i s
one or more d i s c o n t i n u i t i e s i n tube diameter ( F i g .
e x t r e m e l y important. For t h i s reason, c o n s i d e r a b l e
5.3b).
a s i t u a t i o n which o f t e n occurs i n p r a c t i c e .
O r i g i n a l l y t h e model was d e r i v e d f o r t h e
a t t e n t i o n was p a i d a t NLR t o the study o f t h e dy-
system i n s t i l l a i r (Ref. 2 2 ) . Afterwards, when i t
namic behaviour of t h i n c i r c u i a r tubes connected
became apparent t h a t t h e main f l o w across the tube
w i t h small instrument volumes.
entrance had a s i g n i f i c a n t i n f l u e n c e on t h e dynam-
5.2
i c response, t h e model was extended t o t a k e a l s o
THEORETICAL MODEL FOR THE DYNAMIC RESPONSE
t h i s e f f e c t i n t o account (Ref. 7 2 ) .
OF TUBE-TRANSDUCER SYSTEMS
5 . 2 . 1 The p r o p a g a t i o n of pressure waves through
I n o r d e r t o a i d i n t h e design o f o p t i m a l
tube-transducer systems, a t h e o r e t i c a l model was
c y l i n d r i c a l tubes
developed t o c a l c u l a t e t h e i r dynamic response c h a r a c t e r i s t i c s (Refs. c a l model deais w i t h
22 and 7 2 ) . T h i s t h e o r e t i -
To a r r i v e a t t h e dynamic response formulae
N tubes and N voiumes connected
i n s e r i e s ( F i g . 5.2).
f o r tube-transducer systems ( f o r the d e t a i l s o f t h e d e r i v a t i o n , r e f e r e n c e i s made t o Appendix E ) ,
Across t h e e n t r a n c e o f t h e
system, t h e f l o w has a mean v e l o c i t y U , w h i l e t h e
f i r s t t h e s i n u s o i d a l motion o f a f l u i d column i n
p r e s s u r e comprises a s i n u s o i d a l v a r i a t i o n p i ,
a r i g i d tube w i t h c i r c u l a r cross s e c t i o n has t o be
in-
duced by the model o s c i l l a t i o n . For t h e p r e s e n t
considered. T h i s motion i s governed by t h e f o l i o w -
purpose, one i s i n t e r e s t e d i n t h e r a t i o between
ing b a s i c equations:
t h e p r e s s u r e p e r t u r b a t i o n i n volume V pressure v a r i a t i o n p i .
.
and t h e
v,j In g e n e r a l , t h i s r a t i o i s a
(axi-symmetrical),
. . .
complex number, because i t d e s c r i b e s b o t h magnitude and phase angle r e l a t i v e t o t h e i n p u t pressure. The s p e c i a i case o f a s i n g l e system (Fig. 5.3a) can easil.
t h e Navier-Stokes equations
be d e r i v e d . F u r t h e r , t h e model
the equation of c o n t i n u i t y , the equation o f state,
and
t h e energy e q u a t i o n t h a t g i v e s t h e balance between thermal and k i n e t i c energies;
t o g e t h e r f i v e equations f o r t h e same number o f un-
o f f e r s t h e p o s s i b i l i t y t o c o n s i d e r a system w i t h
-31-
known q u a n t i t i e s :
the v e l o c i t y p e r t u r b a t i o n s i n
T 'E . o g 8, I + p,èw'l
a x i a i and r a d i a l d i r e c t i o n and t h e p e r t u r b a t i o n s i n d e n s i t y , temperature and p r e s s u r e .
+u
P, 2R i I + P , P )
As shown i n Appendix 6 , t h e s o l u t i o n t h a t s u f f i c e s f o r the p r e s e n t a p p l i c a t i o n i s the soca l l e d "low- reduced-f requency so1 u t ion",
~
-
.P.(l+p,e'<''li
~
+
A
p
~
eU0"U ' u
l
,
which f o r
t h e p r e s s u r e p e r t u r b a t i o n , p , i n s i d e t h e tube TRAHIDUCER
reads:
@IN
STILL AIR
r
@WITH
caoss
WIND
Boundary c o n d i t i o n s at t h e entrance of t h e tube-transducer system.
Fig. 5 . 4
with
TRANSDUCER
mass l e a v i n g tube j and t h e mass e n t e r i n g tube
= [ J 0 ( i 3 / 2 s ) /J2(i3/2s)]i
+y
n = [I
(y/n)'
j+l.
,
(5.la)
I
J2(i3'*os)
/Jo(i3/20s)]'
,
i n o r d e r t o o b t a i n a r e l a t i o n between t h e
p r e s s u r e p e r t u r b a t i o n and d e n s i t y p e r t u r b a t i o n i n s i d e t h e yoiumes, i t has a l s o been assumed t h a t
(5.Ib)
t h e r e t h e thermodynamic process takes p l a c e p o l y t r o p i c a l l y , w i t h a p o l y t r o p i c constant nv t o be specified.
Here J
A separate c o n d i t i o n has t o be p r e s c r i b e d
and J
a r e Bessel f u n c t i o n s of f i r s t k i n d 2 o f z e r o and second o r d e r , r e s p e c t i v e l y . O
I n the expression ( 5 . 1 ) - ( 5 . i c ) ,
a t t h e entrance o f t h e f i r s t tube ( j = 1 ) . s t i l l air,
x denotes t h e co-
o r d i n a t e i n a x i a l d i r e c t i o n o f the tube, a,
t h i s boundary c o n d i t i o n i s d i c t a t e d by
t h e f a c t t h a t the p r e s s u r e equals the value p i im-
t h e ve-
l o c i t y o f sound. y the r a t i o o f s p e c i f i c heats,
In
posed upon t h e system ( F i g . 5.4a):
I
the square r o o t o f t h e P r a n d t l number, R t h e tube xi
r a d i u s , p s the mean d e n s i t y and U t h e v i s c o s i t y . As
Iand y
g i v e n gas,
= o ,
p = p . .
(5.2)
can be considered a s c o n s t a n t s f o r a the main parameter f o r the p r o p a g a t i o n
T h i s i m p l i e s t h a t t h e r e is no r e s t r i c t i o n f o r the o s c i i i a t o r y f l o w v e l o c i t y u.
o f t h e p r e s s u r e p e r t u r b a t i o n s t h r o u g h t h e tubes i s
i n the tube entrance,
the s o - c a l l e d shear wave number s , d e f i n e d i n
so t h a t a i r i s f r e e t o move i n and o u t of the tube
( 5 . 1 ~ ) . T h i s number i s a measure f o r t h e shearing
opening.
e f f e c t s a l o n g t h e tube w a i i .
entrance, however, t h e a i r l e a v i n g the tube has t o
I n case an a i r f l o w blows across the tube
d i s p l a c e t h e main flow, which l o c a l l y leads t o an a d d i t i o n a l o s c i l l a t o r y p r e s s u r e a t t h e tube en5.2.2
S o l u t i o n f o r complete tube-transducer
t r a n c e ( F i g . 5.4b).
sYstems
T h i s means t h a t ,
i f an o s c i l -
l a t o r y p e r t u r b a t i o n pi e x i s t s i n t h e main flow, t h e boundary c o n d i t i o n a t t h e tube entrance (5.2)
The s o i u t i o n (5.1)
c o n t a i n s two constants, A
has t o be m o d i f i e d i n t o :
and 8, which can be determined by s p e c i f y i n g t h e boundary c o n d i t i o n s a t b o t h ends of t h e tube. t h e general tube-transducer system ( F i g . 5.21,
For
x,-o
1
P'
(5.3)
Pi C A P .
the
f o l l o w i n g boundary c o n d i t i o n s have been introduced: a t t h e e n t r a n c e o f tube j
As argued i n Appendix 8, a l o g i c a l assumption f o r
the pressure perturba-
t h i s e x t r a pressure perturbation i s
t i o n equals t h e p r e s s u r e p e r t u r b a t i o n i n t h e prec e d i n g volume Vv
.j-1
, which
i n i t s t u r n equals t h e Ap = Cpsu0U
p r e s s u r e p e r t u r b a t i o n a t t h e end of tube j - i . Fur-
,
(5.4)
t h e r m r e , f o r each volume V
t h e i n c r e a s e i n mass v,j a t each moment has t o be equal to t h e d i f f e r e n c e i n
w i t h C b e i n g s t i l l an unknown c o n s t a n t and
-32-
U
the
v e l o c i t y o f t h e a i r f l o w across t h e e n t r a n c e open-
.
. a factor given i n (5.lb),
n
ing.
p l a i n e d i n Appendix 8
By a p p l i c a t i o n o f the afore-mentioned bound-
.
nv
,
which r e l a t e s t h e s i n u s o i d a l p r e s s u r e
.
as ex-
can be i n t e r p r e t -
thermodynamic process i n s i d e the tube;
r i v e d f o r t h e general tube-transducer system o f
p e r t u r b a t i o n i n volume V
-
ed as t h e p o l y t r o p i c c o n s t a n t For the
a r y c o n d i t i o n s , a r e c u r s i o n formula has been de-
F i g u r e 5.2,
-
which
t h e p o l y t r o p i c c o n s t a n t f o r t h e transduc e r volume W v ,
t o t h e s i n u s o i d a l pres-
f o r which a v a l u e has t o
be chosen.
VvJ
s u r e p e r t u r b a t i o n i n t h e preceding volume j - i and t h e n e x t volume j+l (formulae 8.47 and 8.51 o f
5.3
THE DYNAMIC CHARACTERISTICS OF
Appendix 8 ) .
TUBE-TRANSDUCER SYSTEMS
For a s i m p l e system t h a t c o n s i s t s o f one t u b e and one t r a n s d u c e r (Fig.
5.3a),
t h e r e s u l t i n g Formula
5.3.1
The dynamic response i n s t i l l a i r
For t h e dynamic response i n s t i l l a i r becomes: The t r a n s f e r f u n c t i o n o f tube-transducer systems i n s t i l l a i r f o r a s i n g l e system can be c a l c u l a t e d w i t h formula (5.5)
and For systems in-
v o l v i n g more than one volume and connecting tubes w i t h formula ( 8 . 4 7 ) OF Appendix 8 . Some ex-
(5.5)
amples t h a t a r e c h a r a c t e r i s t i c f o r the agreement w h i l e t h e f o l l o w i n g formula i s o b t a i n e d f o r the
n o r m a l l y o b t a i n e d between c a l c u l a t e d and measured
s i t u a t i o n w i t h a cross f l o w across t h e entrance:
response c h a r a c t e r i s t i c s a r e shown i n F i g u r e 5.5. Note t h a t i n t h e example o f f i g u r e 5.5a, organ pipes,
(5.6)
A c l o s e r examination o f (5.5) and (5.6)
re-
v e a l s t h a t t h e dynamic response o f a s i m p l e tube-
THEORY
:::t.'
EXPERIMENT
0 INPUT PRESSURE i l k . a INPUT PRESSURE 6SL9
m' m'
2R
ik
I
etsrs:
r
-
2.5- AMPLITUDE RATIO
0.5,
transducer system depends on t h e f o l l o w i n g param-
.
like in
resonance peaks o c c u r where t h e dynam-
O
,
Hi
the p r o p a g a t i o n c o n s t a n t . As f o l l o w s From ( 5 . i a ) , r o n l y depends on t h e shear wave number s = R ( p W/U) f In practice,
IO0
.
S
200
t h i s means t h a t the p r o p a g a t i o n i s determined by the i n t e r n a l tube r a d i u s R, the mean d e n s i t y i n t h e tube p s (which
ioo
no00
i s d i r e c t l y r e l a t e d t o t h e mean steady
p r e s s u r e and the mean temperature), frequency w and t h e v i s c o s i t y
. wL/ao
, ,
u;
OSINGLE TUBE. TRANSDUCER SYSTEM
t h e r a t i o between t h e tube l e n g t h and
R
= 0.7VS nm
L
i
I
""=
T, *
t h e r a t i o between t h e t r a n s d u c e r volume
p,
.
PHASE LAG
the
the wave l e n g t h ao/w;
. Vv/ V t
t
i
1000 28s I.'
m
n"=
WC 10504 k
@)WUBLE TUBE. TRANSDUCER SYSTEM Rn i 0.525 mm R 2 I 0.7POm L,i 500 mm L > i 500 mm V" i 285 3.m
1, = d
p,
1.4
z<*c I O Y I O kg,m2
and t h e volume o f t h e tube; U
,
Fig. 5 . 5
the v e l o c i t y o f the a i r f l o w across the e n t r a n c e opening;
-3 3 -
Experimental and t h e o r e t i c a l t r a n s f e r f u n c t i o n s of tube-transducer systems i n s t i l l air.
i c response shows a maximum. The example o f f i g -
5.3.2
I n f l u e n c e o f the a i r f l o w
which d e a l s w i t h a system w i t h a d i s -
u r e 5.5b,
c o n t i n u i t y i n tube r a d i u s ,
i s damped more h e a v i l y
I n the p r e l i m i n a r y t e s t s a t NLR on o s c i l -
and does n o t e x h i b i t t h e s e pronounced resonance
l a t i n g wind-tunnel modeis i n high-speed flow,
peaks.
was n o t recognized t h a t the main f l o w across t h e
I n t h e l a t t e r example, two v a l u e s o f t h e
it
i n p u t p r e s s u r e a r e used, thereby demonstrating
entrance o f t h e p r e s s u r e tubes was an important
t h a t t h e system behaves l i n e a r .
parameter,and f o r t h e data r e d u c t i o n o f these
- as n o r modeis - and f o r
For g e o m e t r i c a l l y i d e n t i c a l tubes m a l l y a p p l i e d i n t h e wind-tunnel
t e s t s use was made o f t h e t r a n s f e r f u n c t i o n s determined i n s t i l i a i r f o r t h e c o r r e c t values o f
a g i v e n frequency o, t h e t r a n s f e r , f u n c t i o n o n l y depends on t h e f o l l o w i n g q u a n t i t i e s : o s ,
u,
a
O
frequency and steady pressures. Afterwards, when and
s e r i o u s doubts a r o s e a g a i n s t t h e t e s t r e s u l t s ,
n, which f o r normal a i r a r e c o m p l e t e l y determined
t h e i n f l u e n c e o f t h e main f l o w was i n v e s t i g a t e d
by t h e v a l u e o f t h e mean steady p r e s s u r e ps and
i n an e x t e n s i v e experimental program on a two-
t h e mean temperature Ts. The i n f l u e n c e o f t h e s e
dimensional a i r f o i l w i t h o s c i l l a t i n g f l a p ( f o r
two parameters i s shown i n f i g u r e 5.6,
d e t a i l s , see r e f e r e n c e 72). These t e s t s r e v e a l e d
which i n d i -
cates t h a t a v a r i a t i o n i n mean temperature ( F i g .
the c o n s i d e r a b l e i n f l u e n c e t h a t t h e main f l o w may
5.6a) o f 30 OC does n o t have much e f f e c t , whereas
have on t h e dynamic c h a r a c t e r i s t i c s o f t h e tub-
t h e mean pressure has a c o n s i d e r a b l e i n f l u e n c e
i n g system. T h i s i s i l l u s t r a t e d i n f i g u r e 5.7,
( F i g . 5.6b).
which shows t h e t r a n s f e r f u n c t i o n s f o r zero and
The l a t t e r e f f e c t i s o f importance,
because i n high-speed wind-tunnel
tests there are
nonzero f l o w v e l o c i t y f o r two d i f f e r e n t tube
considerable differences i n local airspeed over
lengths.
the a i r f o i l , which r e s u l t s i n a c o n s i d e r a b l e d i f -
f a c t o r y agreement between theory and experiment
ference i n t h e mean steady p r e s s u r e i n the v a r i o u s
can be achieved f o r a wide range o f Mach numbers
tubes. For instance,
and frequencies when the constant C appearing i n
i n an atmospheric wind t u n n e l ,
a change i n l o c a l Mach number between 0.5 and 1.0
I t was a l s o experienced t h a t a v e r y s a t i s -
formula (5.6)
r e s u l t s i n p r e s s u r e changes from 0.84 t o 0.53 ata.
i s taken equal t o 0 . 9 .
In a d d i t i o n , t h e r e s u i t s o f t h e experiments showed t h a t t h e chordwise l o c a t i o n o f the p r e s s u r e o r i f i c e s on a wind-tunnel model does n o t a f f e c t ' the t r a n s f e r f u n c t i o n , w h i l e a d d i t i o n a l p a i r s o f p r e s s u r e tubes and transducers mounted i n t h e s i d e wal1 o f t h e wind tunnel (underneath a r e l a t i v e l y t h i c k boundary l a y e r ) a l s o g i v e the same r e s u l t s . T h i s j u s t i f i e s the c o n c l u s i o n t h a t t h e boundary l a y e r does n o t have a s i g n i f i c a n t e f f e c t ,
a t least
as long as the f l o w remains attached.
5.3.3 V e r i f i c a t o n i n a j o i n t ONERA-NLR investigation @INFLUENCE OF MEAN TEMPERATURE R i
I
0.7s
i
io00
v,* 100 "" * 1.4 9,
Fig. 5.6
i
1
mm "'m
-3
@INFLUENCE OF MEAN PRESSURE
0.75 L I 1000 v " = 100 R
I
nli or0
T,.
The measuring technique i n which use i s made mm
o f p r e s s u r e tubes has been v e r i f i e d i n a program
m"'
-3
o f unsteady p r e s s u r e measurements on an o s c i l l a t i n g
1.1
IS'C
model o f a swept wing, j o i n t l y performed by ONERA
C a l c u l a t e d t r a n s f e r f u n c t i o n s of a tube-transducer system i n s t i l l air, showing t h e i n f l u e n c e of mean tempera t u r e and mean steady pressure.
( O f f i c e N a t i o n a l d'Etudes e t des Recherches Aéros p a t i a l e s ) and NLR. During these t e s t s ,
two d i f f e r -
e n t techniques t o measure unsteady pressures were
-34-
m
u
0.2
FREPUEHCY
IOU 150
TUBE LENGTH U.?-
Fig. 5.7
Fig. 5.8
Comparisons between unsteady pressure distributions measured with in Situ transducers and via tubes (hl,= 0.80) on a model of a swept wing performing oscillations in pitch (Ref. 39) (Note difference in tube lengths).
Fig. 5.9
Comparison between overall moments directly measured and moments from integrated pressures (Ref. 39).
Calculated and measured transfer functions of tube-transducer systems at the same static pressure ps with and without cross wind (M = 0.8).
used s i m u l t a n e o u s l y , namely i n s i t u transducers (ONERA) and the system o f p r e s s u r e tubes and scanning valves
(NLR). The t e s t s were performed i n
t h e High-speed Tunnel (HST) o f NLR; b o t h p a r t i e s used t h e i r own d a t a - r e g i s t r a t i o n system and datar e d u c t i o n procedures.
I n t h e d a t a r e d u c t i o n o f NLR,
the e f f e c t o f t h e main f l o w on t h e t r a n s f e r funct i o n s o f the tube system was determined t h e o r e t i c a l l y . (For d e t a i l s o f t h i s investigation, performed under the auspices o f AGARD,
reference i s
made to r e f e r e n c e 39.) The f i n a l
r e s u l t s o f b o t h measuring methods agree
wel1 f o r a l l t e s t c o n d i t i o n s and f o r w i d e l y d i f f e r e n t tube lengths (see examples i n f i g u r e 5.8),
5.4
PRACTICAL APPLICATION IN WIND-TUNNEL TESTS
5.4.1
Choice and c a l i b r a t i o n o f tube-transducer
which j u s t i f i e s t h e c o n c l u s i o n t h a t e i t h e r techn i q u e g i v e s s a t i s f a c t o r y r e s u l t s i f a p p l i e d carefui ly.
systems As an a d d i t i o n a l check on t h e p r e s s u r e
measurements w i t h t h e tubes,
t h e o v e r a l l unsteady
To make an optimum c h o i c e f o r the dimensions
aerodynamic moments on t h e wing model as o b t a i n e d
o f t h e tube system f o r g i v e n model s i z e and t e s t
by i n t e g r a t i o n o f the unsteady p r e s s u r e d i s t r i b u -
c o n d i t i o n s (Mach number and frequency), a computer
t i o n were compared w i t h t h e aerodynamic moments
program I s used f o r the c a l c u l a t i o n o f t h e dynamic
o b t a i n e d from decay t e s t s o f the complete model.
response o f tube-transducer
The r e s u l t s , g i v e n i n f i g u r e 5 . 9 , agree s a t i s -
based on equations ( 8 . 4 7 )
f a c t o r i l y , which f u r t h e r increased t h e confidence
systems, which i s
and ( 8 . 5 1 )
of Appendix 8.
A p a r t from t h e obvious demand t h a t the pres-
i n t h e NLR technique f o r unsteady p r e s s u r e mea-
s u r e s i g n a l s a r e n o t a t t e n u a t e d t o o much by the
S U remen t 5
tube, o t h e r c r i t e r i a t h a t a r e a p p l i e d a r e t h a t a t
-35-
the t e s t f r e q u e n c i e s the e f f e c t o f t h e mean steady
t h e o r e t i c a l l y and e x p e r i m e n t a l l y .
p r e s s u r e remains s m a l l , and t h a t these frequencies
has proven t o work s a t i s f a c t o r i l y when the c a l i -
do n o t c o i n c i d e w i t h resonance frequencies o f the
b r a t i o n o f a I i m i t e d number o f tubes i s performed
tube-transducer
d u r i n g t h e wind-tunnel
system. The l a t t e r requirement
In practice i t
experiments. For t h i s pur-
i s imposed, because experience has l e a r n e d
pose, a small number o f m i n i a t u r e t r a n s d u c e r s i s
t h a t n o n i i n e a r i t i e s s t a r t t o o c c u r f i r s t around
mounted i n t h e model, which a r e d i s t r i b u t e d over
t h e resonance peaks. F u r t h e r ,
the a i r f o i l
the t h e o r e t i c a l l y
i n such a way t h a t c o n s i d e r a b l e d i f -
predicted t r a n s f e r functions a r e very useful t o
ferences i n l o c a l Mach number can be expected.
support t h e s t i l l - a i r and "in-wind"
Each o f the m i n i a t u r e transducers i s p o s i t i o n e d
calibrations
of t h e tubes d u r i n g t h e a c t u a l wind-tunnel
close t o t h e e n t r a n c e o f a pressure tube ( r e f e r -
program.
ence tubes).
I n o r d e r t o keep t h e data r e d u c t i o n as simple as possible,
D u r i n g t h e wind-tunnel t e s t s , a
d i r e c t c a l i b r a t i o n o f t h e r e f e r e n c e tubes i s ob-
t h e s t i l i - a i r t r a n s f e r func-
t i o n s a r e made i d e n t i c a l by choosing t h e same
t a i n e d by a comparison o f t h e i r o u t p u t w i t h t h e
l e n g t h and diameter f o r a l l tubes. These t r a n s f e r
pressures measured s i m u l t a n e o u s l y w i t h t h e i r c o r -
f u n c t i o n s a r e checked by performing c a l i b r a t i o n s
responding m i n i a t u r e transducers. By c o l l e c t i n g
w i t h a t e s t set-up as sketched i n f i g u r e 5.10.
t h e d a t a f o r t h e reference tubes and by p l o t t i n g
The o r i f i c e a t t h e e n t r a n c e i s covered by a small
them as a f u n c t i o n o f t h e mean steady p r e s s u r e
cap,
( o r i t s e q u i v a l e n t l o c a l Mach number) t h e c a l i -
i n which a s i n u s o i d a l p r e s s u r e v a r i a t i o n i s
-
generated w i t h v a r i a b l e frequency and Y a r i a b l e
b r a t i o n c u r v e can be made. Some examples o f
mean steady pressure. The o s c i l l a t o r y p r e s s u r e a t
c a l i b r a t i o n s o b t a i n e d i n t h i s way and a p p l i e d i n
t h e entrance i s measured d i r e c t l y w i t h t h e r e f e r -
the d a t a r e d u c t i o n o f t h e present i n v e s t i g a t i o n s
ence transducer i n t h e cap. A t t h e e x i t b o t h t h e
a r e shown i n f i g u r e 5 . i i .
mean steady and t h e o s c i l l a t o r y p r e s s u r e a r e determined w i t h t h e t r a n s d u c e r i n t h e scanning
5.4.2
valve. During a high-speed wind-tunnel
test.
Measuring equipment and data r e d u c t i o n
the For t h e steady and unsteady p r e s s u r e mea;
l o c a l Mach number a t t h e e n t r a n c e o f t h e v a r i o u s tubes w i l l i n general be d i f f e r e n t , which i m p l i e s
surements,a
t h a t the p r e s s u r e t r a n s f e r o f t h e tubes w i l l a l s o
(DYDRA)
semi-automatic d a t a - a c q u i s i t i o n system
i s used, which i s designed e s p e c i a l l y f o r
be d i f f e r e n t . Since, however, f o r each tube t h e
t h i s purpose (Ref. 9 5 ) . The most s a l i e n t opera-
l o c a i mean steady p r e s s u r e (and, thus, t h e l o c a l
t i o n s are:
Mach number) i s measured s i m u l t a n e o u s l y w i t h t h e
-
c o n s t a n t a m p l i t u d e f o r model e x c i t a t i o n ;
unsteady p r e s s u r e , i t i s r e l a t i v e l y easy t o de-
-
termine t h e t r a n s f e r o f each i n d i v i d u a l tube,
-=
accelerometers and o t h e r pick-ups;
In p r i n c i p l e ,
- measurement o f
t h i s t r a n s f e r f u n c t i o n can be determined b o t h
the direct-current
(DC)
values
of these s i g n a l s .
PRESSURE GEHERATOR
R
computation o f the f i r s t F o u r i e r component o f t h e s i g n a l s from t h e p r e s s u r e transducers.
p r o v i d e d t h a t t h e i r comnon t r a n s f e r f u n c t i o n as a f u n c t i o n of Mach number i s known.
g e n e r a t i o n o f a 0.1 Hr to 10 kHz s i g n a l w i t h
The system f e a t u r e s a v e r y f l e x i b l e c o n t r o l f o r
T PRESSURE TRANSDUCER
channel s e l e c t i o n and subscanner o p e r a t i o n , and i n c l u d e s a s e l f - t e s t f a c i l i t y t o a s c e r t a i n system i n t e g r i t y b e f o r e a t e s t run. A b l o c k diagram o f the equipment i s shown i n f i g u r e 5 . 1 2 .
A normal t e s t r u n proceeds as f o l l o w s : a t a g i v e n t e s t c o n d i t i o n (Mach number, mean incidence, o r mean f l a p a n g l e ) ,
F i g . 5.10
t h e model i s f o r c e d i n t o a
s i n u s o i d a l m o t i o n a t t h e d e s i r e d frequency and am-
T e s t set-up for s t i l l - a i r c a l i b r a t i o n s .
-36-
u-o
AMPLITUDE RATIO
u-o
----a-
1.0
',
1.0-
*A---
I.
1.2
1.0
0.8
0.6
0.6
0.4
0.2
o
"
/+----
IOC
0.8
I 100-
I =I20 Hz
U 6 C
PHASE LAG ISO.-
200"
i
F i g . 5.11
Examples o f measured t r a n s f e r f u n c t i o n s i n s t i l l a i r and i n f l o w i n g a i r .
M
WIND TUNNEL DATA FINAL DATA
--Fig. 5.12
B l o c k diagram of measuring equipment.
-37-
+./---%
,
O
Ob
b6Ó’Z
5ti
SiiÓ’Z
05
SZP’Z
55
E59’Z
910’1
09
8tii’Z
6óE ‘ I
59
$81 ‘ 2
ti89 ’ I
OL
106’i
616.1
5L
209’1
58
SZ
LSL ‘ I L55‘Z E8Z’Z
08
58Z’i
L96’0
OE
968’2
56 O6
6ti9’0
SE
LL6.Z
O0 I
EIO‘O IEE’O
-8E-
02 SI OL
S’L O’S S’Z
5’0
589.0
S L ‘O
585’0
SZ’I
bEL ‘O
O
dW13 H l l h llO3älW 900W9 W(3WN
1’9
dll-1äS 1531 ûNV Sl3ûOW 1 3 N N I l l - ûNIM 9
ELECTRO DYNAUIC EXCITER
EL 5iOE WALL
CALIBRATION SECTION IN SITU TRANSDUCERS + REFERENCE TUBESI
U*
MEASURING SECTION
Fig. 6.1
-
Schematic view of test set-up of the NACA 64A006 airfoil with flal>.
ACCELEROMETER E DISPLACEMENT PICK UP
t h e s t i f f n e s s c o u l d be a d j u s t e d i n such a way t h a t
O
the resonance frequency o f the f l a p - s p r i n g combinat i o n c o u l d be "matched" quency.
w i t h the desired t e s t f r e -
I n p r a c t i c e t h i s worked reasonably w e l l ,
\d' - 0
be
i t t h a t i t c o u l d n o t be avoided c o m p l e t e l y t h a t t h e Fig. 6.2
mean f l a p a n g l e v a r i e d somewhat w i t h Mach number.
Test set-up and instrumentation o f the NACA 64A006 airfoil with flap.
Both t h e upper and t h e lower surface o f t h e measuring s e c t i o n o f t h e model were p r o v i d e d w i t h
IO
20
10
,1 i 8:
!
,,
S
,
,
I
I 9 pressure o r i f i c e s and tubes ( F i g s . 6.2 and 6.3).
40 d:
.,
72' 77' 50 55 60 65 70:78/80 BI PO 9 5 %
, :
~
~
1, , : I : , / I I
.-. ,
,
, , , , , I , I ---L , i-'
To determine the t r a n s f e r f u n c t i o n s o f t h e tubes
I
,
,
,
. ,
HINGE AX,/
d u r i n g t h e t e s t s , s i x small p r e s s u r e t r a n s d u c e r s (manufactured by K u l i t e ) were p l a c e d i n s i d e t h e
180 mm
I
model, t o g e t h e r w i t h s i x tubes having t h e same IIEASURING IECTION !BOTH UPPER AND LOWER SURFACE)
geometry as t h e tubes i n t h e measuring s e c t i o n .
1-
I
,
< l . .
, I
, , , ,
I
p G ö & G F IUPPER SURFACE ONLY)
To determine t h e m o t i o n o f the f l a p , two accelerometers ( x , , x,)
and two c a p a c i t i v e d i s -
placement p i c k - u p s were used. The c a p a c i t i v e p i c k 5 7 8
ups served a l s o t o measure t h e mean p o s i t i o n o f the f l a p .
In a d d i t i o n , two accelerometers were
.80 I7 I8 I9
.85
.PO .PS
p l a c e d i n t h e main s u r f a c e , c l o s e t o t h e measuring s e c t i o n , t o d e t e c t unwanted o s c i l l a t i o n s o f t h i s
Fig. 6.3
p a r t o f t h e model. A scheme o f the i n s t r u m e n t a t i o n
Location of pressure orifices on the NACA 64ACQ6 airfoil with flap.
o f t h e model i s shown i n f i g u r e 6.2.
6.2
To o b t a i n a t u r b u l e n t boundary l a y e r w i t h a
N L R 7301 AIRFOIL
f i x e d t r a n s i t i o n p o i n t , a 2 . 5 mm wide s t r i p o f carborundum g r a i n s was p l a c e d a t 10 per cent of
The second model under c o n s i d e r a t i o n has an
the chord on b o t h t h e upper and t h e lower s u r f a c e .
NLR
-39-
7301 a i r f o i l s e c t i o n , which i s designed f o r
s h o c k - f r e e f l o w and has a maximum t h i c k n e s s o f
16.5 p e r c e n t
o f the chord ( f o r geometric data,
see t a b l e 2 ) . The model has a chord o f
\
IN SITU TRANSDUCER5
I8 cm and
ORIFICES PRESSURE TUBES
can perform p i t c h i n g o s c i l l a t i o n s about an a x i s a t 40 % o f the chord. The model, made o f Dural,
spanned h o r i z o n t a l l y t h e t e s t s e c t i o n o f the P i l o t tunnel ( F i g s . 6 . 4 and 6 . 5 ) .
TABLE 2 Coordinates o f
N L R 7301* ES WITH MODEL)
U w e r surface
Lower surface HYDRAULIC ACTUATOR
- ,0004
O
.O033
,0196
.O018
,0124
,0079
.O207
,0369 ,0454
.O299
,051 I
,0337
,0554 ,0590 ,0618
,0370 ,0650
,0499
,0600 ,0748
,0180
.IO00
. I300
,0651
.I649
.o998 . I300
,0697 ,0741
.zo00
. i649 ,1995 .2498
.O781 ,0813 ,0847
,2938 ,3499
.2998
,0869
,3497
,0881
,3999 ,4497 ,4998
,3993 ,4492
,0883 ,0876
,5496 ,5996
,4396
,0860
,5493 ,5993
,0832
,6393 ,6791
,6493 ,6993 .7494 ,7982 .E385 ,8786 ,9194
*
Fig. 6.4
,0792 ,0736 ,0661 ,0573 ,0475 ,0388
,9479
,0297 ,0207 ,0140
,9784 I . O000
,0030
,2499
,7193 ,7537 ,7994 ,8377 ,8785 ,9188 ,9487 ,9781
-.O134 -.O247 -.O340 -.O437
Schematic view of test set-up of the NLR 7301 airfoil. AXIS OF ROTATION
-.O525
-.O598 - . 0643 -.O685 -.O718 -.O750 -.O767 -.O770
IFICES
I N SITU TRANSDUCERS ,UPPER SURFACE ONLY,
ORIFICES PRESSURE TUBE5
-.O760 -.O733 -.O604 -.O613 -.O526 -.O447 -.O361 -.O273 -.O105
CCELEROMETERS
1MDVING WITH MODEL,
-.O104 -.O039 .0013
DIMENSIOHS IN m m l
.0043 ,0047 ,0037
.O0 74 Fig. 6.5
f o r a more d e t a i l e d d e s c r i p t i o n , see r e f e r e n c e 78.
-40-
Test set-up and instrumentation of the M R 7301 airfoil.
The mounting o f t h e model d i f f e r e d from the
iiL
system mentioned i n the preceding s e c t i o n . T h i s
s.2smm-
time, use was made o f a h y d r a u l i c a c t u a t o r t h a t
-17.2Sam
DETAILS OF SLOTS
was a b l e t o supply s i m u l t a n e o u s l y a l a r g e timedependent f o r c e as w e l l as a c o n s i d e r a b l e steady f o r c e . T h i s steady f o r c e was necessary, s i n c e t h e model c a r r i e d ,
i n the m a j o r i t y o f t h e t e s t c o n d i -
t i o n s ( l i k e t h e shock-free d e s i g n c o n d i t i o n ) , a c o n s i d e r a b l e mean steady a i r l o a d , which c o u l d n o t
be w i t h s t o o d by a s p r i n g system t h a t a t t h e same t i m e had t o produce t h e r e l a t i v e l y low resonance frequencies r e q u i r e d f o r t h e model e x c i t a t i o n w i t h
Fig. 6.7
Transonic t e s t s e c t i o n of the Pilottunnel.
the electrodynamic e x c i t e r s . An a d d i t i o n a l advantage o f h y d r a u l i c e x c i t a t i o n was t h a t t h e t e s t f r e -
scanning valves o u t s i d e t h e wind tunnel v i a pres-
quencies c o u l d be v a r i e d f r e e l y ,
s ur e tubes.
because i t was no
I 3 m i n i a t u r e transducers
In a d d i t i o n ,
were b u i l t i n . T h i s number,
longer necessary t o match t h e d e s i r e d t e s t frequen-
l a r g e r than necessary
cy w i t h a resonance frequency o f a model-spring
f o r t h e dynamic c a l i b r a t i o n o f t h e p r e s s u r e tubes,
system. Moreover, t h e mean i n c i d e n c e o f the model
c r e a t e d t h e p o s s i b i l i t y t o r e c o r d the a c t u a l time
c o u l d be v a r i e d e a s i l y by changing t h e mean p o s i -
h i s t o r i e s ( c o n t a i n i n g the h i g h e r harmonics) o f t h e
t i o n o f the hydraulic actuator.
chordwise pressure d i s t r i b u t i o n along t h e upper
To keep t h e mounting as simple as p o s s i b l e ,
surface.
t h e model was e x c i t e d a t one s i d e , w h i l e t h e oppo-
F u r t h e r , t o determine t h e motion o f the model, 6
s i t e s i d e was supported by a b e a r i n g j u s t o u t s i d e
accelerometers were used, l o c a t e d i n t h r e e spanwise
the tunnel w a l l ( F i g . 6 . 5 ) .
s t a t i o n s (see f i g u r e 6 . 5 ) .
To a v o i d a c o m p l i c a t e d
s e a l i n g between model and window, t h e window c l o s est t o
The main p a r t o f the wind-tunnel
the a c t u a t o r was a t t a c h e d t o t h e model and
f o l l o w e d i t s motion.
t e s t s was
performed w i t h a t r a n s i t i o n s t r i p o f carborundum
In a d d i t i o n , a c l e a r view on
g r a i n s , l o c a t e d a t both the upper and t h e lower
the model f o r o p t i c a l f l o w s t u d i e s was provided.
s u r f a c e a t 30 per cent
o f the chord.
For f u r t h e r d e t a i l s o f t h e h y d r a u l i c e x c i t a t i o n system, r e f e r e n c e i s made t o Poestkoke (Ref. 96). 6.3
Both t h e upper and t h e lower surface o f t h e
W I N D TUNNEL
NLR 7301 model were p r o v i d e d w i t h 20 p r e s s u r e o r i f i c e s ( F i g s , 6.5 and 6.6).
The t e s t s have been performed i n t h e P i l o t -
connected w i t h two
tunnel o f NLR, which i s an atmospheric, closedAXIS OF ROTATION
c i r c u i t wind tunnel f o r Mach numbers up t o about 1. The dimensions of the t e s t s e c t i o n a r e
0.55 x 0.42 m ( F i g . 6.7). The upper and t h e lower s u r f a c e o f the t e s t s e c t i o n a r e f i t t e d w i t h long i t u d i n a l s l o t t e d w a l l s . The open-area r a t i o o f t h e w a l i s i s O.l,and PRESSURE ORIFICES TUBING SYSTEM (BOTH UPPER A N 0 LOWER SURFACE1
IN SITU TRANSDUCERS (UPPER SURFACE ONLY)
t h e r e i s no connection
between t h e plenum chambers o f f l o o r and bottom. Further d e t a i l s o f the P i l o t t u n n e l are given i n
.6S .20
.li
is 16
r e f e r e n c e 97.
.28
.70
.
i7 8 9 IO
Fig. 6.6
.
I8 I9
zo
.8S .90
6.4
8 .A4
OPTICAL FLOW STUDIES
I The p e r i o d i c a l shock-wave motions on b o t h
L o c a t i o n of pressure O r i f i c e s of the NLR 7301 a i r f o i l .
o s c i l l a t i n g models have been determined i n two
-41-
d i f f e r e n t ways. F i r s t , shadowgraph p i c t u r e s were
l a t i n g model w i t h i t s instantaneous shack p a t t e r n
taken by means o f a s t r o b o s c o p i c l i g h t source
c o u l d be photographed i n every p o s i t i o n d e s i r e d .
t r i g g e r e d by an e l e c t r i c a l s i g n a l from a d i s p l a c e -
The second way o f r e g i s t r a t i o n was o b t a i n e d by
ment p i c k - u p i n t h e model. 8y means o f an a d j u s t -
tak.ing high-speed f i l m s o f t h e time h i s t o r i e s o f
a b l e phase s h i f t
the f l o w ( w i t h a f i l m speed up t o 4000 frames per
i n t h e e l e c t r i c c i r c u i t between
t h e accelerometer and t h e l i g h t source, t h e o s c i l -
second).
7 TEST PROGRAM 7.1
NACA 64A006 AIRFOIL WITH FLAP
TABLE 3 Test program f o r the NACA 64A006 a i r f o i l
The t e s t s on t h e NACA 64A006 a i r f o i l w i t h
w i t h flap
o s c i l l a t i n g f l a p covered t h e Mach number range from 0 . 5 t o 1.0 and t h e frequencies were 30 and 120 HI.
For a few cases, t e s t s were performed a l s o
a t IO, 20 and 90 Hz. The maximum v a l u e o f t h e r e duced frequency achieved d u r i n g t h e t e s t s v a r i e d from 0.39 a t M- = 0.5 t o 0.21 a t H- = I.O. The r e s u l t s f o r t h e l i m i t i n g case o f zero frequency have been d e r i v e d from an e x t e n s i v e s e r i e s o f steady t e s t s . As i t was t h e i n t e n t i o n t o study symmetrical as w e l l as nonsymmetrical f l o w c o n d i t i o n s , t h e model was t e s t e d a t incidences o f O, grees, and mean f l a p angles o f -3,
-2 and - 4 de-
O and
+3
de-
grees, respect i v e l y. Due t o the mechanical p r o p e r t i e s o f t h e mod-
el,
the maximum frequency o f t h e f l a p was l i m i t e d
t o 120 Hz and the a m p l i t u d e o f o s c i l l a t i o n t o about I degree. For mean incidences o f t h e model
perform p i t c h i n g o s c i l l a t i o n s around an a x i s a t
o f -2 and - 4 degrees, t h e maximum Mach number was
40 p e r c e n t
reduced t o about 0.94,
due t o l a c k o f power o f t h e
range between 0.5 and 0.8.
wind t u n n e l . A survey o f t h e t e s t program i s g i v e n
The maximum t e s t f r e -
quency was 80 Hz. which corresponds t o a reduced
i n t a b l e 3. I n cnapters
o f t h e chord covered t h e Mach number
frequency t h a t v a r i e d from 0.26 a t M_ = 0.5 t o
8 and 9 a s u b s t a n t i a l p a r t o f
0.17 a t Mm = 0.8.
t h e t e s t r e s u l t s w i l l be used t o analyze t h e fea-
The combinations o f mean i n c i d e n c e and free-
t u r e s encountered i n high-subsonic and t r a n s o n i c
stream Mach number f o r which t e s t s were performed
f l o w s . For t h e d e t a i l e d t e s t data,
a collec-
were s e l e c t e d i n such a way t h a t the C -H diagram
t i o n o f t a b l e s w i t h steady and unsteady p r e s s u r e
o f t h e a i r f o i l ( F i g . 3.11) was covered reasonably
c o e f f i c i e n t s , t h e reader i s r e f e r r e d t o r e f e r e n c e s
w e l l , so t h a t a good o v e r a l l p i c t u r e o f the un-
98
-
i.e.
a
100.
steady aerodynamic c h a r a c t e r i s t i c s c o u l d be obtained ( t a b l e &a). In addition, four characterist i c c o n d i t i o n s o f Mach number and mean i n c i d e n c e
7.2
NLR 7301 AIRFOIL
were selected,
f o r which e x t r a t e s t runs were con-
ducted t o e x p l o r e t h e i n f l u e n c e o f a m p l i t u d e and The t e s t s on t h e NLR 7301 a i r f o i l t h a t c o u l d
frequency o f o s c i l l a t i o n ( t a b l e 4 b ) .
-42-
E
w
m
ij.
-.
a:
.<
"
PART 111 ANALYSIS OF RESULTS
-45-
8
8.1
THE INTERACTION BETWEEN THE STEADY AND UNSTEADY FLOW FIELD
INTRODUCTORY REMARKS
The comparisons w i t h t h i n - a i r f o i l t h e o r y appear t o be v e r y u s e f u l t o i d e n t i f y t h e t y p i c a l e f f e c t s
A s argued i n s e c t i o n 4 . 1 ,
a s s o c i a t e d w i t h the high-subsonic and t r a n s o n i c
one o f the main
aspects t o be considered i n an a n a l y s i s o f t h e ex-
f l o w regime.
p e r i m e n t a l r e s u l t s should be 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 .
This
i m p l i e s t h a t emphasis should be p l a c e d upon t h e
8.2
e f f e c t s o f a i r f o i l shape, Mach number, mean i n -
THE INFLUENCE OF MACH
NUMBER ON
THE A I R -
LOADS OF THE NACA 64A006 AIRFOIL WITH
cidence,and mean f l a p a n g l e on t h e unsteady a i r -
FLAP
loads and upon t h e development o f these e f f e c t s w i t h frequency.
8.2.1
I n o r d e r t o g e t a c l e a r view o f t h e v a r i o u s
Steady p r e s s u r e d i s t r i b u t i o n s
e f f e c t s as observed i n the p r e s e n t t e s t s , t h e The chordwise d i s t r i b u t i o n o f the steady
f o l l o w i n g p r e s e n t a t i o n i s chosen. F i r s t t h e r e s u l t s o f t h e NACA 64A006 a i r f o i l w i t h f l a p w i l l
pressures a l o n g t h e a i r f o i l w i t h u n d e f l e c t e d f l a p
be described. These r e s u l t s have t h e advantage
i s g i v e n i n f i g u r e 8.1 f o r several values o f the
t h a t the unsteady p r e s s u r e d i s t r i b u t i o n s e x h i b i t
free-stream Mach number.. The c r i t i c a l Mach number
a l a r g e pressure peak a t t h e l e a d i n g edge o f t h e
i s between 0.825 and 0.85.
f l a p which,
number, a weak shock wave appears a t about 4 5 per
f o r a q u a l i t a t i v e d i s c u s s i o n , can be
A t the l a t t e r Mach
i n t e r p r e t e d as a p o i n t d i s t u r b a n c e . T h i s behaviour
cent o f t h e chord. When t h e Mach number i s i n -
makes the r e s u l t s w e l l - s u i t e d f o r p h y s i c a l i n t e r -
creased f u r t h e r ,
p r e t a t i o n s . F u r t h e r , the o p t i c a l f l o w s t u d i e s on
and i s d i s p l a c e d i n downstream d i r e c t i o n . A t about
this airfoil
M_ = 0.92 t h e shock reaches t h e h i n g e l i n e , and
revealed some i n t e r e s t i n g p e r i o d i c a l
t h i s shock grows i n s t r e n g t h
shock-wave motions, which w i l l be analyzed i n a
s l i g h t l y beyond t h i s Mach number t h e f l o w down-
separate chapter (chapter 9 ) .
stream o f t h e shock wave separates from t h e a i r -
I n c h a p t e r 10, an a n a l y s i s w i l l be g i v e n o f
foil.
t h e r e s u l t s f o r t h e s u p e r c r i t i c a l NLR 7301 a i r f o i l .
%o
Here a t t e n t i o n w i l l be p a i d t o r e s u l t s f o r o s c i l l a t i o n s around t h e shock-free d e s i g n c o n d i t i o n as
HACA 6dA006 AIRFOIL
0.4
w e l l a s t o r e s u l t s f o r some t y p i c a l o f f - d e s i g n conditions.
For b o t h the NACA 64A006 and t h e
NLR 7301 a i r f o i l ,
0.5
t h e e f f e c t s o f Mach number, mean
incidence, mean f l a p a n g l e , and reduced frequency w i l l be considered.
0.6
To emphasize t h e dynamic e f f e c t s i n t h e unsteady flow,
i n many cases t h e c o r r e s p o n d i n g
"quasi -steady" ered f i r s t ,
p r e s s u r e d i s t r i b u t i o n s a r e cons i d -
i.e.
t h e "unsteady"
0.7
pressure d i s t r i b u -
t i o n s t h a t occur when t h e o s c i l l a t i o n s a r e i n -
" ~ L*
f i n i t e l y slow. These q u a s i - s t e a d y p r e s s u r e d i s t r i b u t i o n s can
be d e r i v e d e a s i l y from a sequence
o f steady p r e s s u r e measurements. F u r t h e r , t h e t e s t
o
r e s u l t s w i l l be compared w i t h t h e p r e d i c t i o n s o f "thin-airfoil
theory",
i n which a n i n f i n i t e l y
Fig. 8.1
t h i n wing i n a u n i f o r m main f l o w i s considered.
-47-
HINGE AXIS v
0.71
,z I.o
Chordwise pressure distribution 8s B function Of free-stream Mach number.
+
1
O
?,I
3anllldWv
4
hü03Hl l i O d ~ l V - N l H l
3JVJäfl5 ä 3 M O l 33VdäflS ä 3 d d f l i Q=On
1IOdälV 900VP9 VJVN
1
1
P
1
J
m
f
8 . 2 . 2 Unsteady pressure d i s t r i b u t i o n s
The unsteady p r e s s u r e d i s t r i b u t i o n s f o r a f l a p o s c i l i a t i o n w i t h a frequency o f 120 Hr and an -0.25
a m p l i t u d e o f I degree a r e c o l l e c t e d i n f i g u r e 8 . 2 . T h i s f i g u r e c o n t a i n s the mean steady p r e s s u r e d i s t r i b u t i o n s and t h e unsteady p r e s s u r e d i s t r i b u -
O
t i o n s , t h e l a t t e r i n terms o f t h e r e a l and the imaginary p a r t . The
mean steady p r e s s u r e d i s -
a: STEADY
t r i b u t i o n s correspond reasonably w e l l w i t h t h e steady p r e s s u r e d i s t r i b u t i o n s o f f i g u r e 8.1,
obPHASE ANGLE Y
t a i n e d on t h e n o n - o s c i l l a t i n g model. The unsteady p r e s s u r e s a t t h e iower surface a r e d i s p l a y e d w i t h a minus s i g n , t o f a c i l i t a t e the comparison between t h e upper and t h e lower s u r f a c e . Since we a r e d e a l i n g w i t h z e r o incidence and zero mean f l a p angie,
the mean f l o w i s symmetric. T h e r e f o r e , the
unsteady pressures on t h e upper and t h e iower C:UHSTEADI;i:i8
s u r f a c e a r e almost equal i n magnitude, b u t d i f f e r
Fig. 8.3
i 8 0 degrees i n phase a n g l e . A t t h e lowest Mach number, a good agreement
e x i s t s between the measured unsteady d a t a and t h e results of t h i n - a i r f o i l
Hz;i=o.392
Steady, quasi-steady and unsteady Pressure distributions in low-subsonic flow. UACA SPA006
um
theory. T h i s agreement
0.80
i
becomes worse w i t h i n c r e a s i n g Mach number and, -0.50
as soon as t h e f l o w becomes s u p e r c r i t i c a l , the agreement i s good o n l y i n the subsonic p a r t o f t h e f l o w downstream o f the shock wave. Above a f r e e - s t r e a m Mach number o f about 0.875,
the
measured unsteady pressures i n f r o n t o f t h e shock wave decrease s t r o n g l y , and a t M-
= 0 . 9 4 they a r e
zero ahead o f the h i n g e a x i s . The i n f l u e n c e o f the Mach number on the unsteady p r e s s u r e d i s t r i b u t i o n s can be c o n s i d e r e d i n a somewhat d i f f e r e n t way w i t h the h e l p o f f i g -
u r e s 8 . 3 - 8 . 8 , which c o n t a i n the measured and t h e o r e t i c a l unsteady pressure d i s t r i b u t i o n s i n terms o f magnitude and phase a n g l e , t o g e t h e r w i t h the corresponding steady and q u a s i - s t e a d y p r e s s u r e distributions.
0
The f i r s t f i g u r e i n t h i s s e r i e s ( F i g . 8.3) r e p r e s e n t s a t y p i c a l low-speed example. F i g u r e
8.3.3
-1006
shows t h e steady p r e s s u r e d i s t r i b u t i o n s on
t h e upper s u r f a c e o f the a i r f o i l f o r t h r e e f l a p angles, namely - 1 . 5 , tively.
O and 1 . 5 degrees,
respec-
From. ihese steady p r e s s u r e d i s t r i b u t i o n s ,
Fig. 8.4
the quasi-steady pressure
-49-
Steady, quasi-Steady and unsteady pressure distributions in subsonic f l o w .
R e s u l t s f o r t h e same c o n f i g u r a t i o n i n h i g h subsonic f l o w a r e shown i n f i g u r e 8.4.
50 ;er i 5
determined and t h e r e s u l t i n g chordwise d i s -
cent
Around the
chord p o i n t , a bulge s t a r t s t o o c c u r
i n t h e magnitude o f the measured d i s t r i b u t i o n o f
t r i b u t i o n i s shown i n f i g u r e 8.3b.
b o t h t h e q u a s i - s t e a d y and unsteady pressures,
A s mentioned b e f o r e , t h i s q u a s i - s t e a d y p r e s s u r e
which i s n o t p r e d i c t e d by t h i n - a i r f o i l t h e o r y .
d i s t r i b u t i o n can be i n t e r p r e t e d as t h e unsteady
The phase c u r v e a l s o i s p r e d i c t e d less s a t i s f a c -
p r e s s u r e d i s t r i b u t i o n f o r Zero frequency.
t o r i l y by t h e o r y than i n t h e preceding low-speed
f i g u r e 8.3c,
In
t h e unsteady p r e s s u r e d i s t r i b u t i o n
case. A c h a r a c t e r i s t i c f e a t u r e i s t h a t t h e c a l c u -
f o r a frequency o f 120 HZ and an a m p l i t u d e o f
l a t e d phase l a g on t h e f r o n t p a r t o f the a i r f o i l
about 1 degree i s presented. The magnitude o f t h e
i s c o n s i s t e n t l y s m a l l e r than t h e measured value.
unsteady d i s t r i b u t i o n v e r y much resembles t h e
These d i s c r e p a n c i e s between t h i n - a i r f o i l t h e o r y
quasi-steady d i s t r i b u t i o n , because b o t h show t h e
and experiment become more pronounced i n t h e ex-
c h a r a c t e r i s t i c peaks a t t h e l e a d i n g edge o f t h e
ample o f f i g u r e 8.5,
a i r f o i l and a t t h e h i n g e a x i s a t 75 p e r c e n t o f
number i s increased f u r t h e r and t h e f l o w i s almost
the chord.
critical
For t h e unsteady example, a l s o t h e r e -
sult o f thin-airfoil speed,
t h e o r y i s given.
where t h e free-stream Mach
.
A t t h i s low
t h e agreement w i t h t h e experimental magni-
Some c h a r a c t e r i s t i c r e s u l t s f o r a t r a n s o n i c
tude and phase d i s t r i b u t i o n i s s a t i s f a c t o r y .
flow w i t h normal shock waves a r e g i v e n i n f i g u r e s
-
8.6
b:QUASi - STEADY: i
8.8.
I t i s c l e a r , from t h e p r e s s u r e d i s t r i b u -
O
i
MAGNITUDE /ACp/
6 PHASE ANGLE p
PHASE ANGLE Y
IW"
'O0'
o
t z
2
o
O
c:
Fig. 8.5
UNSTEADY; 1-120 Hi: 1-0.248
C:UNSTEADY. I =12OHz; 1.0.240
Fig. 8.6
Steady, quasi-steady and unsteady pressure distributions in highsubsonic flow.
-50-
Steady, quasi-steady and unsteady presaure distilhations in slightly supercritical flaw.
t i o n s i n :hese
f i g u r e s , t h a t a change i n f i a p
h i n g e a x i s and pressure p e r t u r b a t i o n s generared by
a n g l e i s f o i l o w e d by a s h i f t i n shock p o s i t i o n ,
t h e f l a p a r e f e l t only on t h e f l a p i t s e l f . A t s t i l l
which leads t o a h i g h peak i n t h e magnitude o f t h e
h i g h e r Mach numbers, t h e f l o w p a t t e r n becomes much
q u a s i - s t e a d y and unsteady pressures. T h i s peak
more complicated. Then a lambda shock and a
g i v e s a s i g n i f i c a n t c o n t r i b u t i o n t o the o v e r a l l
Prandtl-Meyer expansion a l t e r n a t e l y occur on
unsteady l i f t and moment, which o f course cannot
e i t h e r s i d e o f t h e f l a p (see f i g u r e 3 . 6 ) , w h i l e
be found w i t h t h i n - a i r f o i l
t h e o r y . Another observa-
i n a d d i t i o n t h e f l o w becomes separated. An exam-
t i o n i s that the pressure perturbations i n f r o n t of
p l e o f t h e pressure d i s t r i b u t i o n s measured i n
t h e shock wave a r e s m a l l e r t h a n those p r e d i c t e d by
such a complicated flow p a t t e r n a t near-sonic
theory,and t h a t t h e measured phase curves show a
free-stream c o n d i t i o n s i s shown i n f i g u r e 8.9.
sharp change i n g r a d i e n t i n t h e r e g i o n o f t h e shock wave. T h i s e f f e c t w i l l be discussed i n more d e t a i l i n s e c t i o n 8.3.
When t h e free-stream Mach
8.2.3
Unsteady aerodynamic c o e f f i c i e n t s
number i s increased f u r t h e r , t h e shock s t r e n g t h Examples o f the development w i t h Mach num-
and t h e s i z e o f the supersonic r e g i o n become so l a r g e t h a t p e r t u r b a t i o n s generated by the f l a p a r e
b e r o f t h e o v e r a l l f o r c e and nwment c o e f f i c i e n t s
no l o n g e r f e l t ahead o f the shock wave (compare
f o r quasi-steady and unsteadv f l o w a r e shown i n
f i g u r e s 8.6
-
8.8).
A t about M-
= 0.92,
t h e shock reaches t h e
V A C A 64A006 Urn-0.875
CP
4 1
-0.50
2
-0.25
z
O
o
1.0 b:WASl-iTEADY
i
1.0
b : w a s i -STEADY. i = o
k;0
PHASE ANGLE
Y
% O
x 1.0
W C:UNIIEIDI; I;12PHz;
-C:UNSTEADY: I i 1 2 0 H x : k
L;O.Zld
Fig. 8.8
Fig. 8.7 Steady, quasi-steady and unsteady pressure distributions in transonic flow.
-51-
i
0.226
Steady, quasi-steady and unsteady pressure distributions in transonic flow
( F i g . 8 . 1 1 ) o c c u r s a t a somewhat h i g h e r Nach number than t h e experimental peak v a l u e s . T h i s i s n o t J cp
C.
s u r p r i s i n g , because a c l o s e r l o o k a t f i g u r e 8.2
4
l e a r n s t h a t the measured p r e s s u r e d i s t r i b u t i o n f o r Mrn = 0.825 Mrn =
f i t s b e t t e r w i t h the c a l c u l a t i o n f o r
0.875 than the c a l c u l a t i o n f o r il-= 0.825
does. E v i d e n t l y , a t l e a s t f o r s u b c r i t i c a l f l o w s , a b e t t e r t h e o r e t i c a l p r e d i c t i o n i s o b t a i n e d i f the c a l c u l a t i o n s a r e performed f o r a Mach number averaged over t h e a i r f o i l s u r f a c e i n s t e a d o f f o r the b:OUASl.STEADY;
f r e e - s t r e a m Mach number.
k -0
An u n c e r t a i n t y i s t h e c o n t r i b u t i o n o f PHASE ANGLE
inter-
ference from t h e tunnel w a l l s i n t h e measured r e -
m
s u l t s . U n f o r t u n a t e l y , a r e l i a b l e method t o det e r m i n e t h e c o r r e c t i o n s for the presence o f s i o t t e d tunnel w a l l s i n u n s t e a d y - f l o w t e s t s i s s t i l l l a c k i n g . C o r r e c t i o n s can o n l y be i n d i c a t e d f o r zero frequency ( F i g . 6.101,
f o r which case i t ap-
pears t h a t t h e e f f e c t o f the tunnel w a l l s i s responsible f o r a t least a part of the differences
C: UNSTEADY; i -110 Hz; k =0.211
Fig. 8.9
Steady, quasi-steady a i d unsteady pressure d i s t r i b u t i o n s i n n e a r - s o n i c fIOW.
'"-
f i g u r e s 8.10 and 8 . l i . As m i g h t be expected from
we--
the p r e s s u r e d i s t r i b u t i o n s i n t h e t r a n s o n i c r e -
.
I I
1
NORMAL FORCE
I
I
gime, t h e c o e f f i c i e n t s v a r y s t r o n g l y w i t h Mach number. The sharp decrease of t h e n o r m a l - f o r c e c o e f f i c i e n t , kc, and t h e moment c o e f f i c i e n t , mc, o c c u r r i n g a f t e r the appearance o f a shock wave
"
0 8
(M_:O.85),
can be c o r r e l a t e d w i t h t h e drop ob-
0.5
served i n the magnitude o f t h e p r e s s u r e p e r t u r b a -
C
I
1.0
M'
SHOCK REACHES HINGE AXIS
t i o n s ahead o f t h e shock wave. The hinge-moment coefficient n
0.75
0.05
/./' -----
f a l l s o f f a f t e r t h e shock wave has
passed the h i n g e a x i s , which leads t o t h e more c o m p l i c a t e d f l o w p a t t e r n on t h e f l a p and shockinduced s e p a r a t i o n . The t h e o r e t i c a l curves i n f i g u r e s 8.10 and
8.11 a g a i n a r e based on t h i n - a i r f o i l t h e o r y . When t h e t h e o r e t i c a l curves a r e compared w i t h t h e exp e r i m e n t a l data,
i t appears t h a t t h e t r e n d w i t h
Mach number i s p r e d i c t e d reasonably w e l l ,
o
espeSHOCK REACHES HINGE AXIS
c i a l l y f o r t h e unsteady-flow case ( F i g . 8 . 1 1 ) .
A d i f f e r e n c e i s t h a t t h e maximum v a l u e o f t h e
Fig. 8.10
magnitude o f t h e c a l c u l a t e d moment c o e f f i c i e n t
-52-
Quasi-steady aerodynamic c o e f f i c i e n t s 8s a f u n c t i o n of Mach number.
NACA t i A oot AIRFOIL -0EXPERIMENT
rounding non-uniform a i r f l a w
-THIN-AIRFOIL THEORY
Under these circumstances, a p i c t u r e of t h e a c o u s t i c wave p a t t e r n can be o b t a i n e d w i t h the
NORMAL FORCE
kc!
Weil-known c o n s t r u c t i o n o f Huygens. W i t h t h i s
1.3
method, t h e t i m e h i s t o r y o f a wave f r o n t generated a t a c e r t a i n p o i n t can be c o n s r r u c t e d r a t h e r e a s i l y , because t h e l o c a l propagation speed o f t h e
.i
wave f r o n t
i s equal t o the v e c t o r sum o f the l o c a i
speed o f sound taken normal t o t h e wave f r o n t and o .i
t h e l o c a l f l o w v e l o c i t y ( F i g . 8.12).
1.0
E a r l i e r , Spee
(Ref. 20) s u c c e s s f u l i y a p p l i e d the method t o i n v e s t i g a t e some aspects o f shock-free f l o w around airfoils.
In t h e f o l l o w i n g examples f o r the v e l o c i t y d i s t r i b u t i o n c l o s e t o t h e a i r f o i l s u r f a c e , use i s made of the v e l o c i t i e s measured along t h e a i r f o i l contour, w h i l e t h e v e l o c i t y d i s t r i b u t i o n i n t h e o u t e r f i e l d i s estimated. For s i m p l i c i t y , r e f l e c t .i
.IS
1.0
ed waves from t h e tunnel w a l i s have been neglected. F i g u r e 8.13a
shows t h e p o s i t i o n o f t h e wave
f r o n t s a f t e r equal time i n t e r v a l s A t ,
plotted for
two d i f f e r e n t Mach numbers. The p a r t of the f i g u r e above the a i r f o i l d e p i c t s t h e t i m e h i s t o r i e s o f t h e wave f r o n t s i n the a c t u a l f l o w f i e l d .
Eeiow
t h e a i r f o i l , t h e same wave f r o n t s a r e shown, b u t o .i
.;i
7% ' SHOCK '
M.
Fig. 8.11
now f o r a steady u n i f o r m
I
flow f i e l d , i n which t h e
l o c a l Mach number everywhere i s equal t o t h e
1.0
REACHES n i w E AXIS
free-stream Mach number. The corresponding t r a v e l -
Unsteady aerodynamic coefficients as a function of Mach number (f=120 Hz).
WAVE FRONT AT TIME to
WAVE FRONT AT TIME to+ i t
\
\
between the c a l c u l a t e d and t h e measured r e s u l t s ,
u1
i n p a r t i c u l a r f o r the normal-force c o e f f i c i e n t s . For a more d e t a i l e d account o f t h e e f f e c t o f w a i l
interference,
-+-
r e f e r e n c e i s made t o p a r t i V o f
\
this thesis.
8.3
\
\
GRAPHICAL EXPERIMENT FLOW OIRECTICN
the steady and unsteady f l o w f i e l d s f o r t h e a i r -
\
\
\
The mechanism of t h e i n t e r a c t i o n between
\
f o i l w i t h o s c i l l a t i n g f l a p can be e x p l a i n e d v e r y w e l l by means o f a g r a p h i c a l experiment. For t h i s purpose, t h e a i r f o i l
i s schematized t o an a i r f o i l
w i t h a pulsating pressure disturbance located a t the h i n g e a x i s . From t h i s d i s t u r b a n c e , a c o u s t i c
Fig. 8 . U
waves a r e g o i n g o u t and propagate i n t o t h e sur-
-53-
Upstream propagation of a wave-front element in a non-uniform flow.
---,u
@
= 0.80
upper .. .p a r t s bend around the tol) o f the shock and
NON.UNIFORM FLOW FIELD UNIFORM FLOW FIELO Y,
p e n e t r a t e the supersonic r e g i o n . T h i s i s r e f l e c t -
= 0.875
@
ed i n t h e t i m e l a g c u r v e by a sudden steepening. Since t h e energy c o n t e n t o f t h e wave f r o n t s penet r a t i n g the supersonic r e g i o n has decreased, due t o t h e longer d i s t a n c e they have t r a v e l l e d around the t o p o f the shock, only small pressures can be b u i l t up i n f r o n t o f t h e shock wave. These f i n d i n g s c o r r e l a t e v e r y w e l l w i t h t h e e f f e c t s observed i n t h e wind-tunnel
8 . 7 ~ i.e. ~
r e s u l t s presented i n f i g u r e
r e l a t i v e l y low pressures i n f r o n t o f
t h e shock, a sudden steepening of t h e phase curve, and a p r e s s u r e peak a t t h e shock p o s i t i o n . The main c o n t r i b u t i o n t o t h e l a t t e r peak should, o f course, be a t t r i b u t e d t o t h e o s c i l l a t o r y d i s p l a c e TIME LAG
Fig. 8.13
114
ment o f t h e shock waves, which i s n o t included i n
TIME LAG /S.EI
t h i s s i m p l e g r a p h i c a l experiment.
Upstream propagation of wave f r o n t s generated by a source a t t h e h i n g e a x i s . a: Wave propagation. b : T i m e l a g d e r i v e d from wave p a t t e r n .
A t h i g h e r Mach numbers, b o t h the s t r e n g t h and t h e l e n g t h o f t h e shock a r e increased, which leads t o a c o n t i n u e d decrease i n amplitude o f the
l i n g times ( t i m e l a g s ) a r e g i v e n i n t h e diagrams
pressures measured i n f r o n t o f t h e shock wave
o f f i g u r e 8.13b.
(compare f i g u r e s 8.1 and 8.8).
A t M,
= 0.8,
The c o n s i d e r a t i o n s
where t h e f l o w i s s u b c r i t i c a l ,
t h e wave f r o n t s encounter more "head wind"
on the moving wave f r o n t s
i n high-subsonic f l o w a r e i l l u s t r a t e d i n f i g u r e
i n the
8.14 by shadowgraph p i c t u r e s . The u p s t r e a m - t r a v e l -
a c t u a l f l o w than i n t h e u n i f o r m case, as can be concluded from t h e c l o s e r spacing o f t h e f r o n t s
l i n g wave f r o n t s , which were made v i s i b l e w i t h t h e
and from t h e t i m e l a g curves. Moreover, t h e v e l o c -
h e l p o f a spark exposure (exposure t i m e l o w 6 sec),
i t y g r a d i e n t s normal t o the chord i n t h e a c t u a l
a r e generated by a v o r t e x s t r e e t , shedding from
f l o w cause a forward i n c l i n a t i o n o f t h e wave f r o n t s .
t h e sharp edges a t t h e r e a r o f an a i r f o i l from
When i t i s recognized t h a t t h e spacing o f t h e
which t h e t r a i l i n g - e d g e f l a p was removed for t h i s
wave f r o n t s
i s a measure f o r t h e i n t e n s i t y o f t h e
purpose. The receding motion o f the wave f r o n t s
local pressure perturbation gradient, w h i l e the t i m e l a g i s a measure f o r t h e phase s h i f t ,
i s s m a l l e s t near t h e s u r f a c e a t maximum p r o f i l e
It be-
thickness.
I n t h e spark exposure o f f i g u r e 8.15,
comes c l e a r t h a t t h e high-subsonic e f f e c t s ob-
the n o t i o n o f wave f r o n t s i n a t r a n s o n i c f l o w i s
served i n f i g u r e s 8 . 4 ~and 8 . 5 ~( b u l g e i n magnitude
shown. T h i s p i c t u r e demonstrates how the upstream-
d i s t r i b u t i o n and s h i f t i n phase c u r v e on t h e f r o n t
t r a v e l l i n g wave f r o n t s c o a l e s c e a t t h e f o o t o f
p a r t o f t h e a i r f o i l ) can be a t t r i b u t e d m a i n l y t o
t h e shock and how t h e upper p a r t s p e n e t r a t e the
t h e i n f l u e n c e o f t h e non-uniform steady f l o w f i e l d .
supersonic region.
A t M- = 0.875,
when i n t h e a c t u a l f l o w f i e i d
a s u p e r s o n i c r e g i o n i s p r e s e n t t e r m i n a t e d by a shock wave ( F i g . 8.13),
8.4
the afore-mentioned i n c l i -
THE INFLUENCE OF INCIDENCE AND MEAN
FLAP
n a t i o n o f t h e wave f r o n t s i s e s s e n t i a l f o r en-
ANGLE
a b l i n g t h e waves t o p e n e t r a t e t h e supersonic re-
8.4.1
gion. The g r a p h i c a l experiment f o r t h i s Mach number shows t h a t ,
i n the actual flow f i e l d ,
the I n o r d e r to i n v e s t i g a t e t h e i n f l u e n c e o f
p a r t s o f t h e upstream-moving wave f r o n t s c l o s e t o t h e a i r f o i l s u r f a c e merge i n t o t h e shock,
Unsteady p r e s s u r e d i s t r i b u t i o n s
but the
i n c i d e n c e and mean f l a p angle, a number o f t e s t s
-54-
a
b
CONTINUOUS
LIGHT
S P A R K EXPOSURE j
Fig. 8 . 1 4
SEC. )
Shadowgraph p i c t u r e s in high-subsonic flow.
a
CONTINUOUS LIGHT
b
S P A R K EXPOSURE
Fig. 8.15
-55-
SEC. )
Shadorgraph p i c t u r e s in transonic flos.
were conducted i n which an incidence was g i v e n t o
the c a l c u l a t i o n o f unsteady p r e s s u r e d i s t r i b u -
the steady main s u r f a c e o r i n which the mean a n g l e
t i o n s i t i s n o t p o s s i b l e anymore t o u s e the a n t i -
around which t h e f l a p was o s c i l l a t i n g was v a r i e d .
symmetry o f t h e problem w i t h r e s p e c t t o the chord. An i n t e r e s t i n g p o i n t , which may be o f i m -
Some c h a r a c t e r i s t i c r e s u l t s o f these t e s t s a r e c o l l e c t e d i n f i g u r e s 8.16 and 8.17.
The unsteady
portance when c a l c u l a t i o n methods a r e developed,
pressures measured a t t h e lower surface a r e a g a i n
i s t h a t for p r a c t i c a l a p p l i c a t i o n s t h e i n t e r a c -
p l o t t e d w i t h a minus sign.
t i o n s between t h e steady and unsteady f l o w f i e l d s
F i g u r e 8.16
shows t h e e f f e c t t h a t a small
mean f l a p a n g l e a l r e a d y has
a t t h e upper s u r f a c e and a t t h e lower surface
on t h e development o f
t a k e p l a c e independently o f one another. T h i s i s i l l u s t r a t e d i n f i g u r e 8.18,
t h e mean steady and unsteady p r e s s u r e d i s t r i b u -
which shows a compar-
t i o n s w i t h Mach number. As can be observed from
i s o n of t h e measured r e s u l t f o r a mean f l a p
t h e mean steady p r e s s u r e d i s t r i b u t i o n s ,
a n g l e o f 3 degrees w i t h r e s u l t s for z e r o mean
deflecting
t h e f l a p leads t o an i n c r e a s e i n v e l o c i t y a l o n g
f l a p a n g l e a t a l o w e r and a t a h i g h e r Mach num-
t h e upper s u r f a c e o f t h e a i r f o i l and t o a decrease
ber. For 3 degrees o f mean f l a p angle, b o t h t h e
i n v e l o c i t y a l o n g t h e lower surface. A t t h e lowest
mean steady and t h e unsteady p r e s s u r e d i s t r i b u -
Mach number, M_ = 0.5,
t i o n a t t h e upper s u r f a c e a r e almost t h e same as
the differences in veloci-
t y between t h e upper and t h e lower s u r f a c e a r e n o t
those o b t a i n e d f o r z e r o mean f l a p a n g l e a t t h e
s u f f i c i e n t t o c r e a t e s i g n i f i c a n t d i f f e r e n c e s be-
h i g h e r Mach number, whereas t h e mean steady and
tween t h e unsteady p r e s s u r e d i s t r i b u t i o n s a t b o t h
unsteady p r e s s u r e d i s t r i b u t i o n a t t h e lower s u r -
s i d e s . A t a f r e e - s t r e a m Mach number of about 0.75,
f a c e a r e Very s i m i l a r t o those o b t a i n e d f o r z e r o
however, the d i f f e r e n c e s i n mean steady f l o w a l o n g
f l a p a n g l e a t t h e lower Mach number. E v i d e n t l y ,
the upper and t h e lower s u r f a c e g i v e r i s e t o small
t h e unsteady p r e s s u r e d i s t r i b u t i o n s a t the upper
d i f f e r e n c e s i n t h e corresponding unsteady d i s t r i -
and t h e lower s u r f a c e a r e determined m a i n l y by
b u t i o n s . These d i f f e r e n c e s become, l a r g e r when t h e
t h e i r r e s p e c t i v e steady f l o w F i e i d , independent
free-stream Mach number i s increased f u r t h e r , and
o f t h e steady f l o w f i e l d a t the o t h e r surface.
they become p a r t i c u l a r l y pronounced from about H_ = 0.83,
when a t t h e upper surface a s u p e r s o n i c
r e g i o n appears, t e r m i n a t e d by an o s c i l l a t i n g shock
8.4.2 Unsteady aerodynamic c o e f f i c i e n t s
wave, w h i l e the lower s u r f a c e remains s u b c r i t i c a l .
A s i m i l a r development o f t h e unsteady pres-
From t h e p r e c e d i n g d i s c u s s i o n on the e f f e c t
s u r e d i s t r i b u t i o n s w i t h Mach number can be seen i n
o f i n c i d e n c e and mean f l a p a n g l e on t h e unsteady
f i g u r e 8.17,
pressure d i s t r i b u t i o n s ,
f o r an i n c i d e n c e o f -2 degrees, w h i l e
i t may be expected t h a t
t h e f l a p o s c i l l a t e s around zero mean f l a p p o s i t i o n
t h e o v e r a l l c o e f f i c i e n t s w i l l be s e n s i t i v e t o
w i t h an a m p l i t u d e o f 1 degree. A t about M_ = 0.75,
these parameters a l s o . T h i s i s shown i n f i g u r e s
t h e f l o w a l o n g t h e upper s u r f a c e becomes s u p e r c r i t -
8.19
i c a i , and t h e f i r s t s i g n s o f a (weak) shock can
c o e f f i c i e n t s f o r a frequency o f 120 Hz and v a r -
-
8.21,
which g i v e t h e unsteady aerodynamic
be n o t i c e d i n t h e mean steady and unsteady p r e s s u r e
ious values o f t h e i n c i d e n c e and t h e mean f l a p
d i s t r i b u t i o n s near t h e l e a d i n g edge. W i t h f u r t h e r
angle. Up t o M- * 0.15, t h e e f f e c t o f i n c i d e n c e
increase i n Mach number, t h e shock wave becomes
o r f l a p a n g l e i s n e g l i g i b l e . D e v i a t i o n s occur,
in
s t r o n g e r and i t s e f f e c t on t h e unsteady p r e s s u r e
p a r t i c u l a r i n t h e normal force.and m m e n t about
d i s t r i b u t i o n a l o n g t h e upper s u r f a c e becomes more
t h e q u a r t e r - c h o r d p o i n t , as soon as t h e f l o w be-
pronounced.
comes t r a n s o n i c .
The mean steady and unsteady p r e s s u r e d i s -
l t appears t h a t , due t o a non-
zero mean i n c i d e n c e o r mean f l a p a n g l e , t h e t r a n -
t r i b u t i o n s i n t h e two examples g i v e n above show
sonic effects,
i i k e t h e sharp decrease i n magni-
t h a t a t high-subsonic and t r a n s o n i c speeds t h e
tude i n b o t h t h e n o r m a l - f o r c e c o e f f i c i e n t
unsteady p r e s s u r e d i s t r i b u t i o n s a r e no l o n g e r
8.19)
a n t i - s y m m e t r i c a l , as a consequence o f t h e d i f f e r -
s h i f t t o a lower v a l u e o f t h e free-stream Mach
ences i n steady f l o w f i e l d .
number.
This implies t h a t i n
-56-
(Fig.
and t h e m m e n t c o e f f i c i e n t ( F i g . 8.20),
n
?)
m x
r n
m
n
-85-
I,* 1'") 3nv
AriN3ilb3äi i0 3 3 N 3 f l l i N I 3Hl
5'8
the unsteady p r e s s u r e d i s t r i b u t i o n s i n s l i g h t l y s u p e r c r i t i c a l f l o w i s shown i n f i g u r e 8 . 2 3 ,
shiie
i n f i g u r e 8.24 an example w i t h a r e l a t i v e l y s t r o n g shock wave
i 5
presented.
I n b o t h f i g u r e s the con-
t r i b u t i o n o f t h e o s c i l l a t i n g shock wave can e a s i i y be recognized. Since t h e phase i a g o f the shock motion w i t h respect t o t h e motion o f the f l a p i n creases w i t h frequency,
i c;
o
t h e l a r g e peak induced by
t h e shock wave i n t h e r e a l p a r t becomes s m a l l e r w i t h frequency, whereas t h e peak i n the imaginary
NhCA bPAOO6 AIRFOIL
-
,u O
EXPERIMENT
-IHIN.AIRFOIL TKEORY INCL. #ALL CORRECTICN O U A i I - i i E A C I
o
i Cp'
0.85
i
i c;
Fig. 8.22
Development of unsteady pressure distributions with frequency a t low speed.
A s shown i n f i g u r e 8.22,
a substantial part o f
the d i f f e r e n c e s between t h e o r y and experiment For the l i m i t i n g case o f Z e r o frequency can be a t t r i b u t e d to i n t e r f e r e n c e o f t h e tunnel w a l l s . For c l o s e d tunnei w a l l s , Oe Jager (Ref.
102) showed
t h a t f o r unsteady f l o w t h e i n t e r f e r e n c e e f f e c t s a r e l a r g e r than f o r q u a s i - s t e a d y f l o w w i t h a maximum f o r reduced f r e q u e n c i e s between 0.1 and 0.3. For h i g h e r frequencies,
t h e i n t e r f e r e n c e e f f e c t on
t h e unsteady a i r i o a d s d i m i n i s h e s r a p i d l y and becomes n e g l i g i b l e . A t p r e s e n t i t i s n o t known whether a s i m i l a r e f f e c t may be expected i n a wind tunnel w i t h s l o t t e d r o o f and bottom as a p p l i e d i n t h e present i n v e s t i g a t i o n , s i n c e an adequate
Fig. 8.23
t h e o r y f o r t h i s case i s n o t y e t a v a i l a b l e . The i n f l u e n c e o f t h e reduced frequency on
-59-
Development Of unsteady pressure distributions with frequency in slightly supercritical flow.
p a r t increases. When the supersonic r e g i o n i s s u f f i c i e n t i y l a r g e and i s t e r m i n a t e d by a r e l a t i v e l y strong shock wave, the p r e s s u r e peak r e s u l t i n g from the shock m o t i o n forms a dominant c o n t r i b u t i o n t o the
overall aerodynamic l o a d i n g ( F i g . 8 . 2 4 ) . A s t h i n airfoil
t h e o r y does n o t i n c l u d e t h i s e f f e c t ,
it
i s evident t h a t there e x i s t s a considerable d i s crepancy between t h e t h e o r e t i c a l and experimental pressure d i s t r i b u t i o n s .
O n t h e f l a p , which remains
s u b c r i t i c a l , t h e o r y and experiment show about the same agreement as i n t h e subsonic case. With respect t o t h e c o n t r i b u t i o n o f t h e o s c i l l a t i n g shock wave,
i t f i n a l l y can be n o t e d t h a t the
magnitude o f t h e pressure d i s t r i b u t i o n ( F i g . 8 . 2 5 ) seems t o i n d i c a t e t h a t the shock wave moves over a s h o r t e r t r a j e c t o r y when the frequency i s increased. T h i s w i l l be confirmed by t h e c o n s i d e r a t i o n s on t h e p e r i o d i c a l shock wave motions, g i v e n i n chapt e r 9.
8 . 5 . 2 Unsteady aerodynamic c o e f f i c i e n t s i n t h e preceding S e c t i o n ,
i t was shown t h a t
a t subsonic speed the agreement between the meaNACA 64AOO6 AIRFOIL M,
F i g . 8.24
Development of unsteady p r e s s u r e d i s t r i b u t i o n s w i t h frequency i n transonic flow.
AMPLITUDE 4 . 1 ' eo- 00
= 0.90
.75
Fig. 8 . 2 5
-60-
1.0 X I ,
I n f l u e n c e of frequency on magnitude distribution.
d i f f e r e n t shows up i n the d e v i a t i o n s between the t h e o r e t i c a l and the experimental moment c o e f f i c i e n t s . These d e v i a t i o n s become l a r g e r w i t h i n c r e a s i n g Mach number as a consequence o f the rearward s h i f t o f t h e mean p o s i t i o n o f the shock wave and
i t 5
a s s o c i a t e d p r e s s u r e peak.
F i n a l l y , t h e h i n g e moment shows a tendency t o agree b e t t e r w i t h t h e o r y as the frequency i s increased. T h i s improvement, which was mentioned a l r e a d y i n t h e preceding Section,
.d.. .bv
r e s u l t o f t h e b e t t e r agreement between t h e CalCul a t e d and the measured p r e s s u r e d i s t r i b u t i o n over t h e chord o f t h e f l a p .
i
.-A
i s indeed t h e
i:'
When t h e f i n d i n g s w i t h respect t o t h e i n -
Unsteady aerodynamic coefficients 8s a function of reduced frequency in subcritical flow. sured unsteady p r e s s u r e d i s t r i b u t i o n s and those
Fig. 8.26
f l u e n c e o f frequency a r e summarized,
i t can be
concluded t h a t , a t s u b c r i t i c a l speed, a tendency -i","
c a l c u l a t e d w i t h t h i n - a i r f o i l t h e o r y improves w i t h
O,m OR.,
AInFOlL THEOR" EXPERIMENT
i n c r e a s i n g frequency. T h i s i s r e f l e c t e d a l s o i n t h e o v e r a l I aerodynamic c o e f f i c i e n t s ( F i g . 8.261,
VORUALFORCE
UOMEIT
,c
'WGE uOUEui
which a i i t h r e e show t h e same tendency. I n supercri:ical f l o w ( F i g . 8.27), t h e measured n o r m a l - f o r c e c o e f f i c i e n t s a l s o seem t o show e b e t t e r agreement w i t h t h e o r y when t h e frequency increases. T h i s happens, however, o n l y by c o i n c i dence, a5 may be concluded f o r i n s t a n c e from a c l o s e r l o o k a t F i g u r e 8.24.
In t h e t h e o r y ,
an
in-
crease i n frequency leads a t h i g h Mach numbers t o r a p i d changes i n phase a n g l e on the f r o n t p a r t o f the a i r f o i i , and, as a r e s u l t , t h e r e a i p a r t o f the c a l c u l a t e d pressures upstream o f t h e h i n g e a x i s changes s i g n , which r e s u i t s i n about z e r o n e t f o r c e a:
t h e h i g h e r frequencies. The r e a l
p a r t s o f t h e measured pressures do n o t produce a net force e i t h e r , f o r a q u i t e d i f f e r e n t
reason.
Here t h e presence o f the shock wave p r e v e n t s t h e
pressure p e r t u r b a t i o n s generated by t h e o s c i l l a t i n g f l a p from reaching :he
-r
I
f r o n t p a r t of the
a i r f o i l . The agreement f o r the imaginary p a r t o f t h e normal f o r c e i s o n l y due t o t h e f a c t t h a t t h e i n t e g r a l over t h e imaginary p a r t o f t h e p r e s s u r e peak generated by the o s c i l l a t i n g shock wave has about the same v a l u e as the i n t e g r a l over t h e imaginary p a r t o f t h e c a l c u l a t e d p r e s s u r e d i s t r i bu t ion.
Fig. 8.27 Unsteady aerodynamic coefficients as B function af reduced frequency in transonic flow.
The f a c t t h a t t h e n a t u r e o f t h e c a l c u l a t e d and t h e measured p r e s s u r e d i s t r i b u t i o n s i s q u i t e
-61-
- =::-*..
-19-
fl013 hOW’31S
N I NOIlISOd H3OHS ûNW H13N3öLS X30HS
2’6
-
SäW’W3ü hü013fl00UNI
1‘6
S3hVM XIOHS 40 NOILOW lV31íl01ä3d 3Hl NO 6
PZ/?, = 1
2Y + f+i ( M 2I
-
1)
,
í9.ial
where t h e i n d i c e s I and 2 r e f e r t o the p o s i t i o n s j u s t ahead and j u s t behind t h e shock wave, respect i v e l y . As a consequence, however, of t h e presence o f t h e boundary l a y e r , t h e shock s t r e n g t h s measured
on a i r f o i l s a r e c o n s i s t e n t l y less than those pred i c t e d by t h i s r e l a t i o n . T h i s i s demonstrated i n f i g u r e 9.3,
i n which t h e shock s t r e n g t h s d e t e r -
mined from t h e p r e s e n t steady p r e s s u r e measurements a r e compared w i t h t h e shock s t r e n g t h according t o (9.ia)
and w i t h t h e shock s t r e n g t h s
determined from o t h e r i n v e s t i g a t i o n s (Ref.
14).
The measured shock s t r e n g t h s f o l l o w t h e t r e n d o f t h e t h e o r e t i c a l curve reasonably wel1 u n t i l H F i g . 9.2
Steady Shock p a t t e r n s for zero i n c i d e n c e and v a r i o u s f l a p d e f l e c t i o n s .
about 1.3.
1 Beyond t h i s Mach number, .the exper-
'
imental shock s t r e n g t h s tend t o decrease, which p a t t e r n on t h e a i r f o i l shows a development w i t h
i s caused by the afore-mentioned shock-induced
f l a p a n g l e and Mach number, as i s usual f o r conv e n t i o n a l a i r f o i l s . A steady shock appears as soon
1 EXPERIMENTS ON VARIOUS TYPES OF AIRFOILSI
as the maximum l o c a l Mach number reaches a v a l u e
o f about 1.05,
and shock-induced s e p a r a t i o n occurs
o KACPRZYNSKI-OHLI*H.R. L
when j u s t ahead o f t h e shock wave t h e l o c a l Mach
-1li106
YOIHIH*R*-ZO~ARI-CARIEn.R.
.lOilOb
o PEh7CEY.R. . l x 1 0 6
number i s increased up t o about 1 . 3 .
O FLIGHT.XI.Re r20i106 SINNOTT-OSBORNE, R, r 2 x 106
From f i g u r e 9 . 1 i t f o l l o w s t h a t , a t M_ =
0 . 5 5 and zero f l a p angie, weak shock waves a r e p r e s e n t on b o t h the upper and t h e lower surface o f t h e a i r f o i l a t about 45 per c e n t o f t h e chord. When t h e f l a p i s d e f l e c t e d downward, t h e speed a l o n g the upper s u r f a c e i s increased and the shock i s d i s p l a c e d downstream 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 t h e supersonic r e g i o n a r e increased. A t the same time,
/l
t h e f l o w speed a l o n g
575
t h e lower s u r f a c e i s reduced and t h e development o f t h e f l o w p a t t e r n i s reversed: t h e s i z e o f t h e s u p e r s o n i c r e g i o n decreases and t h e shock wave becomes weaker. Beyond a c e r t a i n f l a p d e f l e c t i o n ,
1.6
O
t h e lower s u r f a c e becomes c o m p l e t e l y s u b c r i t i c a l . A development s i m i l a r t o the one d e s c r i b e d above
I.,
can be observed a l s o a t H_ = 0.875 and M_ = 0 . 9 0 , b u t here l a r g e r f l a p d e f l e c t i o n s a r e r e q u i r e d t o 1.2
suppress t h e shock wave a t t h e lower surface. I n an i n v i s c i d flow,
the strength of a
normal shock wave, which i s t h e p r e s s u r e r a t i o across t h e shock,
1.1
i s determined by t h e Rankine-
Hugoniot r e l a t i o n :
F i g . 9.3
-63-
Pressure r a t i o across t h e shock wave ( p a r t l y reproduced from Ref. 14).
-b9-
E 028’0 = 118‘0
=
006.0 = -wl -0
i
4n
f l O l j A(lW31CNfl N I 03AU3SöO N O I I O W 3AWh-H30HS 30 S 3 d A l
E‘6
TYPE A
TYPE 8
TYPE C
FORMATION O F NEW
A HIGH SPEED FILM o SHADOWGRAPH PICTURES
Fig. 9.5
Observed types of periodical shock-wave motion on the upper surface.
TYPE A
Fig. 9.6
TYPE 0
FLAP MAXIMUM UPWARDS
FLAP UAXIYUU
FLAP MAXIMUM WWHWAROS
FLAP MAXIMUM WINIAROS
Time histories af the periodical Shock-wave motions.
-65-
TYPE
c
type 8:
i n t e r r u p t e d shock-wave motion
ing f l o w model i s i n t r o d u c e d ( F i g . 9 . 7 ~ ) . ( 1 ) The f l o w w i t h a normal shock wave i s con-
The second t y p e o f p e r i o d i c a i shock-wave
s i d e r e d t o be one-dimensional,
m o t i o n , observed a t a somewhat lower free-stream Mach number (M_ = 0.875),
i.e.
pressure
g r a d i e n t s i n a d i r e c t i o n normal t o the f l a w
i s c h a r a c t e r i z e d by the
a r e neglected.
disappearance o f the shock wave i n the dynamic
( 2 ) Upstream o f t h e normal shock wave, a known
case d u r i n g a p a r t o f i t s backward motion (Fig.
pressure d i s t r i b u t i o n w i t h gradients in
In t h e same way as f o r t y p e A, t h e shock
9.5b).
streanwise d i r e c t i o n e x i s t s , which does n o t
wave reaches i t s maximum s t r e n g t h d u r i n g t h e
change when t h e shock wave i s d i s p l a c e d o v e r
upstream motion, as can be n o t i c e d c l e a r l y f r a
a small d i s t a n c e .
t h e s i z e o f t h e shock waves i n t h e successive
wave i s d i s p l a c e d o v e r a d i s t a n c e Ax,
pictures of figure
9.6b.
In o t h e r words, i f t h e shock the
pressure upstream o f t h e shock wave i s assumed t o f o l l o w a known p - l o c u s . 1
type C:
upstream-propagated shock waves The f i r s t assumption i s reasonable when t h e
A t M_ = 0.85,
f l o w around t h i n a i r f o i l s i s described, b u t may b e
where t h e flow i s j u s t super-
c r i t i c a l , a t h i r d t y p e o f p e r i o d i c a l shock-wave
l e s s s a t i s f a c t o r y f o r t h i c k a i r f o i l s , where i n
m o t i o n i s observed ( F i g s . 9 . 5 ~and 9.6c),
general l a r g e p r e s s u r e g r a d i e n t s normal t o t h e
which
d i f f e r s c o m p l e t e l y from t h e preceding types.
s u r f a c e occur.
At
120 Hz a shock wave i s formed on t h e upper
The second assumption
i 5
no r e a i r e s t r i c t i o n
s u r f a c e o f the a i r f o i l a t t h e moment t h e f l a p has
i n t h e quasi-steady case, because then the p,-
j u s t passed i t s maximum d e f l e c t i o n downwards.
locus can be o b t a i n e d from t h e steady pressure
T h i s shock wave moves upstream w h i l e i n c r e a s i n g
d i s t r i b u t i o n s (Figs. 9.7a.b).
i n s t r e n g t h . Then the shock wave weakens again,
case, however,
b u t c o n t i n u e s i t s upstream motion, leaves t h e
p r e s s u r e j u s t i n f r o n t o f the shock wave changes
a i r f o i l from t h e l e a d i n g edge, and propagates
i n t h e same way as i n quasi-steady flow.
upstream i n t o t h e oncoming f l o w as a (weak) f r e e
For the o s c i l l a t o r y
i t i s necessary t o assume t h a t the
Other-
b
a
shock wave. T h i s phenomenon i s repeated p e r i o d i c a l l y and a l t e r n a t e s between t h e upper and lower s u r f a c e ( F i g . 9 . 6 ~ ) . I t i s n o t e d here t h a t i n t h e q u a s i - s t e a d y case t h e shock wave vanishes d u r i n g a p a r t o f the cycle (Fig.
9.5~. f
=
O). As w i l l
be shown l a t e r , t h i s i s e s s e n t i a l f o r t h e occurrence o f a tyQe-C unsteady shock-wave motion. S T A T I C PRESSURE DISTRIBUTIONS, WITHMACHNUMBER FREEZE
STATIC PRESSURE DISTRI0UTIONS. NOMACHNUMBER FREEZE
c
9.4
INTRODUCTION OF AN ANALYTICAL MODEL
9.4.1
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
To understand t h e p h y s i c s behind t h e observed types o f shock-wave motions,
it is suffi-
MEAN SHOCK POSITION x . j s ,
c i e n t t o c o n s i d e r t h e , i n s t a n t a n e o u s shock s t r e n g t h
o f a shock wave which o s c i l l a t e s s i n u s o i d a l l y w i t h
SCHEMATIZED FLOWPATTERN
p r e s c r i b e d a m p l i t u d e and frequency i n a g i v e n
Fig. 9.7
(steady) flow f i e l d . For s i m p l i c i t y , the follow-
-66-
Schematization for a n a l y t i c a l flow model.
wise, a l s o the tine-dependent r e l a t i o n s between
9 . 4 . 2 A p p l i c a t i o n o f the a n a l y t i c a i modei
f l a p p o s i t i o n and t h e p r e s s u r e p e r t u r b a t i o n s j u s t up- and downstream o f t h e shock wave have t o be
For the p r e s e n t a i r f o i l ,
r e l a t i o n (9.2) was
included. T h i s means t h a t f o r unsteady f l o w t h i s
used t o c a l c u l a t e the v a r i a t i o n i n shock s t r e n g t h
simple f l o w model i s f u l l y j u s t i f i e d o n l y i n t h e
t h a t corresponds t o an assumed s i n u s o i d a l motion
case o f a s o - c a l l e d "Mach number freeze'' ( F i g .
o f t h e shock wave. The amplitudes o f the shock-
9.7b):
wave m o t i o n were taken t o be about equal t o the
t h e phenomenon t h a t t h e pressure d i s t r i b u -
t i o n upstream o f t h e shock wave does n o t change
amplitudes observed d u r i n g t h e wind-tunnel
any longer w i t h Mach number o r f l a p angle.
and t h e q u a n t i t i e s p, ( x s ) , H1 and öMl/öx
In t h e s i m p l i f i e d f l o w f i e l d , t h e i n s t a n -
_ - across
taneous shock s t r e n g t h p /p
2 1
a shock wave
which o s ~ i l l a t e ss i n u s o i d a l l y w i t h a small a m p l i tude xo and frequency w Appendix
can be expressed as (see
C):
tests,
were
determined from r h e measured steady p r e s s u r e d i s t r i b u t i o n s (Fig. 9.1).
To o b t a i n agreement w i t h
the steady t e s t s , t h e shock s t r e n g t h i n t h e mean p o s i t i o n was c a l c u l a t e d w i t h e q u a t i o n ( 9 . l b ) , which a f a c t o r A o f 0.7
in
i s used. To o b t a i n agree-
ment f o r t h e quasi-steady r e s u l t s , t h e same f a c t o r has a l s o been used f o r the displacement e f f e c t . pI
I pl I x=xs
Y+I
I ax
al
1
Since any b e t t e r knowledge was l a c k i n g , no reduc-
"
t i o n was a p p l i e d t o t h e term accounting f o r t h e dynamic e f f e c t s .
The q u a n t i t i e s M l ,
aMl/ax
and a l r e f e r t o t h e con-
d i t i o n s j u s t upstream o f t h e shock wave i n i t s steady mean p o s i t i o n ( x = x,).
The f i r s t term o f t h e
r i g h t - h a n d s i d e o f e q u a t i o n (9.2) denotes t h e
The r e s u l t for M_ = 0.90
9.8a.
and can be c a l c u l a t e d w i t h e q u a t i o n ( 9 . i a ) . The
a sinu-
w i t h the shock-wave displacement, In the unsteady case ( f = 120 Hz),
second p a r t denotes t h e change i n shock s t r e n g t h
p a r t , two e f f e c t s can be d i s t i n g u i s h e d .
I n t h e quasi-steady case ( f = O ) ,
s o i d a l shock-wave motion generates a s i n u s o i d a l v a r i a t i o n i n shock s t r e n g t h , which i s i n phase
s t r e n g t h o f t h e shock i n i t s steady mean p o s i t i o n
due t o the o s c i l l a t o r y m o t i o n xoeiwt.
i s given i n f i g u r e
In t h i s
the dynamic e f f e c t g i v e s r i s e
t o two i m p o r t a n t changes: a phase s h i f t and an a m p l i f i c a t i o n o f t h e v a r i a t i o n i n shock s t r e n g t h .
The term
The maximum shock s t r e n g t h i s reached about
c o n t a i n i n g .3Ml/öx accounts f o r t h e change i n
64 degrees a f t e r t h e shock wave has passed i t s
shock s t r e n g t h caused by t h e change i n shock
most backward p o s i t i o n , whereas the minimum
p o s i t i o n (displacement e f f e c t ) , an e f f e c t which e x i s t s a l r e a d y i n t h e quasi-steady case ( o =
O).
The term w i t h i w r e p r e s e n t s the change i n shock s t r e n g t h due t o t h e v e l o c i t y o f t h e shock wave r e l a t i v e t o t h e a i r f o i l (dynamic e f f e c t ) . The advantage o f t h e p r e s e n t model o v e r t h e unsteady shock-wave models presented e a r l i e r by
s t r e n g t h i s reached d u r i n g i t s downstream motion. For t h e g i v e n a m p l i t u d e of the shock-wave motion, the shock s t r e n g t h remains always l a r g e r than 1.0, which means t h a t a shock wave i s p r e s e n t d u r i n g t h e complete c y c l e . These r e s u l t s a r e i n agreement w i t h t h e experimental o b s e r v a t i o n s f o r the shockwave motion o f t y p e A , as discussed i n S e c t i o n 9.3.
Coupry and P i a z o l I i (Ref. l o g ) , Eckhaus (Ref. 1 lo), and Landahl (Ref.
The c a l c u l a t e d r e s u l t s f o r M_
24) i s t h a t t h e Mach number
= 0.875 a r e
upstream o f t h e shock wave i s n o t c o n s t a n t , which
p l o t t e d i n f i g u r e 9.8b. Again a q u a s i - s t e a d y
makes i t v a l i d f o r low f r e q u e n c i e s as w e l l as f o r
shock-wave m o t i o n r e s u l t s i n an in-phase v a r i a t i o n For f = 120 Hz, however, the
t h e quasi-steady f l o w case. Moreover, t h e presence
i n shock s t r e n g t h .
o f a streamwise g r a d i e n t i n Mach number i s essen-
dynamic e f f e c t increases t h e v a r i a t i o n i n shork
t i a l t o e x p l a i n a l l t h r e e types o f shock-wave
s t r e n g t h i n such a way t h a t d u r i n g a p a r t o f the
motions, a s w i l l be demonstrated i n t h e n e x t
c y c l e (1.17
s e c t ion.
-
p2/pl
-67-
c wt c
1.43 n) t h e pressure r a t i o
i s lower than I.O.
T h i s means t h a t the shock
TYPE
9
TYPE
A
u
IHOCK DISAPPEARS
Fig. 9.8
Explanation
O f
d i f f e r e n t types o f shock-wave motion
wave vanishes d u r i n g t h a t p a r t o f t h e c y c l e . T h i s
f o r a shock wave generated by a local explosion:
r e s u l t i s t y p i c a l f o r t y p e 8, and i t
t h e shock wave e n t e r i n g t h e subsonic region con-
i 5
i n qual-
i t a t i v e agreement w i t h t h e experiments. Finally,
t i n u e s i t s upstream m o t i o n as a f r e e shock wave
the shock-wave m o t i o n o f t y p e C
w i t h a speed o f p r o p a g a t i o n n e a r l y equal t o the
can be understood w i t h t h e h e l p o f t h e c a l c u l a t e d r e s u l t s f o r M_ = 0.85,
d i f f e r e n c e ' between the l o c a l v e l o c i t y o f sound and
presented i n f i g u r e 9 . 8 ~ .
t h e l o c a l f l o w v e l o c i t y . From f i g u r e 9 . 8 ~ i t
The a m p i i t u d e o f the assumed shock-wave m o t i o n i n
f o l l o w s a l s o t h a t a new shock wave i s formed on
t h e g i v e n f l o w f i e l d i s so l a r g e t h a t , d u r i n g a p a r t o f t h e c y c l e (0.67 n c w t c "shock"
the a i r f o i l , when t h e shock wave i s on i t s way t o
1 . 3 3 n), the
i t s maximum downstream p o s i t i o n . So the c a l c u i a t e d
i s l o c a t e d upstream o f t h e s o n i c p o i n t 5.
r e s u l t s c o n f i r m t h e r e s u l t s o f t h e present t e s t s
_ _
F o r quasi-steady m o t i o n , t h e shock s t r e n g t h p2/p1
also f o r t h e type-C shock-wave m o t i o n .
i s always s m a l l e r than 1.0 d u r i n g t h e p a r t o f the c y c l e mentioned, o r ,
i n o t h e r words,
no shock wave When the r e s u l t s a r e summarized,
i s p r e s e n t then. T h i s i s i n accordance w i t h the
i t can
be s t a t e d t h a t t h e t h r e e examples discussed above
experiments.
demonstrate the adequacy o f t h e a n a l y t i c a l modei.
The p e r i o d i c a l shock s t r e n g t h as c a l c u l a t e d f o r 120 Hz i s g i v e n i n the bottom c u r v e o f f i g u r e
A c l o s e r look a t e q u a t i o n (9.2)
9 . 8 ~ . The phase s h i f t and a m p l i f i c a t i o n r e s u l t i n g
i t i s n o t s t r i c t l y necessary t h a t these t h r e e
from t h e dynamic e f f e c t l e a d t o a d r a s t i c change
types w i l l o c c u r a t d i f f e r e n t Mach numbers as
i n the situation.
happened i n these cases.
Now the upstream-moving shock
also reveals that
In p r i n c i p l e , a l l three
wave e n t e r s the subsonic r e g i o n upstream o f t h e
types may o c c u r a t t h e same free-stream M x h
s o n i c p o i n t 5 w i t h a c e r t a i n r e s i d u a l s t r e n g t h R.
number, b u t then f o r d i f f e r e n t amplitudes o r f r e -
T h i s means t h a t a s i m i l a r s i t u a t i o n i s c r e a t e d a5
quencies.
-68-
9.5
ADDITIONAL REPARKS
A s f a r as the behaviour a t h i g h frequencies i 5
concerned,
and (9.4)
9 . 5 . 1 Some p r o p e r t i e s o f the unsteady shock
i t can be concluded From both ( 9 . 3 )
that, f o r a given pressure perturbation
Up2, t h e a m p l i t u d e o f the shock-wave motion de-
r e l a t ions
creases w i t h i n c r e a s i n g frequency.
I n o t h e r wards,
high-frequency pressure p e r t u r b a t i o n s cause m a l l e r For the s i m p l i f i e d f l o w model o f f i g u r e 9.7,
amp1 i t u d e s o f the r e s u l t i n g shock-wave motion than low-frequency p e r t u r b a t i o n s do
t h e r e l a t i o n between the s i n u s o i d a l p r e s s u r e p e r t u r b a t i o n Ap2 behind the shock wave and the shockwave motion, a5 d e r i v e d i n Appendix C , reads:
A p a r t i c u l a r p r o p e r t y o f t h e quasi-steady
case i s r e v e a l e d by t h e quasi-steady p a r t o f e q u a t i o n (9.3):
T h i s r e l a t i o n says t h a t an o s c i l l a t o r y pressure p e r t u r b a t i o n behind the shock wave c r e a t e s an
The r i g h t - h a n d s i d e o f t h i s e q u a t i o n becomes zero
o s c i l l a t o r y m o t i o n o f t h e shock wave w i t h such an
for
a m p l i t u d e t h a t the Rankine-Hugoniot
Mach number, the expression changes sign. T h i s
s a t i s f i e d i n s t a n t a n e o u s l y . From
relations are
(9.3)
i t follows
M, = /,'o= 1.483, (aM,/ax i O ) . Above t h i s
behaviour r e s u l t s from two c o u n t e r - a c t i n g e f f e c t s ,
t h a t i n general a phase s h i f t d i f f e r e n t from
which o c c u r when the Mach number ahead a f the
90 degrees may occur between shock-wave d i s p l a c e -
shock wave i s increased: the s t a t i c pressure
ment and p r e s s u r e p e r t u r b a t i o n . T h i s phase s h i f t
ahead o f the shock wave decreases, b u t a t the
i s determined by t h e frequency,
same t i m e the p r e s s u r e r a t i o across the shock wave
t h e Mach number,
and the g r a d i e n t i n Mach number j u s t ahead o f the
increases. E v i d e n t l y , above M, = i . 4 8 3 the f i r s t
shock wave.
e f f e c t predominates, which means t h a t f o r a g i v e n
The phase s h i f t becomes e x a c t l y 90 degrees
i n i t i a l s t a g n a t i o n pressure t h e maximum absolute
when 3 t i / 3 x = O , f o r which case t h e f o l l o w i n g
s t a t i c p r e s s u r e behind a normal shock wave i s
expression a p p l i e s :
o b t a i n e d a t Ml = 1.483.
1
L a i t o n e (Refs.
113, 1 1 4 )
a l s o observed t h i s e f f e c t and argued t h a t t h i s Mach number corresponds t o the l i m i t v e l o c i t y t h a t (9.4)
can e x i s t on a two-dimensional
a i r f o i l j u s t ahead
o f a normal shock t e r m i n a t i n g a l o c a l supersonic r e g i o n . T h i s seems t o be confirmed i n p r a c t i c e , T h i s e x p r e s s i o n corresponds w i t h formulae g i v e n by
s i n c e , t o the a u t h o r ' s knowledge, values o f the
Eckhaus (Ref. 110) and Lambourne (Ref.
l o c a l Mach number h i g h e r than about 1.5 j u s t up-
los),
d e r i v e d f o r a f l o w model w i t h a c o n s t a n t Mach
stream o f a normal shock wave never have been
number ahead o f t h e shock wave. The e x p r e s s i o n
observed i n experiments on two-dimensional a i r -
was g i v e n a l s o by Fiszdon (Ref.
foils.
I l l ) . A higher-
A c t u a l l y , e q u a t i o n (9.5),
o r d e r a p p r o x i m a t i o n o f t h e r e l a t i o n between shock-
w i t h the l i n e a r i -
i s n o t v a l i d f o r Mach
wave m o t i o n and p r e s s u r e p e r t u r b a t i o n was produced
zation i n i t s derivation,
by Dvdrdk (Ref. 1 1 2 ) . A weak p o i n t o f these models
numbers c l o s e t o 1.483,
i s t h a t they f a i l f o r low frequency and i n t h e
indicate that,
quasi-steady case (O = O ) ,
j u s t upstream of t h e shock wave approaches t h i s
from ( 9 . 4 ) ,
s i n c e then, as f o l l o w s
b u t t h e r e s u l t seems t o
i n the case t h a t t h e Mach number
v a l u e , quasi-steady pressure p e r t u r b a t i o n s behind
i n f i n i t e l y l a r g e shock a m p l i t u d e s a r e
r e q u i r e d t o compensate f o r any p r e s s u r e p e r t u r b a -
t h e shock wave w i l l r e s u l t i n r e l a t i v e l y l a r g e
Cion Ap
shock-wave displacements.
2'
-69-
9 . 5 . 2 P o s s i b l e use o f t h e shock-wave nodel
o f the shock wave i s assumed t o v a r y i n the same way a s i n the q u a s i - s t e a d y case. When the modei i s used i n a " s h o c k - f i t t i n g "
The unsteady shock-wave model presented here
procedure, i t
i5
a d v i s a b l e t o r e f i n e the model i n t h i s r e s p e c t and
was i n t r o d u c e d p r i m a r i l y t o understand the p h y s i c s
t o i n c l u d e the time-dependent v a r i a t i o n i n pres-
behind t h e d i f f e r e n t types o f p e r i o d i c a l shock-
s u r e ( o r Mach number) upstream o f the shock wave.
wave motion. For t h i s purpose, t h e model was k e p t
In i t s p r e s e n t form, expression (9.3) may be o f
as simple as p o s s i b l e . For f u r t h e r use, one may
use i n q u a l i t a t i v e s t u d i e s on t h e " w i l l i n g n e s s "
t h i n k o f a p p l i c a t i o n i n unsteady t r a n s o n i c - f l o w
o f a shock wave t o move i n r e a c t i o n upon pressure
theories,
i n which t h e unsteady shock r e l a t i o n s
are introduced e x p l i c i t l y ("shock-fitting").
f l u c t u a t i o n s downstream o f t h e shock, such as
As
generated by f l a p o ~ c i l l a t i o n s , f l o w s e p a r a t i o n
such, the p r e s e n t model i s an improvement i f
( b u f f e t ) o r by noise sources. As mentioned b e f o r e ,
compared w i t h t h e shock models ( c o n s t a n t Mach
parameters o f i n t e r e s t then a r e t h e magnitude and
number ahead o f t h e shock) o f r e f e r e n c e s 24,
109,
frequency o f t h e p r e s s u r e p e r t u r b a t i o n s and t h e
and 110. The model has, however, the disadvantage
Mach-number d i s t r i b u t i o n upstream o f t h e shock
t h a t i n the o s c i l l a t o r y case the pressure ahead
wave.
10 T H E UNSTEADY AERODYNAMIC CHARACTERISTICS O F T H E "SHOCK- F R E E " N L R 7301 AIRFOIL 10.1
INTRODUCTORY REMARKS
11-1:
s l i g h t l y s u p e r c r i t i c a l f l o w on the upper surface, and
11-2:
The r e s u l t s i n t h e preceding chapters con'
I I I : "shock-free"
c e r n a c o n v e n t i o n a l - t y p e a i r f o i l . As mentioned i n c h a p t e r I , however, nowadays t h e r e
f l o w w i t h a well-developed shock wave flow.
a con-
i5
NATURAL TRANSITION R e i 2.1 f IO6
s i d e r a b l e i n t e r e s t i n t h e unsteady aerodynamic characteristics o f so-called supercritical a i r F o i l s . Therefore,
a5
a n e x t s t e p i n t h e NLR p r o -
T H E O R Y , M,
gram on unsteady t r a n s o n i c aerodynamics, a wind-
E R I M E N T . M,
tunnel i n v e s t i g a t i o n was performed on the 16.5 p e r cent t h i c k
= 0.721. C, = 0.595 i
0.747.
C - O 455
e-
N L R 7301 a i r f o i l , designed For
' ( s h o c k - f r e e " f l o w under p r e s c r i b e d condi t i o n s (Fig.
10.1).
On a two-dimensional model o f t h i s
a i r f o i l that could perform p i t c h i n g o s c i l l a t i o n s about an a x i s a t 40 p e r cent o f the chord ( f o r d e t a i l s see S e c t i o n 6 . 2 ) , 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 determined.
In
SONIC L I N E
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 were recorded.
In t h e f o l l o w i n g s e c t i o n s , t h e main unsteady aerodynamic c h a r a c t e r i s t i c s of t h e NLR 7301 a i r f o i l a r e discussed f o r t h r e e d i f f e r e n t flow c o n d i t i o n s , which can be c h a r a c t e r i z e d as f o l l o w s (see f i g u r e 10.2): I : f u l l y subsonic f l o w I I : t r a n s o n i c f l o w w i t h shock wave.
Fig. 10.1
In t h i s
domain two examples w i l l be discussed, namel y
-70-
Theoretical and experimental "Shockf r e e " pressure distributions of the NLR 7301 airfoil (reproduced from Ref. 73).
'
I t w i l l t u r n o u t t h a t the pressure d i s t r i b u t i o n
LIFT COEFFICIENT
r
f o r c o n d i t i o n s I and I I show e s s e n t i a l l y the same f e a t u r e s a s those observed f o r the conventional a i r f o i l w i t h f l a p , w h i l e f o r c o n d i t i o n I l l some new aspects appear.
10.2
UNSTEADY PRESSURE DISTRIBUTIONS
10.2.1
F u l l y subsonic f l o w ( c o n d i t i o n i )
The steady p r e s s u r e d i s t r i b u t i o n s measured .2
i n subsonic f l o w ( c o n d i t i o n i ) f o r two values of
-
t h e a n g l e o f a t t a c k a r e shown i n f i g u r e 10.3a, w h i l e t h e corresponding quasi-steady pressure .e
/ Fig. 10.2
p, k \
.9
coefficients,
MACHNUMBER
Ca-Mm plane for t h e NLR 7301 airfoil.
defined as
C (a-Aa) - C (a+Aa) AC
NLR 7 3 1 AIRFOIL
k -0.80
P
(10.1)
2Aa
.i.7.106
TRANSITION I T R I P AT
a r e g i v e n i n f i g u r e 10.3b. c
I
i n order t o f a c i l i t a t e
.I
the comparison between t h e upper and the lower surface,
the quasi-steady pressures on the iower
s u r f a c e a r e p l o t t e d a g a i n w i t h a reversed sign. The agreement between the measured quasi-steady d a t a and t h e p r e d i c t i o n o f t h i n - a i r f o i l theory i s reasonable. D e v i a t i o n s can be seen over the rear p a r t o f , t h e a i r f o i l , where t h e measured values a r e below the c a l c u l a t e d curve, and near the l e a d i n g edge, where t h e pressures measured on t h e upper s u r f a c e a r e l a r g e r than those p r e d i c t e d NLR 7301 AIRFOIL M,A.M
ib.o.es'Yi-o.s'
IC,. L
O O
IO
PRESSURE TUBE5 IUSITU TRANSDUCERS
X,. O
.O
O
-5
O 'Ir
O
I
O
O
-5
='c
Fig. 10.3
Fig. 10.4
Steady and quasi-steady pressure distributions a t subsonic speed (condition I).
-71-
Unsteady pressure distributions on the upper surface at subsonic speed (condition I).
by the t h e o r y and a i s 0 l a r g e r than the v a l u e s
steady pressures measured d i r e c t l y w i t h the i n -
measured on t h e lower s u r f a c e . As w i i i be d i s -
s i t u transducers and the pressures o b t a i n e d v i a
cussed l a t e r ( c h a p t e r i j ) , these d i f f e r e n c e s have
the p r e s s u r e tubes.
t o be a t t r i b u t e d t o t h e combined e f f e c t o f a i r f o i l t h i c k n e s s ( p i u s i n c i d e n c e ) and the boundary i a y e r .
10.2.2 Transonic f l o w w i t h shock wave
The f i r s t e f f e c t dominates on the F r o n t p a r t o f t h e a i r f o i l , w h i l e t h e boundary-layer e f f e c t
(condition i t )
is
more pronounced on t h e r e a r p a r t .
The n e x t two examples concern o s c i l l a t i o n s
A comparison between t h e unsteady pressures
o f t h e a i r f o i l around t h e o f f - d e s i g n c o n d i t i o n s
measured on t h e upper s u r f a c e o f t h e o s c i l l a t i n g
1 1 - 1 and 11-2 ( F i g . 10.2).
a i r f o i l and t h e corresponding r e s u l t s o f t h i n -
t y p i c a l for ''cIas5icaI" transonic flows,
airfoil
theory (Fig.
10.4) shows d i f f e r e n c e s
i n which
a supersonic r e g i o n terminated by a normal shock
s i m i i a r t o those observed i n t h e q u a s i - s t e a d y ca5e.
These c o n d i t i o n s a r e
wave occurs.
Both cases deal w i t h a free-stream
Mach number o f 0.7,
I n g e n e r a l , t h e agreement between t h e o r y
but w i t h a d i f f e r e n t i n c i d e n c e
and experiment i s reasonable and i s o f t h e same
of the a i r f o i l .
o r d e r a s t h a t observed e a r l i e r f o r t h e a i r f o i l
i n c i d e n c e i s 0.85 degrees and the f l o w aiong the
For c o n d i t i o n 1 1 - 1 ,
t h e mean
w i t h Flap. F u r t h e r , as c o u l d be expected, t h e r e
upper surface i s s i i g h t l y s u p e r c r i t i c a i w i t h a
i s a v e r y s a t i s f a c t o r y agreement between t h e un-
weak shock wave. For c o n d i t i o n 11-2, incidence
i 5
i n which the
3 degrees, the upper s u r f a c e c a r r i e s
a w e l l - d e v e l o p e d supersonic r e g i o n , t e r m i n a t e d by
a r e l a t i v e l y s t r o n g shock wave.
Condition 1 1 - i :
s l i g h t l y supercritical flow
The steady pressure d i s t r i b u t i o n s measured f o r two i n c i d e n c e s c l o s e t o c o n d i t i o n i I - 1 ( F i g . i0.5a)
show the appearance o f a weak shock wave
JCa
M
m
O UPPER SURFACE +JCp D LOWER SURFACE -ACp
10
o
1.0 ' c
Fig. 10.5
Fig. 10.6
Steady and quasi-steady pressure distributions in slightly supercritical flow (condition 11-1).
-72-
Unsteady pressure distributions on the upper surface at slightly supemritical speed (condition 11-1).
on t h e upper surface o f the a i r f o i l , which s h i f t s
i n downstream d i r e c t i o n t o about 20 p e r c e n t of t h e chord as the i n c i d e n c e t o 1.35 degrees.
i 5
increased from 0 . 3 5
NLR ?mi AIRFOIL
In the quasi-steady pressure
d i s t r i b u t i o n on the upper s u r f a c e ( F i g .
10.5b), TRANSITION STRIP
AT
'/<
.I
i
t h i s s h i f t i n shack p o s i t i o n causes a p r e s s u r e peak on t h e f r o n t p a r t o f t h e a i r f o i l , which i s o f the same t y p e as t h a t observed e a r l i e r on t h e a i r f o i l w i t h f l a p ( s e c t i o n 8.2.2).
The lower
s u r f a c e o f the a i r f o i l remains subsonic, and t h e quasi-steady pressure d i s t r i b u t i o n a l o n g t h i s s i d e
i s p r e d i c t e d reasonably w e l l w i t h t h i n - a i r f o i l t h e o r y . T h i s confirms t h e o b s e r v a t i o n s f o r t h e a i r f o i l w i t h f l a p ( s e c t i o n 8.4.1)
t h a t t h e un-
steady p r e s s u r e d i s t r i b u t i o n s a t t h e upper and
O
t h e lower s u r f a c e a r e determined p r i m a r i l y by t h e i r r e s p e c t i v e steady f l o w f i e l d s ,
independent
o f t h e steady f l o w a t the o t h e r s u r f a c e . Unsteady p r e s s u r e d i s t r i b u t i o n s f o r cond i t i o n 1 1 - 1 a r e g i v e n i n f i g u r e 10.6 f o r IO and
80 Hz, r e s p e c t i v e l y . S i n c e t h e r e s u l t s f o r the lower surface behave as usual
i n subsonic f l o w ,
they have been o m i t t e d , The p r e s s u r e d i s t r i b u t i o n s o f f i g u r e 10.6 show about the same c h a r a c t e r i s t i c s as those i n quasi-steady f l o w , and one recognizes e a s i l y the p r e s s u r e peak on t h e f r o n t p a r t o f t h e a i r f o i l , which i s caused by p e r i o d i c a l shock-wave mo t ion.
3 UPPER SURFACE +ACp
0 LOWER SURFACE --KO
C o n d i t i o n 11-2:
flow w i t h a well-developed shock wave
A s the next example o f an unsteady t r a n s o n i c f l o w w i t h shock wave, o s c i l l a t i o n s o f the a i r f o i l about c o n d i t i o n 11-2 ( F i g . be considered.
10.2) w i l l
I n t h i s c o n d i t i o n , t h e upper sur-
face c a r r i e s a supersonic r e g i o n t h a t extends t o about 50 per cent o f t h e chord and i s t e r m i n a t e d by a r a t h e r s t r o n g shock wave. A s shown i n f i g u r e 10.7a, a change i n i n c i d e n c e o f I degree r e s u l t s i n a s h i f t o f the steady shock p o s i t i o n o f about I O p e r c e n t o f the chord. The f l o w a l o n g t h e lower
i
s u r f a c e remains s u b c r i t i c a l .
?Y
From the corresponding quasi-steady pres-
sure d i s t r i b u t i o n s ( F i g . 10.7b), i t can be deduced
Fig. 10.7
t h a t a l o n g the upper surface t h e p r e s s u r e i s domin a t e d by the e f f e c t o f t h e shock displacement
-73-
Steady and quasi-steady pressure distributions in a transonic flow with well-developed shock wave (condition 11-2).
S i m i i a r t o the NACA 64A006 a i r f o i l ( F i g s . 8.7 and
d i t i o n , p r e f e r e n c e i s g i v e n t o t h e data o b t a i n e d
8 . 8 ) , a h ? g h - p r e s s u r e peak
w i t h nacurai t r a n s i t i o n , s i n c e the f l o w i n t h i s
i 5
generated, which o f
course cannot be p r e d i c t e d by t h i n - a i r f o i l
c o n d i t i o n i s found t o be i n f l u e n c e d i n c o n v e n i e n t l y
theory.
by the presence o f the t r a n s i t i o n s t r i p ( a d i s c u s -
A s b e f o r e , the q u a s i - s t e a d y p r e s s u r e d i s t r i b u t i o n on the subsonic lower s u r f a c e i s p r e d i c t e d reason-
s i o n on t h e e f f e c t o f the t r a n s i t i o n s t r i p w i l l be
a b l y wel I .
g i v e n i n s e c t i o n 10.5).
Pressure d i s t r i b u t i o n s on t h e upper s u r f a c e
As shown i n f i g u r e 10.10a, a v a r i a t i o n i n
f o r t h e unsteady case a r e presented i n f i g u r e 10.8,
i n c i d e n c e o f 0.5 degrees about t h e design p o i n t
f o r t h r e e d i f f e r e n t f r e q u e n c i e s . These r e s u l t s
leads t o a c o n s i d e r a b l e change i n the steady pres-
a l s o show t h e dominant e f f e c t o f t h e p r e s s u r e peak
s u r e d i s t r i b u t i o n a l o n g t h e upper surface,
due t o the moving shock wave.
t i c u l a r i n the supersonic region. which ranges
I t i s noted t h a t
i n par-
t h i s p r e s s u r e peak s h i f t s from t h e r e a l p a r t o f
from about 3 p e r c e n t t o 65 per cent o f t h e chord.
t h e p r e s s u r e d i s t r i b u t i o n t o t h e imaginary p a r t
Furthermore, v a r i a t i o n s o f 0 . 5 degree appear t o
w i t h i n c r e a s i n g frequency. T h i s i s caused by the
be s u f f i c i e n t t o g e n e r a t e a shock wave a t about
increased phase l a g o f the p e r i o d i c a l shock m o t i o n
65 per cent o f t h e chord. A t t h e lower surface,
r e l a t i v e t o the m o t i o n o f t h e a i r f o i l , a phenom-
t h e steady pressure d i s t r i b u t i o n changes r e g u l a r l y .
enon t h a t was observed e a r l i e r f o r t h e a i r f o i l
Noteworthy i s t h a t here the v e l o c i t y becomes
w i t h f l a p (see f i g u r e s 8.23 and 8.241,
s l i g h t l y s u p e r c r i t i c a l , b u t s t i l l w i t h o u t shock
and t h a t
w i l l be discussed i n some more d e t a i l i n s e c t i o n
format ion
NLR 7101 AIRFOIL
10.4.
Mp
-o,,
DOSI'
30 .O.P
A IN-SIIU TRANSDUCERS o PRESSURE TUBES
F u r t h e r , by r e p r e s e n t i n g t h e unsteady press u r e d i s t r i b u t i o n s i n terms o f magnitude and phase angle (Fig, i O . 9 ) ,
:.i?":
i t can be shown t h a t the w i d t h
i; 026
and t h e h e i g h t o f t h e p r e s s u r e peak a s s o c i a t e d w i t h t h e shock wave decrease a s t h e Frequency i s
O ',C
increased. T h i s i s caused by t h e decrease o f the
THIN-AIRFOIL
i
a m p l i t u d e o f t h e shock m o t i o n w i t h i n c r e a s i n g frequency. Concerning t h e phase curves i n f i g u r e
10.9,
._
i t can be noted t h a t t h e measurements show
'-60H: 144
a jump o f about 180 degrees j u s t downstream o f t h e
A
mean p o s i t i o n o f the shock wave. T h i s jump i s p r e s e n t a l r e a d y i n q u a s i - s t e a d y f l o w and, thus,
is
b
o .
n o t a dynamic e f f e c t . From t h e comparisons o f t h e measured pres-
s u r e d i s t r i b u t i o n s w i t h t h e d i s t r i b u t i o n s CalCulated with thin-airfoil
t h e o r y ( F i g s . 10.8 and i 0 . 9 ) ,
i t i s e v i d e n t t h a t , as f a r as t h e upper surface i s concerned,
t h i s theory i s not applicable.
10.2.3 The "shock-free"
design condition
(condition I l l )
O f s p e c i a l i n t e r e s t i s t h e unsteady beh a v i o u r o f the a i r f o i l i n i t s "shock-free" c o n d i t i o n ( c o n d i t i o n I l l o f f i g u r e 10.2).
design
Fig. 10.8
In t h e
d i s c u s s i o n o f the r e s u l t s f o r t h i s s p e c i f i c con-
-74-
Development of the unsteady p r e s s u r e distributions on the upper Surface with frequency in transonic flow with shock wave (condition 11-2).
r e g i o n . Probably t h i s wide b u l g e i s c h a r a c t e r -
NLR 7101 AIRFOIL UPPER SURFACE
Mm~070 MAGNITUDE
p\
I JCP I i
11
'1
5OC
"
.ko;lo
1,>~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<. ,102
/I
- THIN-AIRFOIL
r e g i o n o f supersonic f l o w . When t h e quasi-steady d i s t r i b u t i o n o b t a i n e d
THEORY
from the measurements i s compared w i t h the curve determined w i t h t h i n - a i r f o i l
theory,
i t appears
t h a t f o r t h e upper s u r f a c e t h e p r e d i c t i o n i s comp l e t e l y useless. For t h e lower s u r f a c e , where the flow i s o n l y s l i g h t l y s u p e r c r i t i c a l ,
the d i f f e r -
ence between t h e o r y and experiment i s c o n s i d e r a b l y s m a l l e r . Here a small b u l g e i n t h e measured data between 20 and 50 p e r cent o f t h e chord i s found, which i s s i m i l a r t o t h e one observed on the a i r f o i l w i t h f l a p , and which a l s o o c c u r r e d when the flow l o c a l l y became s l i g h t l y s u p e r c r ' i t i c a l ( s e e f o r instance f i g u r e 8.5). A s e r i e s o f f u l l y unsteady pressure d i s t r i b u t i o n s a l o n g the upper surface i n terms o f magnitude and phase a n g l e PHASE A N G L E 0
r------------
i5
g i v e n i n f i g u r e 10.11.
The magnitude curves c l e a r l y e x i i i b i t the l a r g e c o n t r i b u t i o n s a s s o c i a t e d w i t h the changes i n the shape o f t h e p r e s s u r e d i s t r i b u t i o n i n the supers o n i c r e g i o n on the f r o n t p a r t o f the a i r f o i l .
In
a d d i t i o n , a small peak occurs a t about 6 5 per cent o f t h e chord, which i s caused by the p e r i o d i c a l f o r m a t i o n of a weak shock wave i n t h i s r e g i o n (see f i g u r e 10.12). W i t h i n c r e a s i n g frequency the bulge on t h e f r o n t p a r t decreases, and t h e unsteady
pressure d i s t r i b u t i o n shows a tendency t o change i n a d i r e c t i o n towards the p r e s s u r e d i s t r i b u t i o n s found f o r f l o w c o n d i t i o n 11-2. The phase curves i n f i g u r e 10.11 behave v e r y r e g u l a r l y o v e r the f i r s t
60 per c e n t o f the chord. Then a jump i n phase angle o f about 180 degrees occurs, which i s due t o t h e presence o f t h e shock wave. F i n a l l y , a comparison o f t h e measured unsteady p r e s s u r e d i s t r i b u t i o n s w i t h t h i n - a i r f o i l t h e o r y c o n f i r m s what c o u l d be concluded a l r e a d y
Fig. 10.9
on t h e b a s i s of t h e quasi-steady data: one has t o
E f f e c t of shock wave on the unsteady pressure distributions (condition 1 1 - 2 ) .
a p p l y o t h e r p r e d i c t i o n methods f o r these types o f mixed flows.
The changes i n steady p r e s s u r e d i s t r i b u t i o n
10.3
r e s u l t i n a quasi-steady d i s t r i b u t i o n as g i v e n i n
UNSTEADY AERODYNAMIC COEFFICIENTS
f i g u r e 10.iOb. A wide bulge occurs on t h e upper s u r f a c e , which i s caused by t h e above-mentioned
The unsteady aerodynamic c o e f f i c i e n t s ob-
change i n p r e s s u r e d i s t r i b u t i o n i n t h e supersonic
t a i n e d by chordwise i n t e g r a t i o n o f the measured
-75-
-9L -
r-
11
I
CONDITION
1
\
I
M- -0.744
II
MAXIMUM u
INCIDENCE UP
19 TIME I DOWN
m
MINIMUM o
Fig. 10.12
Time history o f the periodical shock-wave motion (condition 111). HLR 7101 AIRFOIL
’-- 0.70 - 0.89
iuro.S’
,io
o
TRANSITION STRIP AT r / c . WATURAL TRANSITION
Rs
1.7
I
oI
.I
,z
.I
THEORY
i ~
TIIEORI
,P’ >-a-- .z
NATURAL TRANSITION
IOb
- ’,t
‘1
TRANSITIOY STRIP AT x,: ~ . I
.i
,THEORY
.z
3 i
L
i
/
,+?”
.I,
.I I
-.zL
,
/’
41
, k
-.2L Fig. 10.13
Unsteady normal-force and moment Coefficients as a function of frequency in subsonic f l o w (condition I).
Fig. 10.14
-77-
Unsteady normal-force and moment coefficients as a function of frequency in slightly supercritical flow (condition 11-1).
2
3AVfi ä30HS 3 H l 30
NOIIOW
3Hl
NO SIöWW3ä
-8L-
$'O1
-i H I N 4 R F I I L iHEORI --- EXPERIUENT
&
.EXPERIMENT *ITHOUT ESTIYATEOl EFFECT OF SMOCK " A V E
i:
?\
'"I
li
'2
EFFECT OF PRESSURE PEAK DUE 10 MOVING SHOCK WAVE SHIFT IN AEROOYNAMIC CENTRE
~
~
'..... \
i i
Fig. 10.17
Qualitative explanation of the effect of the oscillating shock wave on the unsteady aeTodynamic coefficients.
Fig. 10.18
Periodical shock-wave motion in slightly wperciitical flow (condition 11-1).
p e r i o d i c a l m o t i o n o f t h e shock wave i n c o n d i t i o n s 11-1
and 11-2.
F i g u r e 10.18 g i v e s some r e s u l t s f o r
the s l i g h t l y s u p e r c r i t i c a i c o n d i t i o n 1 1 - 1 .
In b o t h
t h e quasi-steady 3nd the unsteady case, the shock wave vanishes d u r i n g a p a r t o f the c y c l e . unsteady case,
In the
the t r a j e c t o r y along which t h i s
happens i s s h i f t e d t o t h a t p a r t o f the motion where t h e shock moves i n downstream d i r e c t i o n . E v i d e n t l y , t h i s i s caused by the dynamic e f f e c t o f the shock motion on t h e s t r e n g t h o f the shock, which r e s u l t s i n a n i n c r e a s e i n shock s t r e n g t h
SHOCK POSIT!ON
d u r i n g the upstream m o t i o n and a decrease i n s t r e n g t h d u r i n g t h e downstream motion. The unsteady shock m o t i o n o f f i g u r e 10.18
i s o f type B ,
d e s c r i b e d i n c h a p t e r 9 . The f l o w c o n d i t i o n s a r e such, however,
i
t h a t f o r h i g h e r frequencies and
M PHASE SHIFT-50'
l a r g e r amp1 i tudes o f t h e a i r f o i I motion shockwave motions o f t y p e
C (upstream-propagating
waves) can be expected.
u From t h e t i m e h i s t o r i e s o f t h e shock d i s -
.30
placement f o r t h e t r a n s o n i c - f l o w c o n d i t i o n 11-2 ( F i g s . 10.19 and 10.20),
Fig. 10.19
it follows that in t h i s
c o n d i t i o n t h e shock wave performs almost s i n u s o i d a l motions, s i m i l a r t o t h e type-A m o t i o n t e r 9 . A s shown i n f i g u r e 10.19,
in
chap-
PHASE SHIFT-SO'
Influence Of the amplitude o f the airfoil oscillation o n Shock-wave motion (condition 11-2).
r e s u l t s i n f i g u r e 10.20,
the a m p l i t u d e o f
which g i v e s the shock
t r a j e c t o r i e s f o r d i f f e r e n t frequencies,
i t can be
t h e shock m o t i o n i s almost p r o p o r t i o n a l t o the
observed t h a t t h e phase l a g o f the shock m o t i o n
amplitude of o s c i l l a t i o n of the a i r f o i l .
r e i a t i v e t o t h e a i r f o i l motion imcreases w i t h f r e -
From t h e
-79-
quency, w h i l e the a m p i i t u d e o f the motion de-
supersonic r e g i o n t e r m i n a t e d b y a shock wave,
creases. The l a t t e r corresponds v e r y wei1 w i t h
because i t i s t h e time p e r i o d a f t e r which major
t h e o b s e r v a t i o n mentioned e a r l i e r concerning the
changes i n f l o w c o n d i t i o n , namely changes i n f l o w
c o n t r i b u t i o n o f t h e moving shock wave t o t h e un-
d i r e c t i o n a t t h e t r a i l i n g edge ( K u t t a c o n d i t i o n ) ,
steady p r e s s u r e d i s t r i b u t i o n ( s e e f i g u r e s 10.8
can be f e l t by the shock wave ( F i g . 10.22).
and 10.9).
The t i m e r e q u i r e d by a s i g n a l t o t r a v e l
A c l o s e r examination o f t h e phase l a g o f
from t h e t r a i l i n g edge t o t h e shock wave i s
the shock m o t i o n r e l a t i v e t o t h e a i r f o i l m o t i o n (Fig.
10.21)
l e a r n s t h a t an almost l i n e a r r e l a -
t i o n s h i p i s found between frequency and phase lag.
A t =-
I n o t h e r words, t h e r e i s a c o n s t a n t t i m e l a g
is
x=c
dx
TI -
Ml o c )
a
loc
(10.2)
'
between t h e motion o f the a i r f o i l and t h e shockwave motion. T h i s corresponds w i t h t h e r e s u l t o f
w i t h Mioc
b e i n g t h e l o c a l Mach number and a
the
E r i c k s o n and Stephenson (Ref. 103), who observed
Ioc l o c a l v e l o c i t y o f sound. Due t o t h e g r a d i e n t i n
a f i x e d r e l a t i o n between t h e phase l a g o f t h e
Mach number normal t o t h e a i r f o i l surface,
shock m o t i o n and the t i m e r e q u i r e d f o r a p r e s s u r e
a ~ ~ ~ swaves t i c propagate a l o n g paths away from t h e
impulse t o t r a v e l from t h e t r a i l i n g edge t o t h e
airfoil.
shock wave. T h i s t r a v e l l i n g t i m e indeed seems t o
upstream d i r e c t i o n w i l l be some average between
be a l o g i c a l parameter f o r an a i r f o i l w i t h a l a r g e
Therefore,
the
t h e p r o p a g a t i o n speed i n
t h e v a l u e o f (l-Mloc)aloc
near t h e a i r f o i l s u r f a c e
and t h e free-stream value. To account for t h i s the f o l l o w i n g v a l u e o f t h e l o c a l Mach num-
effect,
ber i s introduced:
the surface)
= R [Mloc(at
Mloc
-
M-
1+
M-,
NLR 7x1 AIRFOIL U m = O 0 . 7 >.-I'~ii.i.J'
PHASE LAGOF SHOCK-WAVE MOT!ON
.
'
4
m /
/
/
/
m
/
/
(10.3)
EXPERIMENT
R;O.l
/
PREDICTED WITH EWATION 110.2!
1D
U
FREQUENCY
AMPLITUDE OF SHOCK-HAVE WTiON
20
4
U
80
HZ
FREQUENCY
F i g . 10.20
P e r i o d i c a l shock-wave m o t i o n f o r v a r i o u s frequencies ( c o n d i t i o n 11-2).
F i g . 10.21
-80-
E f f e c t of frequency an t h e amplitude and phase l a g of t h e p e r i o d i c a l shockwave m o t i o n ( c o n d i t i o n 11-2).
t o the steady aerodynamicists. The f i n a l c h o i c e f o r the l o c a t i o n o f the t r a n s i t i o n s t r i p a t 30 per
cent o f t h e chord was made f o r t h r e e reasons:
-
t o suppress t h e l a r g e chordwise s h i f t i n t r a n s i t i o n p o i n t a t the upper surface o f the a i r f o i i , which had been observed (ReF. 73) a t the design Mach number f o r incidence between - 1 and +1 de-
c
gree (Fig.
Fig. 10.22
-
The propagation of information from the trailing edge to the Shock wave.
10.23a).
t o a v o i d a t t h e lower surface a boundary l a y e r t h a t remains laminar t o about 60 per cent o f the
where R denotes a r e l a x a t i o n f a c t o r ,
which has a
chord ( F i g .
-
value between O and 1.0. For R Z 0.7,
10.23b), and
t o be c o n s i s t e n t w i t h t h e p r e v i o u s steady wind-
an e x c e l l e n t agreement i s o b t a i n e d
tunnel i n v e s t i g a t i o n s on t h e NLR 7301 a i r f o i l .
between t h e t r a v e l I i n g t i m e o f t h e "Kutta waves" and the c o r r e s p o n d i n g phase l a g o f t h e motion, as
The e f f e c t o f t h e t r a n s i t i o n s t r i p was
i s demonstrated i n f i g u r e 10.21.
10.5
checked by r e p e a t i n g a small p a r t o f the unsteady
THE INFLUENCE OF THE T R A N S I T I O N STRIP
The p r e s e n t i n v e s t i g a t i o n s were conducted i n a m a l l atmospheric wind tunnel and, as a consequence, o n l y r e l a t i v e l y small values o f the Reynolds number c o u l d be a t t a i n e d (about 2 . 2 ~I O 6 , t o be compared w i t h 30 t o 50 x IO6 f o r f u l l - s c a l e flight).
In p r a c t i c e t h i s means t h a t i n t h e wind
tunnel the t r a n s i t i o n o f the boundary l a y e r from
Li
laminar t o t u r b u l e n t takes p l a c e f a r more down-
I -2
stream than i n free f l i g h t , Moreover, t h e . i o c a t i o n
.I
DEIIGW
i
I
o
I
I
2
50
'2,
of the t r a n s i t i o n p o i n t may be v e r y s e n s i t i v e t o
a
changes i n t h e free-stream Mach number or i n c i -
Variation in transition point with incidence at the design Mach number,
.
dence. T h i s m i g h t have l e d t o two unwanted e f f e c t s , one caused by t h e p e r i o d i c a l l y s h i f t i n g t r a n s i t i o n
DESIGN INCIDENCE
it,-.8P
t
p o i n t ( s e e , f o r instance, r e f e r e n c e l i s ) , and t h e o t h e r by t h e i n t e r a c t i o n of a shock wave w i t h a laminar r a t h e r than w i t h a t u r b u l e n t boundary l a y e r . As discussed i n s e c t i o n 3.4
( F i g . 3.8),
the
i a t t e r leads t o a lambda-type shock wave, which
I
i 5
n o t present i n Free f l i g h t .
To a v o i d these e f f e c t s ,
i t was decided t o
Force t r a n s i t i o n a r t i f i c i a l l y by means o f a rough-
UPPER
ness s t r i p . T h i s c r e a t e d , however, a dilemma about t h e chordwise p o s i t i o n o f t h e s t r i p ,
~
%ESIGII
I .2
especially
IURFACE
I .I
I
I .i
I
I
.b
7
t
1
.e
UoI
s i n c e t h e q u e s t i o n about an adequate s i m u l a t i o n o f
b
high-Reynolds-number e f f e c t s on r e a r - l o a d e d a i r f o i l s by means o f a r t i f i c i a l t r a n s i t i o n i s s t i l l
Variation in transition point with Mach number at the d e s i p incidence.
Fig. 10.23
q u i t e open and forms a source o f c o n s t a n t concern
-81-
Location of natural transition point (from Ref. 73).
HLR 7101 AIRFOIL o I R A H I I T I O N iTRiP A T D
./c.
( c o n d i t i o n I ) , t h e e f f e c t a f the s t r i p on t h e
1
NATURAL TRAMSITION
unsteady pressure d i s t r i b u t i o n s and the c o r r e spond ing unsteady aerodynamic c o e f f i c i e n t s ( F i g . 10.13)
i s n e g l i g i b l e . A i s 0 f o r the " c l a s s i c a l "
t r a n s o n i c flow c o n d i t i o n s 1 1 - 1 and l i - 2 ,
JC ;
the
effects are very small. This i s not s u r p r i s i n g , s i n c e , as f a r as t h e upper s u r f a c e i s concerned, the n a t u r a l t r a n s i t i o n i n c o n d i t i o n 1 1 - 1
takes
p l a c e upstream o f t h e l o c a t i o n o f t h e t r a n s i t i o n s t r i p ( F i g . 10.23b), w h i l e f o r c o n d i t i o n 11-2 n a t u r a l t r a n s i t i o n o c c u r s a t about t h e same l o c a t i o n as t h e s t r i p .
In c o n t r a s t herewith, t h e f l o w i n the d e s i g n c o n d i t i o n appears t o be v e r y s e n s i t i v e t o t h e presence o f t h e t r a n s i t i o n s t r i p ( F i g s .
10.24 and
10.25). The a d d i t i o n o f t h e s t r i p causes a d r a s t i c increase i n unsteady pressures between 30 and 60 per cent o f t h e chord, w h i l e the weak shock wave i s s h i f t e d t o a more upstream p o s i t i o n . T h i s l a t t e r e f f e c t can be i n f e r r e d v e r y w e l l from t h e i o c a t i o n o f t h e "180 degrees" jump i n the phase curves o f f i g u r e 10.25.
The d i f f e r e n c e s a r e caused
by the o b l i q u e compression wave, which emanates from t h e t r a n s i t i o n s t r i p and i s r e f l e c t e d as an expansion wave from t h e s o n i c l i n e (see f i g u r e 10.26). When t h e a i r f o i l o s c i l l a t e s , t h e l o c a t i o n o f t h e o b l i q u e compression wave AB and the s o n i c l i n e change p e r i o d i c a l l y . As a r e s u l t , the l o c a l c i o n o f t h e p o i n t s where the successive expansion waves impinge on t h e a i r f o i l a l s o changes p e r i o d -
t
i c a l l y . T h i s leads t o t h e complicated f l o w p a t t e r n s shown i n t h e shadowgraph p i c t u r e s o f f i g u r e 10.27.
A comparison o f these p i c t u r e s w i t h
t h e ones f o r n a t u r a l t r a n s i t i o n ( F i g . 10.12) c l e a r l y shows t h a t t h e d i s t u r b a n c e o f the f l o w f i e l d by t h e t r a n s i t i o n s t r i p i s an improper element i n the d e l i c a t e c o n d i t i o n o f "shock-free" flow.
I n the a n a l y s i s o f t h e r e s u l t s f o r the
18shock-freeoLdesign c o n d i t i o n ( s e c t i o n 10.2.3), p r e f e r e n c e t h e r e f o r e was g i v e n t o t h e data ob-
Fig. 10.24
t a i n e d d u r i n g t h e wind-tunnel
E f f e c t of t r a n s i t i o n s t r i p on t h e unsteady p r e s s u r e d i s t r i b u t i o n s ( f = 80 Hz, Au i 0 . 5 0 ) .
transition,
tests with natural
be i t a t t h e c o s t of an unknown e f f e c t
caused by p o s s i b l e l a r g e changes i n l o c a t i o n o f the t r a n s i t i o n Doint.
t e s t s w i t h n a t u r a l t r a n s i t i o n . The r e s u i t s o f these a d d i t i o n a l t e s t s can be summarized as f o l lows (see f i g u r e 10.24): A t low subsonic speed
-82-
10.6
SOME ADDITIONAL EFFECTS
10.6.1 The e f f e c t o f Mach number
The i n f l u e n c e o f t h e free-stream Mach numb e r on t h e unsteady n o r m a l - f o r c e c o e f f i c i e n t i l l u s t r a t e d i n f i g u r e 10.28. t h e d e s i g n incidence, a. (Fig.
=
i0.28a) a h i g h peak
I t appears t h a t f o r
0.85', i 5
is
a t low frequency
found i n the magni-
tude o f ka, which c o i n c i d e s w i t h t h e design Mach number. A t h i g h e r frequencies ( F i g . i0.28b),
this
peak decreases c o n s i d e r a b l y , which i s i n accordance w i t h t h e development of t h e unsteady pressure d i s t r i b u t i o n s shown p r e v i o u s l y ( F i g .
i0.li).
The n o r m a l - f o r c e c o e f f i c i e n t f o r an incidence o f
3 degrees a l s o shows a peak v a l u e , b u t now a t a lower Mach number, a t which a r e l a t i v e l y strong o s c i l l a t i n g shock wave i s p r e s e n t . An increase i n frequency,
l e a d i n g t o a decrease i n the amplitude
o f t h e shock motion, has l e s s e f f e c t on the peak v a l u e of t h e normal f a r c e i n t h i s case than i n t h e 'lshock-free" Finally,
fiow condition.
i t can be observed from f i g u r e 10.28
t h a t , a t l e a s t a t low frequency,
the t r a n s o n i c
e f f e c t s on t h e unsteady a i r l o a d o f the a i r f o i l i n P H A S E ANGLE
I
"
i t s design c o n d i t i o n a r e o f the same o r d e r o f
p-----O I
magnitude as those o c c u r r i n g i n the " c l a s s i c a l "
e
*
t r a n s o n i c f l o w c o n d i t i o n s w i t h shock wave f o r O
c L o = 3 .
10.6.2
The e f f e c t o f the amplitude o f o s c i l l a t i o n
I n r e l a t i o n t o the p o s s i b i l i t y o f I i n e a r ization,
i t i s an important q u e s t i o n whether 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 amplitude o f t h e a i r f o i l motion and the r e s u l t i n g unsteady aerodynamic loading. For t h i s r e a s o h a number o f a d d i t i o n a l t e s t s was performed, i n which the amplitude o f t h e model o s c i l l a t i o n was v a r i e d . Because i n t h i s respect t h e p o s s i b i l i t i e s o f the t e s t set-up were l i m i t e d , o n l y amplitudes up t o i deg r e e c o u l d be r e a l i z e d . F i g . 10.25
The r e s u l t s o f these a d d i t i o n a l t e s t s (as an
I n f l u e n c e o f t r a n s i t i o n s t r i p on t h e unsteady p r e s s u r e d i s t r i b u t i o n for o s c i l l a t i o n s around t h e d e s i g n p o i n t ( c o n d i t i o n 111).
example, the d a t a f o r the maximum frequency o f
80 Hz a r e summarized i n t a b l e 5 ) show no system-
-83-
t e s t d a t a o f the NLR 7301 a i r f o i l
i s summarized,
i t can be concluded t h a t the r e s u l t s f o r h i g i i -
subsonic and t r a n s o n i c f l o w w i t h normal shock waves c o n f i r m very w e l l the f i n d i n g s a f the experiment on t h e NACA 64A006 a i r f o i l w i t h f l a p . I n high-subsonic flow,
a reasonable agreement
w i t h the r e s u l t s o f t h i n - a i r f o i l
theory exists.
For t r a n s o n i c f l o w w i t h a shock wave, t h e p e r i o d -
i c a l motion o f t h e shock g i v e s r i s e t o h i g h local loads, which determine the o v e r a l l unsteady a i r TRANSITION STRIP
Fig. 10.26
loads t o a l a r g e e x t e n t . The c o n t r i b u t i o n o f the
Sketch showing effect of transition strip (condition 111).
o s c i l l a t i n g shock wave t o the l o a d i n g d i m i n i s h e s a s the frequency i s increased, which i s the r e s u i t
a t i c changes, which can be i n t e r p r e t e d as a con-
o f t h e decrease i n amplitude o f t h e shock motion.
s i s t e n t e f f e c t o f amplitude. Only small v a r i a t i o n s
I t can, thus, be expected t h a t f o r v e r y h i g h Fre-
i n t h e o v e r a l l c o e f f i c i e n t s a r e found, which F a l l
quencies the motion o f the shock wave w i l l become
c o m p l e t e l y w i t h i n t h e accuracy o f t h e t e s t s , t o be
l e s s important f o r t h e unsteady a i r l o a d s . F o r the
e s t i m a t e d a t about 5 t o 10 p e r cent i n magnitude
present a i r f o i l ,
and 3 t o 6 degrees i n phase angle.
quencies w e l l above the range o f p r a c t i c a i i n t e r -
however, t h i s w i l i happen a t Fre-
est f o r F l u t t e r investigations.
10.7
A new Feature i s observed For o s c i l l a t i o n s
CONCLUDING REMARKS
o f the a i r f o i l about
i t 5
"shock-free"
design :on-
d i t i o n . Then t h e unsteady pressure d i s t r i b u t i o n i s
When the outcome o f t h e a n a l y s i s o f t h e
CONDITION El
I
1
ru Fig. 10.27
II
MAXIMUM (I
INCiDENCE
LY
MINIMUM n
Time histgry of the periodical shock-wave motion showing effect of transition Strip (condition 111).
-84-
Fig. 10.28
Influence of Mach number on t h e magnitude of t h e normal-force coefficient. TABLE 5
E f f e c t o f a m p l i t u d e o f o s c i l l a t i o n on the unsteady aerodynamic c o e f f i c i e n t s
0.5
0.85'
.44O
.263
0.7
0.85'
0,745
O . 745
3O
0.85'
0.85O
0.4'
.32
73'
2.1
75 76 76 78
.77
1.45 1.35
2.0
.94
1.37
2.5
.33
.25O
4.7
1.62
-5.9'
.39
78'
.60
1.43
.70
1.52 1.54
-6.7 -4.3
.36 .40
80 84
-2.2
.43
82
.192
.87 0.7
1.36 1.36
.33 .36 .32
.55 .76
1.83
-32.7'
.23
23'
.58
1.76
-33,7
.21
27
.70 .83
1.86 1.72
-34.7 -33.9
.23 .22
21
1.64
-28.4'
.35
59'
.48
i.54
.52 .69 1.13
1.70 1.68 1.46
-27.4 -26.1
.35 .40
52 52
-26.9 -31.3
.37 .24
51 57
.55O
.27O
,132
,181
Condition 11-1
C o n d i t i o n 11-2
21 Condition I l l
NLR 7301 a i r f o i l Frequency: 80 Hz
I . 56
-22.9O
.33
55'
Condition I l l
.61
1.61
-21.3
.35
47
( N a t u r a l t r a n s it i o n )
.76
1.66
-23.9
.35
43
.56O
,181
Condition I
drminated by t h e l a r g e changes o c c u r r i n g i n t h e
low frequency,
supersonic r e g i o n on t h e f r o n t p a r t o f t h e a i r f o i l .
those observed f o r " c l a s s i c a l "
The r e s u l t i n g unsteady a i r i o a d s are, a t l e a s t a t
w i t h a w e l l - d e v e l o p e d shock wave.
-85-
o f t h e same o r d e r o f magnitude as t r a n s o n i c flovis
11 SOME CONSIDERATIONS ON A LINEARIZED TREATMENT OF UNSTEADY TRANSONIC FLOWS I I. I
INTRODUCTORY REMARKS
As o u t l i n e d i n c h a p t e r 4,
i t i s an impor-
t a n t q u e s t i o n t o what e x t e n t t h e l i n e a r r e l a t i o n s h i p between t h e unsteady aerodynamic loads and the a i r f o i l motion holds. This question i s o f i n t e r e s t , n o t o n l y w i t h r e s p e c t t o t h e develop-
t h e terms c o n t a i n i n g t h e reduced frequency k a r e
ment o f methods t o c a l c u l a t e t h e aerodynamic loads,
linear i n
m.
T h i s suggests t h a t w i t h i n c r e a s i n g k
b u t a l s o f o r t h e a p p l i c a t i o n o f t h e aerodynamic
t h e l i n e a r p a r t o f t h e e q u a t i o n becomes more
data i n a e r o e l a s t i c c a l c u l a t i o n s , since caicula-
dominating, w h i l e f o r low frequency and quasi-
t i o n methods f o r t h e s o l u t i o n o f n o n l i n e a r equa-
steady f l o w ( k =
t i o n s o f m o t i o n on a r o u t i n e b a s i s a r e n o t w e l l
i m p o r t a n t . O f course, an i m p o r t a n t q u e s t i o n i s
developed.
In t h i s chapter i t i s attempted t o
O) t h e n o n l i n e a r term i s more
a150 whether o r n o t the decrease o f t h e n o n l i n e a r
assess t h e p o s s i b i l i t y o f l i n e a r i z a t i o n i n a
c h a r a c t e r happens monotonously w i t h frequency.
p r a c t i c a l sense. For t h i s purpose i t i s supposed
d e f i n i t e c o n c l u s i o n i n t h i s r e s p e c t can be drawn
t h a t l i n e a r i z a t i o n o f t h e unsteady a i r l g a d s about
from t h e present experiments.
a g i v e n mean p o s i t i o n o f an a i r f o i l
i5
No
justified
as l o n g a s s m a l l , b u t s t i l l p r a c t i c a l changes i n
11.2
i n c i d e n c e o r f l a p a n g i e (say i n t h e o r d e r o f 0.5 and 1 degree,
FLOW CONDITIONS WITH AN OSCILLATING SHOCK WAVE
r e s p e c t i v e l y ) around t h i s mean p o s i -
t i o n g i v e r i s e t o l i n e a r l y d e v e l o p i n g changes i n the
flOW
field.
i n t h i s r e s p e c t i t i s a l s o o f in-
11.2.1
Locai e f f e c t s o f a shock wave
t e r e s t which procedure has t o be f o i l o w e d t o s e l e c t a minimum number o f s u i t a b l e mean steady
The l o c a l e f f e c t s i n t r o d u c e d by a shock
flow conditions.
wave can be i l l u s t r a t e d c o n v e n i e n t l y w i t h diagrams
When c o n s i d e r i n g t h e l i n e a r i t y o f t r a n s o n i c
i n which the steady pressures a t c o n s t a n t chord-
f l o w s , one can t h i n k o f t h e f o l l o w i n g f l o w condi-
w i s e s t a t i o n s a r e p l o t t e d as a f u n c t i o n o f i n c i -
t i o n s i n which l i n e a r i t y may be v i o l a t e d :
dence ( o r f l a p a n g l e ) . Diagrams o f t h i s type,
-
f l o w 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,
which c l e a r l y show the d i f f e r e n c e between a sub-
flow conditons,
s o n i c f l o w and a t r a n s o n i c f l o w w i t h shock wave,
condition,
l i k e t h e "shock-free"
design
a r e g i v e n i n f i g u r e s 1 1 . 1 and 1 1 . 2 ,
i n which a small change i n i n c i d e n c e
respectively.
In subsonic f l o w ( F i g . l l . l ) , t h e
i s f o l l o w e d by a r a t h e r a b r u p t change o f t h e flow pat tern.
curves o f
t h e l o c a l - p r e s s u r e c o e f f i c i e n t v a r y almost l i n e a r -
l y w i t h incidence, which i m p l i e s t h a t t h e i r slopes
In t h e subsequent s e c t i o n s , these condit i o n s w i l l be c o n s i d e r e d i n some more d e t a i l . The
dC /da and, thus, the quasi-steady pressures a r e P c o n s t a n t and independent o f t h e a c t u a l v a i u e o f
c o n s i d e r a t i o n s w i l l be m a i n l y based on the q u a s i -
t h e incidence. So i t can be concluded t h a t , a t
steady behaviour o f t h e a i r f o i l s used i n t h e
l e a s t as long as the f l o w remains a t t a c h e d ,
present investigation. A quasi-steady analysis
l i n e a r i z a t i o n i s no problem i n t h i s f l o w regime.
seems j u s t i f i e d , s i n c e i t may be expected t h a t
T h i s o b s e r v a t i m i s confirmed i n unsteady flow,
n o n l i n e a r e f f e c t s w i l l be l a r g e s t i n t h e q u a s i -
where t h e a c t u a l r e c o r d i n g s o f t h e pressures f e l t
steady case and a t low f r e q u e n c i e s . T h i s expecta-
by t h e i n - s i t u K u l i t e transducers i n t h e o s c i l -
t i o n i s based on the o b s e r v a t i o n t h a t i n t h e
l a t i n g model show almost s i n u s o i d a l l y v a r y i n g
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 e q u a t i o n (eq. 2.15 o f
s i g n a l s (see f i g u r e 2 . 2 ) .
chapter 2), r e w r i t t e n f o r sinusoidal perturbations
r e p r e s e n t i n g t h e unsteady l i f t and moment, behave
as:
i inearly.
-86-
A l s o t h e summed s i g n a l s ,
N L R 7101 A I R F O I L UPPER SURFACE)
PRESSURE COEFFICIENT C.
= 0.50 R e = 1.7 IO6 TRANSITION STRIP ' I C =.I Mm
PRESSURE COEFFICIENT
.
NLR 7301 AIRFOIL (UPPER SURFACE)
= 0.70 Re = 2.1 I 106 TRANSITION STRIP '/< Mm
-2.0
=
,
Cp
I -1.5
=.Is
-1.0 -0.50
! -0.5 -
x$
=.70
a
INCIDENCE
Fig. 11.2 -I
O
1
2
1
4
SO
Steady pressures at constant chordwise positions as a function o f incidence in a transonic flow with shock wave.
INCIDENCE U
Fig. 11.1
Steady pressures at constant chordwise positions as a function of incidence in subsonic flow.
The diagram f o r the t r a n s o n i c f l o w c o n d i t i o n CHORDWISE POSiTlON
CP
w i t h a shock ( F i g . 11.2) shows t h a t t h e passage
%
o f the shock wave across a f i x e d p o i n t on t h e a i r -
PASSAGE OF SHOCK
f o i l surface g i v e s r i s e t o a sudden jump i n t h e
\
curves o f the l o c a l p r e s s u r e a g a i n s t incidence. As indicated schematically i n f i g u r e 11.3, i m p l i e s t h a t a quasi-steady,
QUASI. STEADY PRESSURE SIGNAL
?.
1
this
sinusoidal v a r i a t i o n
i n i n c i d e n c e r e s u l t s no longer i n a s i n u s o i d a l pressure perturbation, but i n a pressure v a r i a t i o n e x h i b i t i n g p e r i o d i c a l jumps a t t h e i n s t a n t t h e shock wave passes. Quasi-steady pressures f o r a mean i n c i d e n c e o f 3 degrees and an a m p l i t u d e o f 0.5 degrees have been d e r i v e d from f i g u r e 11.2.
They a r e p l o t t e d as
a f u n c t i o n o f t i m e i n f i g u r e 11.4.
The passage of
the shock wave i s f e i t o n l y i n a l o c a l r e g i o n between 35 and SO p e r c e n t o f t h e chord.
Fig. 11.3
In spite
o f t h e s t r o n g n o n l i n e a r i t y o f t h e pressures i n
-87-
Determination of quasi-steady pressure signals (transonic flow with shock wave).
11.2.2
\
&xe INCIDENCE
C o n t r i b u t i o n o f a shock wave t o the o v e r a l l aerodynamic loads
,NLR M =7101 0.70AIRFOIL
The phenomenon t h a t t h e o s c i l l a t i n g shock
,>
wave leads t o an almost l i n e a r c o n t r i b u t i o n t o the o v e r a l l unsteady a i r l o a d s can be made p l a u s i b l e a s follow5.
;1Cp=.2
I n f l o w p a t t e r n s w i t h a well-developed
PRESSURES
shock wave,
the shock m o t i o n i s observed t o t a k e
'/c-.iO
p l a c e almost s i n u s o i d a l l y , and t h e a m p l i t u d e o f
.20
t h e shock motion appears t o be almost p r o p o r t i o n a l
.I5
t o the amplitude o f the sinusoidal motion of the
.40
a i r f o i l (see f i g u r e s 10.19 and 1 0 . 2 0 ) . T h i s makes i t p o s s i b l e t o i n t r o d u c e t h e schematic model o f
f i g u r e 11.5, .4s
i n which t h e change i n pressure a t a
f i x e d p o i n t A i s considered as generated by a s i n u s o i d a l shock-wave m o t i o n o f amplitude x 0 . As f o l l o w s from t h e c o n s i d e r a t i o n s i n chapter 9, t h e l o c a l shock s t r e n g t h when t h e shock passes a
.so
p o i n t A l o c a t e d w i t h i n t h e shock t r a j e c t o r y can be
.70
w r i t t e n as:
LIFT
MOMENT .Ob
F i g . 11.4
Histograms o f p r e s s u r e s and overall c o e f f i c i e n t s f o r a quasi-steady p i t c h motion.
Here Ap denotes t h e v a r i a t i o n i n shock s t r e n g t h
2 r , = ' ~TRAJECTORY ~ ~ ~ ~
t h i s r e g i o n , t h e c o r r e s p o n d i n g o v e r a l l l i f t and
M
moment v a r y almost s i n u s o i d a l l y . E v i d e n t l y , t h e n o n l i n e a r i t i e s i n t r o d u c e d by t h e shock wave have
only a l o c a l e f f e c t and do n o t i n f l u e n c e t h e overa l l loads. From t h e time h i s t o r i e s o f t h e p r e s s u r e s i g n a l s f o r the o s c i l l a t i n g model (see f i g u r e 2 . 4 ) , i t becomes apparent t h a t t h i s c o n c l u s i o n remains
v a l i d f o r u n s t e a d y ' f l o w . The unsteady p r e s s u r e s i g n a l s show s i m i l a r c h a r a c t e r i s t i c s as t h e q u a s i steady histograms, and a l s o t h e unsteady l i f t v a r i e s almost s i n u s o i d a l l y . The o v e r a l l moment shows i r r e g u l a r i t i e s , b u t i t s a m p l i t u d e i s v e r y small and has been s t r o n g l y a m p l i f i e d .
From these
r e s u l t s i t can be concluded t h a t i n t h e present example t h e o v e r a l l unsteady aerodynamic c o e f f i -
ASSUME0 SHOCK DISPLACEMENT
PRESSURE VARIATION IN POINT A ( p A l
c i e n t s behave l i n e a r l y f o r a m p l i t u d e s o f o s c i l l a t i o n up t o about a t l e a s t 0 . 5 degrees,
in spite of
t h e presence o f an o s c i l l a t i n g shock wave.
F i g . 11.5
C o n t r i b u t i o n af a p e r i o d i c a l moving shack *ave t o t h e p r e s s u r e s i g n a l i n a f i x e d observation point.
duringthe ,hoci<-wave motion. For s t r o n g shock
i s spread o u t over the shock t r a j e c t o r y a s a con-
amplitude m o t i o n s , t h e l a s t term
w a v e 5 and
sequence o f the o s c i l l a t o r y m o t i o n O F the shock
in the above expression can be d i s c a r d e d r e l a t i v e
wave. The d i s t r i b u t i o n o f the F o u r i e r component
t o (?,-?,) Kg. When the l o c a l pressure i n p o i n t A i s de-
w i t h t h e same frequency as t h e a i r f o i l motion has
s c r i b e d as a f u n c t i o n o f time, a b l o c k - t y p e s i g n a l
jump (pz-pI)xs.
a ma%imum v a l u e o f 2 / r i times t h e steady pressure I n t e g r a t i o n o f t h e v a r i o u s compo-
occurs ( s e e f i g u r e 11.5), o f which the F o u r i e r
nents o v e r the shock t r a j e c t o r y , t o o b t a i n the
decomposition y i e l d s :
c o n t r i b u t i o n s t o the o v e r a l l l i f t and moment,
p(x,~ =
[p,(x,) -(p2-
-
(p2-
arccosíxA/xo}
pl) X
[$
pl),
l e a r n s t h a t the l i f t c o n t a i n s only a c o n t r i b u -
1+
t i o n o f t h e fundamental frequency. The moment c o n t a i n s , however, a l s o a term l i k e
s
sin(
a r c c o s í x A / x o ~ } coswt
+
5 i; l
+
2 ZIT
sin{2 arccosíxA/xoÌ]
cos2wt+
sin13 a r c c o s i x A / x o ~ lc o s j u t +
Thus,
i t can be expected t h a t t h e e f f e c t s o f the
second harmonic f i r s t show up i n t h e unsteady
+-
4
moment and n o t i n t h e l i f t .
,
From t h e c o n s i d e r a t i o n s g i v e n above, (11.3)
it
a l s o f o l l o w s t h a t measuring t h e f i r s t F o u r i e r component o f t h e p r e s s u r e s i g n a l s , a5 i s done i n
where / x A
1 <x .
the present t e s t s v i a t h e t u b i n g system, g i v e s
O
The corresponding d i s t r i b u t i o n s o f t h e f i r s t F o u r i e r components a l o n g t h e t r a j e c t o r y o f the shock m o t i o n a r e shown i n f i g u r e 11.6.
The
d i s t r i b u t i o n o f t h e mean v a l u e i l l u s t r a t e s t h a t
a f t e r chordwise i n t e g r a t i o n a c o r r e c t value o f the unsteady l i f t . A s f a r as t h e moment i s concerned, the second harmonic of o r d e r x z cannot be d i s t i n O
g u i shed.
the p r e s s u r e jump as o c c u r r i n g i n steady f l o w
-
----7,.o
11.3
1.0
SPECIAL FLOW CONDITIONS
E1st
HARMONIC
I n t h e present i n v e s t i g a t i o n i t was found
OSCILLATING AIRFOIL
t h a t f l o w c o n d i t i o n s i n which a small change i n AIRFOIL AT REST'
i n c i d e n c e ( o r f l a p angle)
A----I
o
-I %o
1
-I
0
," 0
SHOCK TRAJECTORY
i s f o l l o w e d by a r a t h e r
a b r u p t change i n a s u b s t a n t i a l p a r t o f the flaw f i e l d can be t r a c e d e a s i l y by c o n s i d e r i n g t h e beh a v i o u r o f t h e steady aerodynamic c o e f f i c i e n t s .
In p a r t i c u l a r t h e behaviour o f t h e moment c o e f f i c i e n t appears t o be an e x c e l l e n t i n d i c a t o r f o r l i n e a r i t y , a s w i l l be i l l u s t r a t e d w i t h 3ome examples.
11.3.1
"Shock-free''
flow
F i g u r e 11.7 shows t h e s t e a d y - l i f t and
Fig. 11.6
D i s t r i b u t i o n of t h e f i r s t f o u r F o u r i e r components a f t h e unsteady a i r l o a d due to an oSCi11atOlY shock-wave motion.
moment c o e f f i c i e n t o f t h e NLR 7301 a i r f o i l a s a function of
i n c i d e n c e i n t h e neighbourhood o f t h e
"shock-free"
design condi t i o n . Besides the mea-
f i q u r e 10.23).
Q u a n t i t a t i v e l y , there i s a i a r g e
sured data, t h i s f i g u r e a150 c o n t a i n s the t h e o r e t -
discrepancy between t h e t h e o r e t i c a l and the exper-
i c a l curves c a l c u l a t e d w i t h t h e f i n i t e - d i f f e r e n c e
imentai r e s u l t s o f f i g u r e i i . 7 . T h i s discrepancy
method o f Bauer, Korn, and Garabedian (Ref.
should be a t t r i b u t e d t o the e f f e c t o f t h e boundary
ll6),
who i n c l u d e the e f f e c t o f t h i c k n e s s and i n c i d e n c e
l a y e r , which i n t h e t r a n s o n i c regime i s v e r y i m -
b u t n o t the viscous e f f e c t s .
p o r t a n t . ( A more d e t a i i e d d i s c u s s i o n of these
I t appears t h a t t h e r a t h e r a b r u p t change i n t h e
differences
p r e s s u r e d i s t r i b u t i o n around t h e design c o n d i t i o n
i s g i v e n i n chapter i 3 . )
I t i s o f i n t e r e s t t o note t h a t t h e s l o p e o f
i s r e f l e c t e d i n t h e c a l c u l a t e d moment c u r v e by a
t h e moment curves ( F i g . 11.7)
n o n l i n e a r behaviour between O and 2 degrees o f
a small range o f incidences near t h e "shock-free"
incidence. In c o n t r a s t h e r e w i t h , the c a l c u l a t e d
flow condition.
l i f t curve shows t h a t , f o r i n v i s c i d t h e o r y , a
cause a s e l f - e x c i t i n g o s c i l l a t i o n o f l i m i t e d am-
l i n e a r r e l a t i o n i s found between l i f t and i n c i -
p l i t u d e . Whether t h i s occurs i n r e a l i t y w i l l i a r g e -
dence.
l y depend on t h e e l a s t i c and i n e r t i a p r o p e r t i e s
I n o t h e r words, n o n l i n e a r i t i e s i n t h e f l o w
p a t t e r n do n o t show u p i n t h e quasi-steady coefficients.
lift
becomes n e g a t i v e i n
I n p r i n c i p l e , t h i s phenomenon may
o f t h e s p e c i f i c s t r u c t u r e under c o n s i d e r a t i o n .
E v i d e n t l y , t h e aerodynamic moment
Therefore, no c o n c l u s i o n can be i n f e r r e d from t h i s
i s much more s e n s i t i v e t o t h e d e t a i l s of t h e
w i t h o u t a proper ( n o n l i n e a r ) c a l c u l a t i o n o f t h e
p r e s s u r e d i s t r i b u t i o n than t h e o v e r a l l l i f t .
aeroelastic characteristics o f t h i s Structure.
In a q u a l i t a t i v e
sense one can say t h a t ,
Another u n c e r t a i n t y i n t h i s respect i s whether the
u n t i l f l o w s e p a r a t i o n occurs, t h e steady aerody-
n e g a t i v e s l o p e i n t h e measured moment curve
namic c o e f f i c i e n t s as measured w i t h a t r a n s i t i o n
remains p r e s e n t f o r f u l l - s c a l e values o f t h e
s t r i p show about t h e same c h a r a c t e r i s t i c s a s those
Reynolds number.
p r e d i c t e d by the theory. The d a t a o b t a i n e d w i t h
From t h e o b s e r v a t i o n s o f the quasi-steady
n a t u r a l t r a n s i t i o n show a much deeper l o c a l d i p ,
aerodynamic c o e f f i c i e n t s , i t can be concluded t h a t
e s p e c i a l l y i n the moment curve, which has t o be
a n o n l i n e a r behaviour o f the aerodynamic moment
a t t r i b u t e d t o a large extent t o the rapid varia-
can be expected f o r t h e p a r t i c u l a r c o n d i t i o n o f
t i o n i n t h e l o c a t i o n o f the t r a n s i t i o n p o i n t ( s e e
"shock-free''
flow.
For unsteady f l o w , t h i s behav-
i o u r i s confirmed by t h e time h i s t o r i e s o f the
NLR 7101 A.IRFDIL
l o c a l pressures and the corresponding aerodynamic CORRECTEDFOR WALL
loads recorded w h i l e the a i r f o i l was o s c i l l a t i n g (Fig.
11.8).
These r e c o r d i n g s indeed show t h a t
,,-. \ \
I
x
/
/
\
1st
0.9
J
li Fig. 11.7
o
i
ia
-
2
0
2
1
Steady lift and moment coefficients as B function af incidence at the design Mach number.
INCIDENCE
Fig. 11.8
UNSTEADY
PRESSURES
LIFT
UOUENT
Unsteady pressure signals for the
NLR 7301 airfoil oscillating around it8 design condition.
-90-
the p r e s s u r e s a r e n o n l i n e a r o v e r a s u b s t a n t i a l
NLR ,911 AIRFOIL
s e c t i o n o f the chord. A i s 0 the o v e r a l l moment
Mm -0.70
e x h i b i t s a n o n l i n e a r behaviour, w h i l e t h e r e s u l t +THEORY
i n g l i f t i s l e s s influenced.
(REF 1161
EXPERIYEHT.
-----
TRANIITIOH STRIP AT ‘/e = . I
--+-NATURAL TRANSITIOH
11.3.2
Re
Flow w i t h a double shock
-
2.1.
ioh
C.
A second example o f a n o n l i n e a r i t y can be t r a c e d from t h e behaviour of t h e aerodynamic coeff i c i e n t s f o r t h e NLR 7301 a i r f o i l a t (Fig.
11.9).
M- = 0.7
Again i t appears t h a t i n t h e o r y 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 l i f t and i n c i dence, .independent of t h e d e t a i l s of the f l o w o r t h e presence o f shock waves. The c a l c u l a t e d aerodynamic moment shows a r e l a t i v e l y s t r o n g change i n s l o p e i n a small range o f i n c i d e n c e s between 2 and
*
I
3 degrees. T h i s r a p i d change i n s l o p e i s asso-
-
2
0
2
4
u
c i a t e d w i t h a d r a s t i c change i n f l o w p a t t e r n ; a t LI
= 2.5 degrees,a
Fig. 11.9
f l o w p a t t e r n w i t h even two shock
waves behind each o t h e r occurs. O u t s i d e t h i s range,
Steady lift and moment coefficients as a function of incidence in transonic flow.
l i n e a r i z a t i o n over p r a c t i c a l ranges o f i n c i d e n c e
extends over a p a r t o f t h e f l a p . Then the change
v a r i a t i o n s (say i n t h e o r d e r o f 0.5 degrees) seems
i n a i r f o i l contour a t the l e a d i n g edge o f the f l a p
t o be p o s s i b l e again.
leads t o w i d e l y d i f f e r e n t f l o w p a t t e r n s o c c u r r i n g
The measured data i n f i g u r e 11.9 f o i l o w t h e
d u r i n g one c y c l e o f o s c i l l a t i o n ( F i g .
ii.lO). A
t r e n d s p r e d i c t e d by the i n v i s c i d t h e o r y i n a
f a n o f expansion waves i s formed a t one side, t e r -
q u a l i t a t i v e sense. Q u a n t i t a t i v e l y , t h e r e a r e l a r g e
minated by a normal shock wave, w h i l e s i m u l t a -
d e v i a t i o n s again, r e s u l t i n g from viscous e f f e c t s
13).
The b u l g e i n t h e measured
i i f t c u r v e between -0.5
and 2 . 5 degrees of i n c i -
(see a i s 0 chapter
n e o u s l y a t the o t h e r s i d e a lambda shock i s gen-
e r a t e d . The f l o w p a t t e r n i s complicated f u r t h e r by a f l o w s e p a r a t i o n near t h e t r a i i i n g edge. The
dence i s caused (as shown i n r e f e r e n c e 117) by a s h i f t i n the t r a n s i t i o n p o i n t o f the boundary
NACA 61 A O06 AIRFOIL
WTH FLAP
M,.0.96
l a y e r . I n t h i s range t h e shock wave i s l o c a t e d
L>a.O~
upstream o f the t r a n s i t i o n s t r i p a t 30 p e r cent o f
PRUIDTL -#EYER EXPANSION,
t h e chord, which causes t r a n s i t i o n ahead o f t h i s
HINGE MOMENT
c,
.
102
strip. The r e s u l t s i n f i g u r e 11.9 show t h a t f o r p r a c t i c a l purposes l i n e a r i z a t i o n i s p o s s i b l e , w i t h t h e e x c e p t i o n o f a small range t h a t can be d e t e r -40
mined from t h e behaviour o f t h e moment curve.
1i.3.3
c=j_
Flow around an a i r f o i l w i t h f l a p
\
-2
d
.#=
d, p 9
P‘
The f i n a l example of a s p e c i a l f l o w condit i o n f o r which l i n e a r i z a t i o n i s n o t j u s t i f i e d concerns an a i r f o i l w i t h f l a p . T h i s c o n d i t i o n o c c u r s when a wel I-developed supersonic r e g i o n
nonlinear behaviour.
-91-
‘I-.-
2
4-
FLAF ANGLE
.o5 .
steady h i n g e moment a s a f u n c t i o n o f f l a p a n g l e ,
regimes, where t h e unsteady-flow problem can be
shown a l s o i n f i g u r e 11.10, r e v e a l s t h a t the s l o p e
t r e a t e d independently o f the mean p o s i t i o n .
o f t h e h i n g e moment i s n e g a t i v e f o r f l a p angies
I n o r d e r t o s e l e c t the mean p o s i t i o n s i n
between +iand - i degrees. T h i s behaviour o f the
t r a n s o n i c f l o w and i n o r d e r t o l o c a t e t h e non-
h i n g e moment i s one o f t h e c o n d i t i o n s f o r which i n
I i n e a r ranges, use can be made o f measured o r
practice self-exciting oscillations o f limited
c a l c u l a t e d steady aerodynamic c o e f f i c i e n t s , I n
a m p l i t u d e o f c o n t r o l surfaces ( s o - c a l l e d a i l e r o n
t h i s r e s p e c t , the c a l c u l a t i o n method f o r steady
buzz) have been encountered.
t r a n s o n i c f l o w of Bauer, Korn, and Garabedian (Ref.
116) and i t s extended v e r s i o n , which i n -
c l u d e s t h e e f f e c t of t h e boundary l a y e r (Ref. i i 8 L
11.4
CONCLUDING REMARKS
can form a u s e f u l t o o l .
The c o n s i d e r a t i o n s on the p o s s i b i l i t y o f
tended t o unsteady flow.
I t may be expected t h a t
t h e c o n c l u s i o n s f o r quasi-steady f l o w can be exFor a d e f i n i t e answer,
l i n e a r i z a t i o n i n quasi-steady f l o w i n d i c a t e t h a t
however, f u r t h e r i n s i g h t i s needed i n the way i n
the normal f o r c e a l l o w s a l i n e a r i z e d t r e a t m e n t
which t h e n o n l i n e a r regions change w i t h frequency.
around a g i v e n mean p o s i t i o n o f t h e a i r f o i l , a t
I n t h e c o n s i d e r a t i o n s g i v e n above,
the
l e a s t as long as t h e f l o w remains a t t a c h e d and
a e r o e l a s t i c i a n was assumed n o t t o be i n t e r e s t e d i n
o t h e r severe v i s c o s i t y e f f e c t s a r e absent. The
t h e d e t a i l e d n o n l i n e a r behaviour o f t h e pressures,
moment, b e i n g more s e n s i t i v e t o t h e d e t a i l s o f
as f o r i n s t a n c e i n t r o d u c e d by shock-wave o s c i l l a -
the p r e s s u r e d i s t r i b u t i o n , shows an e s s e n t i a l
t i o n s . One may, however, t h i n k o f problems (such
non1 i n e a r c h a r a c t e r i n o n l y r e l a t i v e l y small
a5 panel f l u t t e r )
ranges o f i n c i d e n c e o r f l a p angle. Beyond these
m a r i l y i n t h e l o c a l behaviour o f the pressures.
ranges,
Here t h e l i n e a r behaviour o f t h e o v e r a l l c o e f f i -
I i n e a r i z a t i o n around a few s u i t a b l y chosen
mean p o s i t i o n s o f the a i r f o i l
i n which one i s i n t e r e s t e d p r i -
c i e n t s d e s c r i b e d above i s o f no a v a i l .
i s possible. This
c o n t r a s t s w i t h the subsonic and s u p e r s o n i c f i o w
-92-
PART IV
THE C U R R E N T STATUS OF UNSTEADY FLOW
THEORY AND E V A L U A T I O N O F SOME NEW METHODS FOR UNSTEADY TRANSONIC FLOW
-93-
12 REVIEW OF CALCULATION METHODS FOR TWO 12.1
- DIMENSIONAL
UNSTEADY FLOW
CLASSIFICATION OF THE VARIOUS METHODS
t a i n l y the most a c c u r a t e way o f s o l v i n g a f i u i d -
During the l a s t few years, q u i t e a number
f l o w problem, b u t a l s o the most c o m p l i c a t e d one.
o f c a i c u l a t i o n methods f o r high-subsonic and
t h e f i r s t successful computations f o r
t r a n s o n i c f l o w have been developed, which t a k e
Recently,
i n t o account t h e i n t e r a c t i o n between t h e steady
steady t r a n s o n i c flow,
and the unsteady f l o w f i e l d s and a r e present
-
-
i f shock waves
were r e p o r t e d by Deywert e t a l .
also the e f f e c t o f t h e p e r i o d i c
(Refs.
146-148).
T h e i r computations r e q u i r e d an enormous amaunt of
shock-wave motion. A summary o f these new methods i s given i n t a b l e
based on these equations,
computer t i m e ( u p t o I O hours on a
6, w h i l e the equations forming
puter!),
COC 7600 com-
which i n d i c a t e s t h a t c a l c u l a t i o n s of t h i s
t h e b a s i s o f t h e v a r i o u s methods a r e c o l l e c t e d i n
t y p e a r e s t i l l f a r from r o u t i n e a p p l i c a t i o n .
f i g u r e 12.1.
S i m i l a r a t t e m p t s f o r unsteady f l o w a r e j u s t be-
T h i s f i g u r e shows the e q u a t i o n s ob-
t a i n e d i n successive stages o f a p p r o x i m a t i o n o f
g i n n i n g t o be e x p l o r e d (Ref.
149).
The most advanced c a l c u l a t i o n methods now
the f u l i Navier-Stokes equations. For completeness, a l s o the c o r r e s p o n d i n g equations f o r steady f l o w
a v a i l a b l e f o r unsteady flows (Hagnus and Yoshihara,
are g i v e n .
Refs.
Ref. 123; and L e r a t and S i d e s ,
S o l v i n g t h e f u l l Navier-Stokes equations, t o g e t h e r w i t h s u i t a b l e t u r b u l e n c e models,
119, 120; Beam and Warming, Ref. 122; Lavai, Ref. 124) a r e based
on the E u l e r equations ( 7 2 . 1 ) , which f o l l o w from
i s cer-
TABLE 6 Review o f c a l c u l a t i o n methods f o r 2-0 h i g h subsonic and t r a n s o n i c f l o w EOUATIONS SOLVE0 UAVIER
-
AUTHORIS1
NOT YET EXISTING
,
MAGNUS 6 YOSHIHARA
o
BEAM 6 WARMING
197b
LAVAL
1975
, LERAT a
UETHOOS FOR F U L L POTENTIAL EO.
D
-
SIDES
FINITE DIFFERENCES
I977
I 1
1 j
122
FINITE OIFFERENCES
123
FINITE OIFFERENCES
124
FINITE DIFFERENCES
THE METHOD IS EXTENDED TO INCORPORATE VISCOUS EFFECTS ( R E F . I 2 1 1
ISOGAI B A L L H A U S 6 LOMAX
TRANSONIC SMALL PERTURBATION EO.
'NTH SHOCK WAVES
I REF. 1 METHOD OF SOLUTION I REMARKS
STOKES EOS
EULER EOS
TRANSONIC FLOW
'EAR
D
BALLHAUS 6 STEGER
D
CHAN 6 CHEN
o
EHLERS
D
TRACI, FARR 6 ALBANO
KRUPP 6 COLE
LINEARIZED TRANSONIC SMALL-PERTURBATION EO.
-
-
~
o
METHODS FOR NEAR SONIC FLOW ' M m
i
METHODS FOR HIGH
STAHARA bi SPREITER
o
ISOGAI
a
KIMBLE 6
o
OOWELL 6 PARK
I1
WITHOUT SHOCK WAVES
CHAN 6 BRASHEAR
D
ZWAAN
o
NIXON
o
ISDGAI
WU
SUBSONIC FLOW
EO FOR 5.BSOhiC L SEAR (Mm CM']
L.FT hG-I.RFACE
TIJOEMAN 6 ZWAAN. ROO
INTEGRAL METHOD
GIESING. KALMAN. ROOD1
INTEGRAL METHOD
-hEO?I e
-95-
LOCAL MACH NUMBER CORRECTION LOCAL MACH NUMBER CORRECTION
DECREASING DEGREE
O F APPROXIMATION
I
COMPLETE
I
NAVIER-STOKES EQUATIONS
4 EULER EQUATIONS:
NO VISCOSITY
==o
with
e = (y-I)
-1
p + &P (U2+Vz)
(12.1)
I I RROTATI ONAL ; I SENTROPIC
FULL POTENTIAL EQUATION:
small p e r t u r b a t i o n s
--
TRANSONIC SMALL-PERTURBATION
SMALL THICKNESS, INCIDENCE AND AMPLITUDE
~
+
_
_
non-uniform s t e a d y f l o w s m a l l unsteady p e r t u r b a t i o n s
(TSP) EQUATION:
I-
@ = Oe
+$e
k = & ;
iwt
f.=
I
u-
LINEARIZED TRANSONIC SMALL-PERTURBATION EQUATION:
AMPLITUDE < c THICKNESS AND INCIDENCE
(I
-
M2
-
Mi
r+l u;b mox)
$ xx
+ myy
-
2ikM:$x
I
+
-
fl2
k2M2$ - ( y + l )
@oxx$x =
O
(12.4)
CLASSICAL TRANSONIC EQUATION:
1 1
-96-
EQUATION FOR SUBSONIC AND SUPERSONIC LINEAR LIFTING-SURFACE THEORY:
Basic equations of the various calculation
Fig. 12.1
methods for two-dimensional flow
i
FULL POTENTIAL EQUATION FOR
(aZ-üz)mxx
with
-3=
at
SOLUTIONS FOR STEADY FLOW ( A . O . ) : NIEUWLAND (Ref. 9Jt
FLOW:
*
O
+ (aZ-m2)m
az = a i
Y
-
YY
- ~
~ = Om
~
~ ~ (Ref. BOERSTOEL O
+ k(OZf$)]
(y-1)
m 0
10)
BAUER, KORN E GARABEDIAN ( R e f . JAMESON (Ref. l 5 ) * *
(12.2a)
STEADY TRANSONIC SMALL-PERTURBATION EQUATION:
0
*
(1
-
N2 . M Z y + l m
m
a_
“)@xx
+
myy
=o
MURMAN & COLE (Ref.i60)’* (12.3a)
LINEAR LIFTING-SURFACE THEORY FOR AND SUPERSONIC FLOW:
(1-112)
mx, + ayy
=
STEADY SUBSONIC
o (12.6a)
* v i a hodograph t r a n s f o r m ** finite-difference methods
-97-
lI6)*‘
t h e Navier-Stokes e q u a t i o n s by assuming t h e f l o w
steady f l o w f i e i d .
t o be i n v i s c i d . T h i s i m p l i e s t h a t the e f f e c t
(12.4)
Of
The r e s u i t i n g l i n e a r e q u a t i o n
possesses s p a t i a l l y v a r y i n g c o e f f i c i e n t s ,
the boundary l a y e r i s n o t i n c l u d e d , which may lead
which depend on t h e v e i o c i t y d i s t r i b u t i o n o f the
t o a change i n e f f e c t i v e a i r f o i l c o n t o u r and,
mean steady f l o w f i e l d , T h i s steady f l o w f i e l d has
thus,
t o be known e i t h e r from steady computations o r
t o a change i n c i r c u l a t i o n , shock s t r e n g t h ,
and shock p o s i t i o n . T h i s drawback i s removed t o
from experiment. T h i s i m p l i e s t h a t t h e e f f e c t o f
a c e r t a i n e x t e n t i n an e n g i n e e r i n g t y p e o f proce-
shock waves can be included only, i f t h e shocks
dure d e s c r i b e d by Magnus and Yoshihara (Ref.
do e x i s t a l r e a d y i n the steady f l o w f i e l d .
121),
From
who i n c l u d e d t h e displacement e f f e c t o f t h e bound-
t h e methods based on t h e l i n e a r i z e d t r a n s o n i c
a r y l a y e r i n an a p p r o x i m a t i v e way.
s m a l l - p e r t u r b a t i o n e q u a t i o n (Refs. 130-142), only those o f E h l e r s (Ref.
A f u r t h e r s i m p l i f i c a t i o n o f t h e unsteady-
(Ref.
f l o w equations i s o b t a i n e d i f one assumes t h e f l o w
i 3 0 ) , T r a c i , F a r r and Albano
l3l), and Chan and Brashear (Ref.
132) a r e
t o be i s e n t r o p i c and i r r o t a t i o n a l . T h i s has t h e
a b l e t o t a k e t h e e f f e c t o f shock waves i n t o
advantage t h a t t h e f l o w problem can be d e s c r i b e d
account. The o t h e r methods a r e l i m i t e d t o near-
by a p o t e n t i a l e q u a t i o n w i t h only one unknown. A
s o n i c flows w i t h o u t shock waves o r t o s u b c r i t i c a l
s e r i o u s drawback, however,
flows (see t a b l e 6 ) .
i s t h a t the shock
strength included inherently i n the potential
As a f i n a l s t e p i n t h e successive stages o f approximation,
theory i s no l o n g e r c o r r e c t . A S p o i n t e d o u t by Van der Vooren and S l o o f f (Ref.
150), i n p o t e n t i a l
(12.5), a p p l i e d by Landahl (Ref. 2 4 ) , o r
f l o w the v e l o c i t y ( o r p r e s s u r e ) jump across a
equation
normal shock wave i s l a r g e r than t h a t d e r i v e d from
t o e q u a t i o n (12.6),
the Rankine-Hugoniot
t h e p o t e n t i a l e q u a t i o n (12.4)
can be reduced t o t h e c l a s s i c a l t r a n s o n i c - f l o w
used i n i i n e a r i z e d i i f t i n g -
s u r f a c e t h e o r y . The i a t t e r t h e o r y , t o g e t h e r w i t h a
r e l a t i o n s (see f i g u r e 12.2).
The d i f f e r e n c e increases w i t h Mach number and, a s
semi-empirical
correction f o r the e f f e c t o f the
a consequence, p o t e n t i a l t h e o r y i s i i m i t e d t o weak
mean steady f l o w f i e l d ,
i s used i n the l a s t two
and moderately s t r o n g shock waves, f o r which t h e
methods, g i v e n i n t a b l e 6 (Refs.
upstream Mach number remains l e s s than about 1.3.
methods based on t h e c l a s s i c a l t r a n s o n i c - f l o w
So f a r , o n l y one method i s r e p o r t e d i n which t h e f u l l p o t e n t i a l e q u a t i o n (12.2) ( I s o g a i , Ref.
125).
e q u a t i o n (12.5) a r e n o t considered here, s i n c e
i s solved
they a r e only a p p l i c a b l e i n t h e case o f reduced
In the o t h e r methods, use i s
frequencies,
made o f more s i m p l i f i e d v e r s i o n s of t h i s e q u a t i o n . B a l l h a u s and h i s c o l l e a g u e s (Refs. Krupp and Cole (Ref.
143-145). The
which n o r m a l l y a r e t o o h i g h f o r
p r a c t i c a l a p p l i c a t i o n s i n a e r o e l a s t i c probiems.
126, 1 2 7 ) ,
128), and Chan and Chen ( R e f .
130) developed methods t o s o l v e t h e t r a n s o n i c
0 ,
s m a l i - p e r t u r b a t i o n e q u a t i o n (12.3), which i s based
/’ /
on the a d d i t i o n a l assumption t h a t t h e steady and
/’
t h e unsteady p e r t u r b a t i o n s o f t h e f l o w f i e i d remain small. These methods a r e t h e r e f o r e l i m i t e d t o t h i n a i r f o i l s and small amplitudes o f o s c i l l a tion.
In addition,
the description of the f l o w i s
\\
RANKINE -HUGOHIOT
less a c c u r a t e near l o c a t i o n s where t h e f l o w has t o a d j u s t i t s e l f t o a b r u p t changes i n s u r f a c e s l o p e s , such as t h e l e a d i n g edge of t h e a i r f o i l o r t h e l e a d i n g edge o f a d e f l e c t e d f l a p . As i n d i c a t e d i n f i g u r e 12.1 (and discussed e a r l i e r i n s e c t i o n 2.2.3).
the t r a n s o n i c s m a l l 1.1
p e r t u r b a t i o n e q u a t i o n can be l i n e a r i z e d by con-
Fig. 12.2
s i d e r i n g t h e unsteady f l o w f i e l d as a small p e r t u r b a t i o n superimposed upon a non-uniform mean
-98-
1.2
1.1
l.&
Relation between the Mach number upstream and d c m s t r e a m of B normal shock wave (from Ref. 150).
LI,
1.5
I n the subsequent s e c t i m 5 o f t h i s c h a p t e r ,
The c o m p l i c a t e d Kernel f u n c t i o n Kíx ;k,M-)
(for
O
t h e v a r i o u s c a l c u i a t i o n methods a r e b r i e f l y de-
more d e t a i l s , see r e f e r e n c e s 2 5 , 2 7 ,
s c r i b e d . T h i s d e s c r i p t i o n i s preceded by an o u t -
be i n t e r p r e t e d i n a p h y s i c a l sense a s an aerody-
l i n e o f 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 theory,
namic i n f l u e n c e f u n c t i o n , g i v i n g the dawnwash
which i s i n widespread use f o r a e r o e i a s t i c inves-
induced i n p o i n t x due t o a pressure jump o f u n i t
5.
151,
152) can
t i g a t i o n s and forms an i m p o r t a n t b a s i s o f compar-
strength in point
ison f o r the r e s u l t s o f the present investigation.
depends on t h e reduced frequency, k , the f r e e -
Moreover, t h i s t h e o r y i s used i n t h e semi-empirical
stream Mach number, M-,
method f o r high-subsonic f l o w developed a t
NLR
and
This influence function
and the d i s t a n c e between
the sending and r e c e i v i n g p o i n t , xo. The downwash,
d e s c r i b e d i n s e c t i o n 12.3.
w(x).
i s determined by t h e requirement t h a t a t
each i n s t a n t t h e f l o w i s t a n g e n t i a l t o the surf a c e of t h e o s c i l l a t i n g a i r f o i l . LINEARIZED SUBSONIC LIFTING-SURFACE THEORY
12.2
I f the a i r f o i l
o s c i l l a t i o n i s d e s c r i b e d by
12.2.1 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 load d i s t r i b u t i o n t h e tangency c o n d i t i o n y i e l d s : The essence of the m a j o r i t y of t h e methods used i n subsonic l i f t i n g - s u r f a c e t h e o r y i s t h a t the sinusoidally o s c i l l a t i n g t h i n a i r f o i l
is
represented by a sheet o f p r e s s u r e d o u b i e t s , w i t h
where k i s t h e reduced frequency,
t h e i r axes normal t o the f l o w ( F i g .
reference length
t h i s way,
12.3).
In
e
based on a
= I.
I n t h e l i t e r a t u r e , several methods have
the p o t e n t i a l e q u a t i o n can be t r a n s -
formed i n t o an i n t e g r a l e q u a t i o n r e l a t i n g t h e
been p u b l i s h e d t o s o l v e t h e i n t e g r a l e q u a t i o n
known downwash d i s t r i b u t i o n , w,
(12.7),
a t the a i r f o i l
t h e main d i f f e r e n c e between the v a r i o u s
s u r f a c e (assumed a t y = O ) t o t h e unknown load
methods being t h e way i n which the numerical i n -
distribution AC
t e g r a t i o n s a r e performed. I n the c o n t e x t o f the
P'
For two-dimensional
flow,this
p r e s e n t review, t h e K e r n e l - f u n c t i o n method and the
e q u a t i o n reads:
D o u b l e t - L a t t i c e method, nowadays being the most popular methods, w i l l be b r i e f l y described. For d e t a i l s o f the o t h e r methods, r e f e r e n c e
i 5
made
t o the well-known textbooks on a e r o e i a s t i c i t y with
x
O
= x
-
(Refs. 25, 27,
1 5 1 , 152).
5
12.2.2
The K e r n e l - f u n c t i o n method
I n t h e K e r n e l - f u n c t i o n method, IHEETOF ACCELERATION DOUBLETS (PRESIURE WUBLETSI SENDING
mini
it is
assumed t h a t t h e unknown pressure d i s t r i b u t i o n
VIBRATION MODE
appearing i n t h e i n t e g r a l e q u a t i o n (12.7) can be
RECEIVIUG POINT
d e s c r i b e d by a s e r i e s of p r e s e l e c t e d loading
%.U,
functions: N
Fig. 12.3
Representation of an oscillating airfoil in linearized Subsonic lifting-surface theory.
The f u n c t i o n s H.(x) a r e chosen i n such a way t h a t J
-99-
each f u n c t i o n s a r i s f i e s the edge c o n d i t i o n s known
p r o p e r e v a l u a t i o n o f the i n t e g r a l s K . .
f o r t h e ioad d i i t r i b u t i o n On a f i a t p l a t e , i . e .
which can be i n t e r p r e t e d a s the downwash i n
(IZ.il),
'i
the r o o t s i n g u l a r i t y a t the l e a d i n g edge and the
p o i n t x . due t o a chordwise load d i s t r i b u t i o n
K u t i a c o n d i t i o n a t t h e t r a i l i n g edge (see t h e
H.(X).
J
example i n f i g u r e 12.4). In the case o f a f l a p , o t h e r l o a d i n g f u n c t i o n s have t o be added t o take c a r e o f the s i n g u l a r behaviour o f the load d i s -
12.2.3
The D o u b i e t - L a t t i c e method
t r i b u t i o n a t t h e l e a d i n g edge o f t h e f l a p .
A s a n e x t s t e p , (12.10) i s s u b s t i t u t e d i n
(12.7),
I n t h e D o u b l e t - L a t t i c e method, developed
and t h e i n t e g r a l s
o r i g i n a l l y by Albano and Rodden (Ref. 153), t h e i n t e g r a l e q u a t i o n (12.7)
1
K.. IJ =
#
dividing the a i r f o i l Hj(S)K(xi-f,;k,Mm)
dg
(12.11)
i s d i s c r e t i z e d by
i n t o a l a r g e number o f
elements. A t t h e q u a r t e r - c h o r d p o i n t o f each
O
element, a p r e s s u r e doublet of y e t unknown s t r e n g t h i s placed, w h i l e a t the t h r e e - q u a r t e r -
a r e e v a l u a t e d f o r a s e t o f N s u i t a b l y chosen p o i n t s xi,
t h e s o - c a l l e d c o l l o c a t i o n p o i n t s . By p r e -
s c r i b i n g t h e downwash wi
i n t h e ii c o l l o c a t i o n
p o i n t s , a c c o r d i n q t o (12.91,
chord p o i n t o f each element a c o n t r o l p o i n t i s chosen.
By e q u a l i z i n g f o r each c o n t r o l p o i n t the
downwash i n t r o d u c e d by a l l d o u b l e t s t o the down-
a set o f N s i m u l t a -
wash a c c o r d i n g t o (12.9),
neous a l g e b r a i c equations i s o b t a i n e d , which has
the following set of
a l g e b r a i c e q u a t i o n s i s obtained:
t o be s o l v e d f o r the unknown c o e f f i c i e n t s a.: J wi/Um
=
z j
K..(k,M,) '1
a.
J
.
wi/U,
(12.12)
=
1D..(k,M,) IJ i
AC
. ,
PsJ
(12.13)
where 0 . . i s the aerodynamic i n f i u e n c e c o e f f i c i e n t , IJ r e p r e s e n t i n g t h e downwash i n the c o n t r o l p o i n t o f
The most l a b o r i o u s p a r t o f the method i s the
element i due t o a d o u b l e t o f u n i t s t r e n g t h a t the q u a r t e r - c h o r d p o i n t o f element j . From (12.13)
the
s t r e n g t h o f t h e doublets and, hence, o f the pres-
sure c o e f f i c i e n t s A C
can be computed. p,j An important d i f f e r e n c e between the Kernel-
f u n c t i o n method and the D o u b l e t - L a t t i c e method i s t h a t the aerodynamic i n f l u e n c e c o e f f i c i e n t s
i n the
f i r s t method i n v o l v e complete chordwise pressure l o a d i n g f u n c t i o n s , whereas i n the second method a point-to-point
r e l a t i o n i s used. T h i s l o c a l r e l a -
t i o n s h i p makes the D o u b l e t - L a t t i c e method w e l l s u i t e d f o r the i n t r o d u c t i o n of semi-empirical effects,
such as t h e e f f e c t o f l o c a l Mach number,
t o be discussed i n the n e x t s e c t i o n .
12.3
METHODS FOR HIGH-SUBSONIC FLOW
12.3.1 Local-Mach-number c o r r e c t i o n s 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 theory
Fig. 12.4
Example of p ~ e s s u r eloading functions used in the Kernel-function method.
A r e l a t i v e l y simple procedure t o i n t r o d u c e
-100-
the e f f e c t - o f the n o n - u n i f o r m i t y o f t h e steady
phase s h i f t between t h e shock-wave motion and the
f l o w f i e l d i n an a p p r o x i m a t i v e way has been
p i t c h i n g m o t i o n o f the NLR 7301 a i r f o i l .
developed a t NLR (Tijdeman and Zwaan, Ref. 1 4 3 ;
A p a r t from the semi-empirical c o r r e c t i o n
Roos, Ref. 1 4 4 ) . I n s p i r e d by t h e g r a p h i c a l exper-
o f t h e i n f l u e n c e c o e f f i c i e n t s , a l s o a local-Mach-
iments d e s c r i b e d i n s e c t i o n 8 . 3 ,
number e f f e c t I s i n t r o d u c e d i n the boundary con-
the idea was
conceived t o c o n s i d e r t h e d o u b l e t s i n l i n e a r i z e d
d i t i o n , by changing (12.9)
into
l i f t i n g - s u r f a c e t h e o r y as eiements e m i t t i n g waves t h a t induce t h e downwash i n the c o n t r o l p o i n t s . Then t h e e f f e c t o f t h e non-uniform main f l o w can be i n t r o d u c e d by making t h e i n f l u e n c e c o e f f i c i e n t s , D..,
IJ
o f e q u a t i o n (12.13)
functions
where M . denotes t h e l o c a l Mach number i n t h e
o f the local
c o n t r o l p o i n t i. T h i s l o c a l c o r r e c t i o n o f t h e
Mach number between t h e sending p o i n t and t h e
boundary c o n d i t i o n was a p p l i e d e a r l i e r by Ashley
r e c e i v i n g p o i n t . For s i m p l i c i t y , D.. i s made a IJ f u n c t i o n o f M . . , which i s t h e l o c a l Mach number 11 averaged between p o i n t i and p o i n t j . A c c o r d i n g l y . r e l a t i o n (12.13)
and Rowe (Ref.
154) and by Rowe, Wlnther and
155), without further corrections,
Redman (Ref.
however, o f t h e aerodynamic i n f l u e n c e f u n c t i o n s .
becomes:
R e c e n t l y , Giesing, Kalman and Rodden (Ref.
145) r e p o r t e d a m o d i f i c a t i o n o f t h e NLR
approach. The b a s i c idea o f t h e i r method i s t o transform t h e d i s t a n c e between the sending and the r e c e i v i n g p o i n t , depending on the t i m e re-
i n which
q u i r e d by t h e p r e s s u r e s i g n a l s t o t r a v e l t h a t M..
IJ
= R [ M . . ( a t the s u r f a c e ) IJ
-
M-}
+
M-,
d i stance.
(12.136)
and
12.3.2
k . . = kM-/M.. IJ
IJ
Methods based on the l i n e a r i z e d t r a n s o n i c small-perturbation equation
.
(12.13~) C a l c u l a t i o n methods f o r unsteady h i g h -
The c o n s t a n t , R ,
i n (12.13b)
i s a r e l a x a t i o n fac-
subsonic f l o w , based on t h e l i n e a r i z e d t r a n s o n i c
t o r ( O c R < 1.0) t h a t i s i n t r o d u c e d t o t a k e c a r e
s m a l l - p e r t u r b a t i o n e q u a t i o n ( i 2 . 4 ) , were developed
o f the f a c t t h a t the waves propagate a l o n g t h e
by Zwaan (Ref. i 3 9 ) , Nixon (Refs.
1 4 0 , i 4 1 ) , and
surface o f the a i r f o i l a s w e l l as along paths
i s o g a i (Ref.
a t a c e r t a i n d i s t a n c e away from t h e a i r f o i l
assumption t h a t t h e local Mach number of t h e
surface (see f i g u r e 8.13).
This implies that the
142). Zwaan i n t r o d u c e d the a d d i t i o n a l
steady f l o w f i e l d o n l y depends on t h e x - c o o r d i n a t e .
average "head wind" o f t h e upstream-propagating
By a p p l y i n g t h e matched-asymptotic-expansions
waves i s somewhere between t h e v e l o c i t y a t t h e
technique, he o b t a i n e d a s o l u t i o n f o r an a i r f o i l
a i r f o i l s u r f a c e and t h e f r e e - s t r e a m v e l o c i t y .
w i t h o s c i l l a t i n g f l a p i n high-subsonic and low-
R e l a t i o n ( 1 2 . 1 3 ~ ) i n d i c a t e s t h a t t h e reduced
supersonic f l o w . Nixon and Isogai a l s o s o l v e d equat i o n (12.4)
frequency v a r i e s l o c a l l y w i t h Mach number. For p r a c t i c a l a p p l i c a t i o n s , t h e values
known i n t e g r a l - e q u a t i o n methods f o r steady f l o w ,
M . . can be determined from e i t h e r experimental o r
11 t h e o r e t i c a l steady p r e s s u r e d i s t r i b u t i o n s .
i n an a p p r o x i m a t i v e way. T h e i r methods
should be c o n s i d e r e d as an extension o f the w e l l -
Of
as i n i t i a t e d by O s w a t i t s c h (Ref.
I t was
among o t h e r s ,
found a t NLR t h a t s a t i s f a c t o r y r e s u l t s a r e obt a i n e d i f t h e r e l a x a t i o n f a c t o r R i s s e t equal t o
156) and extended,
by Nixon and Hancock (Ref.
157).
The r e s u l t s o b t a i n e d by a l 1 t h r e e a u t h o r s
0.7 (Roos, Ref. 1 4 4 ) . T h i s v a l u e agrees w i t h t h e
demonstrate t h a t i n i n v i s c i d high-subsonic f l o w
r e l a x a t i o n f a c t o r t h a t , f o r t h e same reason, was
t h e t h i c k n e s s o f an a i r f o i l may have a c o n s i d e r -
used i n s e c t i o n 10.4 f o r t h e e x p l a n a t i o n o f t h e
a b l e e f f e c t on t h e unsteady a i r l o a d s . Nixon, f o r
-101-
i n s t a n c e , s h o w s - t h a t the aerodynamic c o e f f i c i e n t s
e s t a b l i s h e d f i n i t e - d i f f e r e n c e merhods f o r steady
I< and
t r a n s o n i c f l o w (Refs. 1 1 - i s ,
m ( w i t h the e x c e p t i o n o f t h e c o n t r o l - s u r c face c o e f f i c i e n t n c ) f o r an NACA 0012 a i r f o i l w i t h
a t r a i l i n g - e d g e f l a p o f 40 p e r c e n t o f the chord
118, 159-161).
Concerning t h e f i n i t e - d i f f e r e n c e
methods
a r e 1 5 t o 35 p e r cent l a r g e r than t h e correspond-
f o r steady flow,
ing i i n e a r values. The d i f f e r e n c e s a r e l a r g e s t
d i s t inguished: t ime-progress ing methods (Ref. 159)
f o r l o w values o f t h e reduced frequency.
and r e l a x a t i o n methods (Refs. i 4 ,
-
two d i f f e r e n t approaches can be
116, 118, 160,
161). I n t h e former approach, the time-dependent e q u a t i o n s a r e used and t h e d e s i r e d s t e a d y - f l o w
12.4
METHODS FOR NEAR-SONIC FLOW \JITHOUT
c o n d i t i o n i s o b t a i n e d a s y m p t o t i c a l l y w i t h a proper
SHOCK WAVES
f i n i t e - d i f f e r e n c e procedure marching i n time. The second approach i 5 based on t h e s t e a d y - f l o w equa-
For Mach numbers near u n i t y , t h e e f f e c t of
t i o n s and t h e corresponding f i n i t e - d i f f e r e n c e
a i r f o i l t h i c k n e s s has been taken i n t o account i n t h e methods o f Stahara and S p r e i t e r (Ref. I s o g a i (Refs.
analogue i s s o l v e d by a r e l a x a t i o n process. As
1331,
f a r as computer t i m e i s concerned,
134, i 3 5 ) , Kimble and Wu (Ref. 136),
Dowell and Park (Refs. 137,
the l a t t e r
methods a r e much more e f f i c i e n t than t h e time-
138). In a l !
p r o g r e s s i n g methods.
f o u r methods, the l i n e a r i z e d t r a n s o n i c smal!-pert u r b a t i o n e q u a t i o n (12.4)
i s s o l v e d by employing
a concept analogous t o t h e l o c a l l i n e a r i z a t i o n
For t h e f i n i t e - d i f f e r e n c e
methods f o r
method developed f o r steady f l o w about twenty
unsteady f l o w , a l 5 0 a c h o i c e can be made between
years ago by S p r e i t e r and Alksne ( Ref .
t i m e - p r o g r e s s i n g methods and r e l a x a t i o n methods.
158).
T h i s c h o i c e depends l a r g e l y on t h e type o f un-
For t h e unstesdy case, c a l c u l a t e d r e s u l t s a r e presented f o r a number o f o s c i I l a t i n g para-
steady equations considered. As p o i n t e d o u t by
b o l i c - and c i r c u l a r - a r c + i r f o i l s w i t h several
Beam and Warming (Ref. 122), time-progressing
t h i c k n e s s r a t i o s . The examples c l e a r l y i n d i c a t e
methods probably a r e most e f f i c i e n t f o r the non-
t h a t f o r low reduced f r e q u e n c i e s ( k
l i n e a r time-dependent equations,
i
0.1)
thick-
the Euler
e q u a t i o n s , the full p o t e n t i a l equation, and the
ness has an important i n f l u e n c e , w h i l e f o r h i g h
reduced frequencies ( k l a r g e r than I )
i.e.
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.
the r e s u i t s
For these
converge t o t h e r e s u l t s t h a t can be o b t a i n e d w i t h
e q u a t i o n s , r e l a x a t i o n methods a r e less e f f i c i e n t
the I i n e a r t r a n s o n i c t h e o r y based on t h e c l a s s i c a l ,
since,
t r a n s o n i c e q u a t i o n (12.5).
( t i m e ) i s r e q u i r e d than i n steady flow.
T h i s l a t t e r behaviour
For the
l 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
c o u l d be expected a l r e a d y on t h e b a s i s o f t h e a n a l y s i s g i v e n by Landahl (Ref.
i n t h e unsteady case, one more dimension
( s m a l l unsteady p e r t u r b a t i o n s ,
24).
superimoosed upon
a non-uniform main f l o w ) , t h e c h o i c e
i 5
e v i d e n t . According t o Beam and Warming,
12.5
HETHOOS FOR TRANSONIC FLOW WITH SHOCK WAVES
12.5.1
General remarks
less relaxation
schemes appear r o s t p r o m i s i n g i f s t a t i o n a r y s i n u s o i d a l motions a r e considered, whereas timep r o g r e s s i n g methods seem t o be b e t t e r s u i t e d f o r t h e s o - c a l l e d i n d i c i a l - f u n c t i o n approach.
I n the
l a t t e r approach, which i s v a l i d only f o r l i n e a r
W i t h t h e e x c e p t i o n o f t h e methods of Chan and Chen (Ref. 129) and Chan and ürashear (Ref.
systems', a s t e p change i s imposed i n t h e normal
1 3 2 ) , a l l c a l c u l a t i o n methods developed r e c e n t l y
v e l o c i t y component on t h e a i r f o i l , which c o r r e -
f o r unsteady t r a n s o n i c f l o w w i t h shock waves make
sponds t o t h e d e s i r e d mode o f v i b r a t i o n . The
use o f a f i n i t e - d i f f e r e n c e technique f o r t h e
r e s u l t i n g aerodynamic response ( i n d i c i a 1 func-
501U'
t i o n o f t h e unsteady-flow e q u a t i o n s (see t a b l e 6 ) .
tion)
They can be considered as e x t e n s i o n s o f t h e w e l l -
required a i r l o a d s f o r sinusoidal o s c i l l a t i o n s are
-102-
i s compuL-Ld as a f u n c t i o n o f t i m e , and t h e
o b t a i n e d by a F a u r i e r t r a n s f o r m a f the i n d i c i a i
convex a i r f o i l w i t h a p e r i o d i c a l l y moving shock
response.
wave. The complete E u l e r equations were solved a l s o by L e r a t and Sides (Ref.
12.5.2 Nethods based on the E u l e r e q u a t i o n s
1 2 4 ) . They used a
time-dependent mapping technique, which has the advantage t h a t the boundary c o n d i t i o n s can b e
As i n d i c a t e d i n t a b l e 6, a number o f
imposed on t h e a i r f o i l c o n t o u r i n i t s a c t u a l
methods have been produced t h a t a r e based on t h e E u l e r equations.
Beam and Warming (Ref.
p o s i t i o n , r a t h e r than on some mean p o s i t i o n . L e r a t
1 2 2 ) con-
and Sides presented unsteady p r e s s u r e d i s t r i b u t i c n s
s i d e r e d small p e r t u r b a t i o n s i n a non-uniform
on an NACA 0012 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 . The
steady f l o w and s o l v e d the corresponding unsteady
reduced f r e q e n c y o f t h i s example i s v e r y h i g h ,
E u l e r equations w i t h a t i m e - p r o g r e s s i n g method.
and as a r e s u l t t h e shock wave remains i n an a l -
They demonstrated t h e a p p l i c a b i l i t y o f t h e
most s t a t i o n a r y pos i t ion.
i n d i c i a l - f u n c t i o n approach f o r a f l a t p l a t e i n compressible f l o w and showed f u r t h e r an example 12.5.3 Methods based on t h e p o t e n t i a l . e q u a t i o n
of a biconvex 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 i n t r a n s o n i c flow.
S o l u t i o n s o f the f u l l p o t e n t i a l equation
Magnus and Yoshihara s o l v e d the E u l e r equat i o n s w i t h o u t f u r t h e r assumptions concerning f r e -
were r e p o r t e d by I s o g a i (Ref. 1251, who used a
quency o r a m p l i t u d e o f o s c i l l a t i o n . The boundary
time-progressing f i n i t e - d i f f e r e n c e
c o n d i t i o n s a r e imposed a l o n g a c o n t o u r c o i n c i d e n t
a p p l i e d the t a n g e n t i a l - f l o w c o n d i t i o n on the a i r -
w i t h the mean p o s i t i o n of t h e a i r f o i l surface.
In
f o i l contour i n i t s mean p o s i t i o n .
r e f e r e n c e 120, d e t a i l e d unsteady p r e s s u r e d i s t r i b u -
technique and
I s o g a i pre-
sented numerical r e s u l t s f o r t h e NACA 64A006 a i r -
t i o n s a r e p u b l i s h e d f o r the NACA 64A410 a i r f o i l
f o i l w i t h f l a p and f o r a s u p e r c r i t i c a i a i r f o i l
performing o s c i l l a t i o n s i n p i t c h around i t s mid-
developed by Bauer e t a l . (Ref.
chord p o i n t . S i m i l a r r e s u i t s f o r the NACA 64A006
forms o s c i l l a t i o n s i n p i t c h around i t s "shock-
a i r f o i l w i t h f l a p were g i v e n i n references 1 2 1 and
f r e e " design c o n d i t i o n . The l a t t e r examples show
162. F a r the l a t t e r a i r f o i l , a l s o an a p p r o x i m a t i v e
t h a t t h e n o n l i n e a r behaviour o f the a i r f o i l i n
118), which per-
method was i n t r o d u c e d t o t a k e the displacement
i t s design c o n d i t i o n d i m i n i s h e s w i t h i n c r e a s i n g
e f f e c t o f t h e boundary i a y e r i n t o account. . A l l
frequency.
computations s t a r t e d w i t h t h e steady-flow
solution The s o l u t i o n a f the unsteady-transonic-
and w e r e c o n t i n u e d u n t i l a complete c y c l e o f the p e r i o d i c f l o w was o b t a i n e d . T h i s i s a time-consuming process,
f l o w problem a s g i v e n by Bailhaus and Lomax
i n p a r t i c u i a r a t low reduced f r e -
(Ref.
quency, where the shock wave e x h i b i t s l a r g e excur-
126) i s based on t h e 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.
In a d d i t i o n , they considered the
s i o n s . The f i r s t r e s u l t g i v e n i n r e f e r e n c e 120
low-frequency v e r s i o n o f t h i s equation,
took about 7 hours on a CDC 7600 computer. A f t e r
t h e term
et,
i n which
i s p u t equal t o z e r o. Computed re-
refinement o f t h e computer program, t h i s c o u l d be
s u l t s were g i v e n f o r i m p u l s i v e l y s t a r t e d a i r f o i l s .
reduced t o about 2 hours.
L a t e r on, t h e low-frequency method was extended
I t should be noted,
however, t h a t i t was n o t intended t o develop a
t o the l i f t i n g case (Ref. i 6 3 ) , and the e f f i c i e n c y
method f o r r o u t i n e use, but r a t h e r t o g e n e r a t e
o f the method was increased c o n s i d e r a b l y by the
some s o l u t i o n s t h a t m i g h t reveal t h e n a t u r e o f t h e
implementation o f a v e r y e f f i c i e n t a l g o r i t h m
f l o w o r serve as t e s t cases f o r more a p p r o x i m a t i v e
developed by Ballhaus and Steger (Ref.
b u t F a s t e r methods.
s o l v e t h e f i n i t e - d i f f e r e n c e equations. I n references 163 and 164, some r e s u l t s a r e shown
An approach v e r y s i m i l a r t o t h a t o f Magnus and Yashihara i s f o l l o w e d by Laval.
127) t o
f o r sinusoidally oscillating a i r f o i l s .
I n reference
I n the
second reference, a comparison i s made between
123 he p r e s e n t s an example o f an o s c i l l a t i n g b i -
-103-
r e s u l t s obtained w i t h the smaii-perturbation
bation equation appropriate for a helicopter r o t o r
e q u a t i o n i n the low-frequency l i m i t and r e s u l t s
i n forward f l i g h t ,
from the procedure o f Magnus and Yoshihara, b o t h
dimensional computations, showing a type-C shock-
a p p l i e d t o the p i t c h i n g NACA 64A410 a i r f o i l . The
wave motion (Ref. 169). They o b t a i n e d t h e i r
agreement i s n o t v e r y s a t i s f a c t o r y , t h e main
Lions w i t h a f i n i t e - d i f f e r e n c e scheme, which i s an
d e f i c i e n c y being the improper c a p t u r e o f t h e shock
e x t e n s i o n o f t h e t e c h n i q u e o f Murman and Cole
wave i n the s m a l i - p e r t u r b a t i o n method.
(Ref.
I t must be
they gave a i 5 0 r e s u l t 5 o f two-
501"-
160) f o r steady f i o w .
n o t e d , however, t h a t t h e a m p l i t u d e s of o s c i l l a t i o n As mentioned p r e v i o u s l y . a s i m p l i f i c a t i o n
as a p p l i e d i n t h i s comparative study were 2 degrees, which i n t r a n s o n i c f l o w c e r t a i n l y cannot
of t h e 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 i s
be considered as a small p e r t u r b a t i o n . Moreover,
o b t a i n e d i f t h e unsteady flow i s considered a s a
during a large p a r t o f a cycle of o s c i l l a t i o n the
smal I p e r t u r b a t . i o n upon a known non-uniform steady
Mach number j u s t upstream o f t h e shock wave on t h e
f l o w f i e l d (Fig.
NACA 64A410 example i s l a r g e r than 1.3,
which
1 2 . 1 ) . A f i n i t e - d i f f e r e n c e solu-
t i o n o f the l i n e a r i z e d small-perturbation equation
s e r i o u s l y v i o l a t e s t h e assumptions o f p o t e n t i a l
was g i v e n by E h l e r s (Ref. 130). T r a c i , F a r r and
f i o w (see s e c t i o n 12.1).
Albano (Ref.
Ballhaus and G o o r j i a n (Refs.
i 3 l ) presented a method f o r the
s o l u t i o n o f a low-reduced-frequency approximation
165, 166)
o f t h e equation.
p u b l i s h e d c a l c u l a t e d r e s u l t s f o r t h e t h i n n e r NACA
I n b o t h methods, a r e l a x a t i o n
64A006 a i r f o i l w i t h f l a p . A l s o f o r t h i s a i r f o i l
technique i s a p p l i e d t h a t i s v e r y s i m i l a r t o t h e
comparisons were made w i t h r e s u l t s o f t h e method
one developed by Murman and Cole (Ref. 160). Both
o f Magnus and Yoshihara, and t h i s t i m e a much
E h l e r s and T r a c i e t a l . presented unsteady p r e s -
b e t t e r agreement was achieved (see a i s 0 c h a p t e r 13).
s u r e d i s t r i b u t i o n s f o r an a i r f o i l w i t h o s c i l l a t i n g
I n t h i s respect, i t i s o f
flap.
i n t e r e s t t o know t h a t on
a CDC 7600 computer t h e method o f B a l l h a u s r e q u i r e d
8 seconds per c y c l e o f o s c i l l a t i o n , whereas t h e
A q u i t e d i f f e r e n t approach t o the s o l u t i o n
procedure o f Magnus and Yoshihara r e q u i r e d 1500
o f t h e t r a n s o n i c m a l i - p e r t u r b a t i o n equation and
seconds p e r c y c l e .
the l i n e a r i z e d v e r s i o n o f i t was f o l l o w e d by Chan and Chen (Ref.
Recentiy, the method o f B a l l h a u s e t a l . was
129) and Chan and Brashear
extended w i t h a "shock-f i t t i n g " procedure (Vu,
(Ref. 1321, who used a f i n i t e - e l e m e n t
Seebass and B a l l h a u s ; Ref.
method. Here t h e f l o w f i e l d i s d i v i d e d i n t o a
167), which means t h a t
the unsteady-shock r e l a t i o n s a r e f e d e x p l i c i t l y
l a r g e number o f small subregions ( t r i a n g i e s , f o r
i n t o t h e c a l c u l a t i o n process. T h i s procedure i n -
i n s t a n c e ) . The v e l o c i t y p o t e n t i a l i n each sub-
creases f u r t h e r t h e e f f i c i e n c y o f t h e o r i g i n a l
r e g i o n i s approximated by piecewise polynomials,
method, i n which t h e shock wave evolves automat-
and t h e polynomial c o e f f i c i e n t s i n t h e v a r i o u s
i c a l l y as a p a r t o f t h e numerical s o l u t i o n .
subregions a r e determined by a m i n i m i z a t i o n pro-
A i s o Krupp and Coie (Ref. a finite-difference
cedure. An advantage of t h e method i s the easy
128) formulated
way o f t a y l o r i n g t h e a i r f o i l contour and t h e shape
method f o r t h e low-reduced-
frequency v e r s i o n o f t h e t r a n s o n i c s m a l l - p e r t u r -
o f t h e shock wave. According t o references 129 and
b a t i o n equation. They produced an i n t e r e s t i n g
130, t h i s method o f f e r s a l s o t h e promise o f a v a s t
example of a p l u n g i n g NACA 0012 a i r f o i l , which
r e d u c t i o n i n computation times and, t h e r e f o r e ,
shows a moving shock wave t h a t disappears d u r i n g
c e r t a i n l y deserves f u r t h e r a t t e n t i o n .
a p a r t o f the c y c l e (type-B shock m o t i o n ) .
To conclude t h e d e s c r i p t i o n of f i n i t e 12.6
d i f f e r e n c e methods based on t h e t r a n s o n i c small-
ROLE OF THE NLR RESULTS
p e r t u r b a t i o n e q u a t i o n , a l s o t h e work of Caradonna
Frm t h e review i n t h e preceding s e c t i o n s ,
and lsom (Refs. 168, i 6 9 ) , s h o u l d be mentioned.
i t i s apparent t h a t i n t h e p a s t few years con-
A l t h o u g h they d e r i v e d and s o l v e d t h e s m a l l - p e r t u r -
-104-
-SOL-
MO13 3 1 N O S N V ö l A a V ä l S N n äOd S û O H 1 3 W N O I l V l n 3 l V 3 MàN 3WOS 30 N O l l V l l l V A 3 El
steady wind-tunnel
t e s t s a reasonable e s t i m a t e can
be given o f the amount o f i n t e r f e r e n c e w i t h the s l o t t e d tunnel w a l l s , a s i t u a t i o n which has n o t y e t been achieved f o r t h e unsteady measurements.
13.2
COMPARISONS BETWEEN THEORY AND EXPERIMENT I N STEADY AND QUASI-STEADY FLOW
13.2.1
Correction f o r tunnel-wall
interference
To determine t h e amount of w a l l i n t e r f e r ence i n t h e steady wind-tunnel
tests,
i t was
assumed t h a t t h e main e f f e c t of t h e tunnel w a l l s
i s a change i n e f f e c t i v e incidence, which f o r the NLR P i l o t t u n n e l can be taken as (Ref.
I F i g . 13.1
Comparison o f t h e s t e a d y - f l o w f i e l d s as p r e d i c t e d b y a number o f f i n i t e d i f f e r e n c e methods.
Au = C C,
where C,
(degrees),
175): (13.1)
i s t h e steady l i f t c o e f f i c i e n t , and
C a
I j l ) , which i s based on a n o n - c o n s e r v a t i v e t r e a t -
c o e f f i c i e n t t h a t v a r i e s w i t h the free-stream Mach
ment of the s m a l l - p e r t u r b a t i o n e q u a t i o n , d e v i a t e s
number ( F i g .
c o n s i d e r a b l y from t h e o t h e r ones. It
i5
steady f l o w f i e l d as shown i n f i g u r e 1 3 . 1 a l s o
It is,
the quasi-steady
r e s u l t s as o b t a i n e d from the measurements a r e
i n e v i t a b l e t h a t d i s c r e p a n c i e s i n the
show up i n the unsteady r e s u l t s .
13.2).
I n t h e f o l l o w i n g examples,
presented b o t h w i t h and w i t h o u t c o r r e c t i o n f o r the tunnel w a l l s . The u n c o r r e c t e d v a l u e s a r e
therefore,
a d v i s a b l e t o compare n o t only t h e unsteady r e s u l t s ,
g i v e n a l s o , because they a r e c o n s i s t e n t w i t h the
b u t a l s o the corresponding r e s u l t s o f t h e c a l c u -
f u l l y unsteady r e s u l t s , on which no t u n n e l - w a l l
l a t i o n s f o r the a i r f o i l a t rest.
c o r r e c t i o n c o u l d be a p p l i e d .
The comparative c a l c u l a t i o n s f o r the sym-
A major s o u r c e o f discrepancy between
m e t r i c a l NACA 64A006 a i r f o i l a r e performed f o r
t h e o r y and experiment i s t h a t a l 1 c a l c u l a t i o n
zero mean i n c i d e n c e and zero mean f l a p p o s i t i o n ,
methods developed so f a r f o r unsteady t r a n s o n i c
w h i l e t h e c a l c u l a t i o n s f o r t h e l i f t i n g NLR 7301
f l o w deal w i t h i n v i s c i d f l o w and thus n e g l e c t t h e
a i r f o i l concern a mean incidence, aC, which i n -
boundary l a y e r ( e x c e p t f o r a r e c e n t a t t e m p t i n r e f e r e n c e 1 2 1 ) . Moreover, t h e experimental d a t a a r e contaminated by t u n n e l - w a l l
interference.
To g a i n some i n s i g h t i n t o t h e i n f l u e n c e of t h e boundary l a y e r , t h e d i s c u s s i o n o f t h e unsteady r e s u l t s i s preceded by a c o n s i d e r a t i o n o f q u a s i -
-1.0 -'s
steady r e s u l t s for b o t h the NACA 64A006 and t h e
t
NLR 7301 a i r f o i l . Considering t h e q u a s i - s t e a d y case f i r s t has t h e g r e a t advantage t h a t f o r t h e
n =a,.Aa A n . C.C&
comparative c a l c u l a t i o n s use can be made of t h e e x i s t i n g r o u t i n e method f o r steady f l o w , which
(SUBSCRIPT~DENOTESUNCORRECTED,MEASUREDVALUESI
i n c l u d e s a procedure t o t a k e t h e boundary l a y e r i n t o account
(DEGI
F i g . 13.2
(Ref. 118). I n a d d i t i o n , f o r t h e
-106-
Wall-interference correction f o r the NLR P i l o t t u n n e l .
cludes rhe.corrsc:ion
For w a I i
aerodynamic l o a d i n g i s even l a r g e r than i n sub-
i n t e r f e r e n c e . No
fui-ther c o r r e c t i o n i s a p p l i e d on the v a r i a t i o n s
s o n i c flow. T h i s i s demonstrated f o r the NACA
i n f l a p a n g i e o r i n c i d e n c e around the r e s p e c t i v e
64A006 a i r f o i l i n f i g u r e
1 3 . 5 , and Far the NLR f i g u r e 1 3 . 6 . Both f i g u r e s show a
mean c o n d i t i o n s , which a r e necessary t o determine
7301 a i r f o i l i n
the t h e o r e t i c a l q u a s i - s t e a d y r e s u l t s .
c o n s i d e r a b l e improvement i n the t h e o r e t i c a l p r e d i c t i o n i f t h i c k n e s s and boundary-layer e f f e c t s a r e taken i n t o account simultaneously.
1 3 . 2 . 2 Subsonic f l o w
HACA MAW6 AIRFOIL WITH FLAP U .,
@
1"G
The f i r s t Comparison concerns t h e measured
INVISCID WTEHTIAL FLOW IREF. l W
and c a l c u l a t e d subsonic p r e s s u r e d i s t r i b u t i o n s f o r t h e NACA 64A006 a i r f o i l ( F i g . 1 3 . 3 ) .
a*.0 .
0.50
STEADY lSo.l'l L I N C L . BOUNDARY LAIER
The steady
p r e s s u r e d i s t r i b u t i o n s on t h e a i r f o i l w i t h deflected f l a p (Fig.
i3.3a) c l e a r l y indicate t h a t a
i a r g e p a r t o f the d i f f e r e n c e between the r e s u l t s o f p o t e n t i a l t h e o r y and experiment can be a t t r i b u t e d t o t h e presence o f t h e boundary l a y e r . From t h e corresponding q u a s i - s t e a d y p r e s s u r e d i s t r i b u t i o n s (Fig.
13.3b).
i t can be concluded t h a t , com-
pared w i t h t h i n - a i r f o i l t h e o r y ,
the i n c l u s i o n o f
t h i c k n e s s increases t h e p r e s s u r e ahead o f t h e h i n g e a x i s , whereas the pressures on t h e f l a p show a s l i g h t decrease. F u r t h e r , the a d d i t i o n o f the o
boundary l a y e r has a s i g n i f i c a n t e f f e c t , which
EXPERIMENT EXPERIMENT CORRECTED FOR WALL INTERFERENCE
compensates t o a l a r g e e x t e n t the c o n t r i b u t i o n due
I
1
t o thickness. The e f f e c t o f t h e boundary l a y e r i s a l s o s T g n i f i c a n t f o r the 1 6 . 5 p e r cent t h i c k NLR 7301 a i r f o i l ( F i g . i 3 . 4 a ) . The quasi-steady
I I I
resuits
g i v e n i n f i g u r e 1 3 . 4 b demonstrate t h a t t h e d e v i a t i o n s o f the t e s t r e s u l t s from t h i n - a i r f o i l
@
DUASI-SIEAOI
I
theory
S
a r e due t o the combined e f f e c t s o f thickness,
THIN4IRÇOIL THEORY INCL. THICKNESS
incidence, and v i s c o s i t y . The e f f e c t o f t h i c k n e s s
INCL. IHICXNESS
I
BOUNDARY LAIER
and i n c i d e n c e dominates on t h e f r o n t p a r t of t h e a i r f o i l , and t h e e f f e c t o f t h e boundary l a y e r i s more important towards t h e r e a r (see a l s o t h e d i s c u s s i o n i n s e c t i o n i0.2.l). F u r t h e r i t can be noted t h a t the presence o f t h e tunnel w a l l s has a i a r g e i n f l u e n c e on b o t h t h e NACA 64A006 and t h e NLR 7301 a i r f o i i .
t
1 3 . 2 . 3 Transonic f l o w w i t h shock wave I n transonic flow,
t h e presence o f the
O
boundary. l a y. e r leads t o a c o n s i d e r a b l e change i n
Fig. 13.3 Effect of thickness and boundary layer an the steady snd quasi-steady pressure distribution in subsonic flow.
shock p o s i t i o n and shock s t r e n g t h . As a consequence,
i t s e f f e c t on t h e steady and q u a s i - s t e a d y
-107-
@
c o l l e c t e d i n t a b l e 7. T h i s t a b l e shows, Far i n -
I T E S O Y MEAN PCSITION,
stance, t h a t a t bi_ = 0 . 8 5 the i n c l u s i o n o f wing
NLR i101 AIRFOIL
t h i c k n e s s increases the n o r m a l - f o r c e c o e f f i c i e n t , k , a s obtained w i t h t h i n - a i r f o i l theory, w i t h
EXPERIMENT t ~I o .85 RE=1.7 I 106
.
INVISCID
about 55 per c e n t , whereas the boundary l a y e r i s
TRANSITION STRIP AT s i c =.I
r e s p o n s i b l e f o r a decrease o f about 10 per cent.
WITH BCUNDARY LAYER CORRECTION
The i n t e r f e r e n c e w i t h t h e tunnel w a l l s c o n t r i b u t e s f o r r o u g h l y 20 p e r c e n t . A f t e r t h e v a r i o u s c o r r e c t i o n s , t h e agreement between t h e o r y and experiment i s improved c o n s i d e r a b l y . The aerodynamic c o e f f i c i e n t s f o r the NLR 7301 a i r f o i l a t ,M
= 0.7
(table
8)
reveal t h a t
a l s o f o r t h i s case t h i c k n e s s and i n c i d e n c e p r o v i d e
in
@
QUAII-STEADY [UPPER SURFACE1
25.
O EXPERIMENT
u~=.8S~.Ae..IO
EXPERIMENTCORRECTEDFGR
15 THIN-AIRFOIL THEORY INCL. THICKNEIS AND INCIDENCE INCL. THICKNEIS. INCIDENCE ANOBOUNDARY LAYER
10
5
O
O
.IC
~ i g .13.4
Effect of thickness, incidence and boundary layer on the steady and quasi-steady pressure distribution in subsonic flow (condition I).
The f a c t t h a t f o r t h e NACA 64AU06 a i r f o i l w i t h f l a p t h e e f f e c t o f t h i c k n e s s i s f u l l y com-
Fig. 13.5
pensated by t h e e f f e c t o f t h e boundary l a y e r i s i l l u s t r a t e d a l s o by t h e o v e r a l l c o e f f i c i e n t s ,
-108-
Effect of thickness and boundary layer an the steady and quasi-steady pressure distribution in transonic flow.
an increa5e o f the t h i o - a i r f o i l v a i u e o f the n o r m a l - f o r c e c o e f f i c i e n t , ko, o f more than 50 p e r c e n t . Accounting f o r t h e boundary l a y e r leads t o a decrease o f about 35 p e r cent,
,-
as f o i l o w s from
INCL.THICXNESS WITH B.L.CORRECTION
,-
b o t h values o b t a i n e d w i t h t h e non-conservative
IHCL.THICKHESS
c a l c u l a t i o n scheme. The c o n s e r v a t i v e scheme, which guarantees t h e b e s t numerical s o l u t i o n , d i d n o t converge f o r i n v i s c i d flow, so t h i s v a l u e i s n o t included. The l a s t two columns show t h a t i n t h e present t e s t s t h e tunnel w a l l s a r e r e s p o n s i b l e f o r a c o r r e c t i o n of about 30 p e r cent. A t M_ = 0.5, the e f f e c t s mentioned above a r e l e s s than a t t r a n s o n i c speed,
but s t i l l s i g n i f i c a n t .
,-
WITH B.L.CORRECTION IOUASI-CONSERVATIVE) INVISCIO [NON-CONSERVATIVE1
r
U,
THIN4IRFOIL THEORY
.I
=.=P.R s i L \ i i \ 0 6 TRANSITIOH STRIP AT
-1.0
.-
-.
20 THIN-AIRFOIL
O O
THEORY
.Ir
‘o I \I,INCL.THICKNESS
O
Fig. 13.1% Steady pressure distribution in the mean position (condition 11-1).
Thin-airfoil theory
H
kC
m
C
Fig. 13.6b Quasi-steady Pressure distribution in transonic flow (condition 11-1).
t
Theory o f Ref. 118 I n c l . thickness Incl.thickness + bound. l a y e r kC
m
C
r,<
LOWERSURFACE
kC
m
C
Experiment i n c l . tunnelUncorrected wall correction kc
m
C
kC
m
C
0.8
2.03
0.69
2.44
0.80
1.88
0.65
1.66
0.61
1.32
0.61
0.85
2.31
0.79
3.61
1.29
1.94
0.60
1.82
0.75
1.41
0.75
- 1og-
TABLE 8 Quasi-steady aerodynamic c o e f f i c i e n t s (NLR 7301 a i r f o i l ) ~
Theory o f Ref. l i 8 I n c l . thickness + bound. l a y e r
Thin-ai r f o i I
Mm
m
1O
ka
m
a
ka
Experiment I n c l . thickness I n c l . tunnel+ bound. l a y e r w a l l c o r r e c t i o n Uncorrected
m
ka
m
m
ka
ka
0 . 5 0.85'
2.31
O
2.73
0.04
2.53
-0.04
2.53
-0.04
2.55
-0.10
2.18
-0.09
0 . 7 3.00'
2.80
O
4.24
0.11
3.21
0.00
3.92
-0.22
4 . IO
-0.44
3.20
-0.34
Non-conservative F-O scheme
Conservative F-D scheme
Non-conservative F-O scheme
F-D = F i n i t e D i f f e r e n c e c o n t r i b u t e d by Kooi, Ref.
176). The reason f o r
t h i s i s t h a t i n c r e a s i n g the Reynolds number produces two c o u n t e r a c t i n g e f f e c t s . One e f f e c t i s t h e decrease o f t h e boundary-layer t h i c k n e s s , which changes t h e p r e s s u r e d i s t r i b u t i o n i n t h e d i r e c t i o n of
i n v i s c i d theory. The o t h e r i s t h a t the t r a n s i -
t i o n p o i n t , i n which t h e boundary l a y e r changes from laminar t o t u r b u l e n t , s h i f t s towards the l e a d i n g edge. T h i s r e s u l t s i n a t h i c k e n i n g o f the boundary l a y e r , which c o u n t e r a c t s the F i r s t effect.
From t h e examples discussed so Far, i t can be concluded t h a t i n v i s c i d - f l o w t h e o r i e s t h a t i n c l u d e a proper r e p r e s e n t a t i o n o f t h i c k n e s s and i n c i d e n c e produce i n a q u a l i t a t i v e senqe c o r r e c t c h a r a c t e r i s t i c s o f the pressure d i s t r i b u t i o n s . Representation o f t h e boundary i a y e r i s , however, e s s e n t i a l t o o b t a i n q u a n t i t a t i v e agreement between t h e o r y and experiment.
F i g . 13.7
I n f l u e n c g of Reynolds number and l o c a t i o n of t r a n s i t i o n p o i n t on t h e steady m e s s u r e d i s t r i b u t i o n ( c o n d i t i o n 11-2).
1 3 . 2 . 4 "Shock-f ree" f l o w To conclude t h e e v a l u a t i o n o f t h e c a p a b i l i t y o f advanced t h e o r i e s on t h e b a s i s o f quasi-
The experiments and c a l c u l a t i o n s g i v e n above deal w i t h a r e l a t i v e l y low Reynolds number
steady flow,
(about 2 m i l l i o n ) , w h i l e f o r f u l l - s c a l e a i r c r a f t
t h e NLR 7301 a i r f o i l w i l l be considered. For t h i s
one i s i n t e r e s t e d i n much h i g h e r values o f t h i s
purpose, a comparison i s made between r e s u l t s from
parameter ( 2 0
-
the "shock-free"-flow
condition o f
c a l c u l a t i o n s w i t h t h e method o f Bauer, Korn
40 m i l l i o n ) . I t may be expected,
however, t h a t a t h i g h Reynolds numbers t h e i n -
and Garabedian (Ref. 116)
f l u e n c e o f t h e boundary l a y e r remains of t h e same
"shock-free"
o r d e r o f magnitude a s shown here (see f i g u r e 1 3 . 7 ,
r e s u l t s f o r t h e c o n d i t i o n a t which "shock-free('
-110-
for the theoreticai
design c o n d i t i o n , and experimental
f l o w i s o b t a i n e d i n the wind t u n n e l .
I n t h i s way
the f a c t t h a t the experimental design c o n d i t i o n ( i . e . Mach number and i n c i d e n c e ) d i f f e r s from the i n v i s c i d t h e o r e t i c a l d e s i g n c o n d i t i o n can be d i s carded, which assures t h a t b o t h t h e o r y and experiment deai w i t h the c a r e f u l l y balanced c o n d i t i o n o f "shock-free"
flow.
The steady p r e s s u r e d i s t r i b u t i o n s computed f o r incidences a t and around t h e d e s i g n c o n d i t i o n ( F i g . l 3 . 8 a ) e x h i b i t t h e same c h a r a c t e r i s t i c changes i n shape i n the s u p e r s o n i c r e g i o n a t t h e upper surface as those observed i n t h e measure-
10.10).
ments ( F i g .
The lower surface behaves v e r y
r e g u l a r 1y . The comparison between t h e corresponding quasi-steady pressure d i s t r i b u t i o n s given i n f i g u r e 13.8b shows a c o n s i d e r a b l e improvement over thin-airfoil
theory. The t y p i c a l c h a r a c t e r o f t h e
d i s t r i b u t i o n aiong t h e upper surface i s p r e d i c t e d reasonably w e l l , and a l s o the p r e d i c t i o n f o r the lower surface i s improved. T h i s j u s t i f i e s t h e e x p e c t a t i o n t h a t methods based on i n v i s c i d theory are able t o predict, a t least qualitatively, the main c h a r a c t e r i s t i c s o f t h e unsteady f l o w f o r o s c i i l a t i o n s aroung t h e "shock-free" dition.
design con-
NLR 7301 AIRFOIL
I t has t o be noted, however, t h a t t h i s
c o n c l u s i o n i s based on comparisons w i t h s o l u t i o n s o f the f u l l p o t e n t i a l e q u a t i o n . Whether s i m i l a r
THIN-IIIRFOIL THEORY
improvements can be achieved w i t h s m a l l - p e r t u r b a -
THEORETICAL DESIGN CONDITION
t i o n theories i s questionable.
EXPERIMENTAL OEIIGN CONDITION
NLR 7301 AIRFOIL
-.-
LOWER SURFACE
Fig. 13.8b
13.3
Quasi-steady pressure distributions for the theoretical and experimental "shock-free" design condition (condition 1x1).
COMPARISONS BETWEEN THEORY AND EXPERIMEN1
I N UNSTEADY FLOW
13.3.1 Fig. 13.8a
Calculated steady pressure distributions st and about the theoretical "shock-free" design condition.
Pressure d i s t r i b u t ions
As mentioned before, t h e comparison between t h e o r y and experiment f o r f u l l y unsteady f l o w has
-I 1l -
t o be i i m i t e d
to
the NACA 64A006 a i r f o i l w i t h f i a p ,
sponds v e r y w e l l w i t h the observed d i f f e r e n c e s i n
s i n c e , a t t h e moment o f w r i t i n g , t h i s a i r f o i i i s t h e
quasi-steady f l o w ( s e c t i o n 13.2.2).
only one f o r which r e s u l t s b o t h from measurements
The t h e o r e t i c a l magnitude d i s t r i b u t i o n s o b t a i n e d
and from c a l c u l a t i o n s a r e a v a i l a b l e . The comparisons
by the d i f f e r e n t a u t h o r s f o r 120 Hz a r e s t i l l f a r
a r e l i m i t e d t o i n v i s c i d - f l o w t h e o r i e s ; no c o r r e c -
from unanimous. The r e s u l t s of Chan and Brashear
tions are applied for wall interference.
look somewhat s u s p i c i o u s , s i n c e t h e i r magnitude d i s t r i b u t i o n on t h e f r o n t p a r t o f t h e a i r f o i l i s
(a)
Pressure d i s t r i b u t i o n s i n high-subsonic f l o w
I n f i g u r e 13.9. f o r high-subsonic flow.
c o n s i s t e n t l y l e s s than t h a t p r e d i c t e d by t h i n a i r f o i l theory, which c o n f l i c t s w i t h what should
some comparisons a r e shown
-
happen p h y s i c a l l y .
For t h e sake o f compiete-
ness, a l s o t h e quasi-steady
results ( f
O),
(As shown i n r e f e r e n c e 132,
t h i s d i f f e r e n c e e x i s t s a l r e a d y f o r quasi-steady
ob-
flow.)
t a i n e d w i t h t h e method o f r e f e r e n c e 118, a r e added
W i t h r e s p e c t t o t h e phase curves i n f i g u r e
13.9 i t can be observed t h a t , compared w i t h t h i n -
A comparison between t h e chordwise magni-
a i r f o i l theory,
tude d i s t r i b u t i o n s f o r 30 Hz c a l c u l a t e d w i t h t h i n -
t h e phase curves upstream o f the
h i n g e a x i s have changed i n a d i r e c t i o n t h a t c o u l d
a i r f o i l t h e o r y , by E h l e r s and by Roos, shows t h a t
be expected from t h e g r a p h i c a l experiment .shown i n
i n f r o n t o f t h e h i n g e a x i s t h e more advanced
f i g u r e 8.13.
theories are less accurate i n predicting the
I t i s n o t understood why f o r 30 Hz the
phase c u r v e c a l c u l a t e d by E h l e r s does n o t behave i n
measurements than t h i n - a i r f o i l t h e o r y . On t h e flap,
t h e same sense.
a s l i g h t improvement can be n o t i c e d . T h i s c o r r e -
A comparison between experiment, t h e two ,üAüER.KORH LLGARABEOIAN íREF.IIBI
c a l c u l a t i o n methods f o r high-subsonic f l o w developed a t NLR, and t h e high-subsonic method o f I s o g a i (Ref. NACA 64A006 AIRFOIL WITH FLAP
U,.0.80
O
1 4 2 ) . g i v e n i n f i g u r e 13.10,
iilus-
t r a t e s t h a t t h e r e l a t i v e l y simple D o u b l e t - L a t t i c e
.moo''O
method w i t h local-Mach-number c o r r e c t i o n g i v e s
EXFERIMEHT THIN-AIRFOIL THEORY
n*c*
U A W ~AIRFOIL WITH FLAP
u,.o.azs
u..o-
LI ZO^^:^.^.^
PHASE ANGLE
L...'
Pig. 13.9
Fig. 13.10
Comparison between calculated and measured unsteady pressure distributions in high-subsonic flow.
-112-
Comparison between calculated and measured unsteady pressure distributiona in hi&h-subaonic flow.
about the same r e s u l t 5 as the methods o f Zwaan (Ref.
139)
and i s a g a i . S i m i i a r r e s u i t s have been
r e p o r t e d by Giesing, Kaiman and Rodden (Ref.
i45),
who f o i i o w e d a c l o s e l y r e l a t e d approach. (b)
Pressure d i s t r i b u t i o n i n t r a n s o n i c f l o w
The results i n f i g u r e i 3 . i I
for the s l i g h t -
l y s u p e r c r i t i c a l f i o w c o n d i t i o n a t N_ = 0.85 a r e taken from E h i e r s (Ref. G o o r j i a n (Refs.
i 3 0 ) , B a i l h a u s and
165, 166),and
I s o g a i (Ref.
125).
As can be observed, t h e d e v i a t i o n s between t h e o r y and experiment a r e o f t h e same o r d e r o f magnitude as those i n quasi-steady fiow.
Compared w l t h t h i n -
a i r f o i l theory, t h e r e i s a d r a s t i c improvement i n a q u a l i t a t i v e sense, e s p e c i a l l y t h e peak i n t h e magnitude d i s t r i b u t i o n t h a t occurs near the mean shock p o s i t i o n i s p r e d i c t e d v e r y w e i l . A l s o the c h a r a c t e r of the phase curves i s improved, i n p a r t i c u l a r i n f r o n t o f t h e h i n g e a x i s , where t h e t y p i c a l steepening e f f e c t c o n f i r m s the observat i o n s from the g r a p h i c a l experiment ( F i g . 8.13).
In q u a n t i t a t i v e sense t h e r e i s s t i l l a l a r g e d i s crepancy between t h e o r y and experiment, which
-
as
was argued i n t h e preceding s e c t i o n f o r q u a s i steady f l o w (see f i g u r e 13.5)
-
should be a t t r i b -
u t e d m a i n l y t o the e f f e c t s o f the boundary l a y e r and t h e tunnel w a l l s . S i m i l a r remarks as t h o s e made f o r the examples o f f i g u r e 1 3 . 1 1 a p p l y t o t h e comparisons i n f i g u r e 13.12,
which deal w i t h a t r a n s o n i c f i o w
w i t h a well-developed shock wave. The discrepanc i e s between the r e s u i t s o f t h e d i f f e r e n t t h e o r e t i c a l methods can t o a l a r g e e x t e n t be a t t r i b u t e d t o the d i f f e r e n c e s t h a t do e x i s t a l r e a d y i n t h e corresponding s t e a d y - f l o w f i e l d , in figure 13.1:
which were shown
The s t e a d y - f l o w f i e l d o f Magnus
and Yoshihara, f o r instance, g i v e s a s t r o n g e r shock wave a t a more rearward p o s i t i o n than the
Fig. 1 3 . 1 1 Comparison between c a l c u l a t e d and measured unsteady pressure d i s t r i b u -
s m a l l - p e r t u r b a t i o n method. Thus i t i s n o t surp r i s i n g t h a t i n t h e i r r e s u l t s t h e h i g h peak i n t h e
t i o n s i n s l i g h t l y s u p e r c r i t i c a l flow.
unsteady p r e s s u r e d i s t r i b u t i o n generated by t h e shock wave ( F i g . 13.12)
i s h i g h e r and l o c a t e d more
downstream than i n the o t h e r c a l c u l a t i o n s . Moreover,
13.3.2
Aerodynamic c o e f f i c i e n t s
t h e increased shock s t r e n g t h makes i t more d i f f i c u l t f o r t h e pressure p e r t u r b a t i o n s generated by t h e
As f a r as t h e r e s u l t i n g o v e r a i i unsteady
o s c i l l a t i n g f l a p t o t r a v e l around t h e t o p o f t h e
forces and moments a r e concerned, T r a c i , F a r r and
shock wave, which r e s u l t s i n a lower p r e s s u r e i e v e i
Aibano (Ref.
upstream o f the shock wave.
t h e behaviour o f t h e c o e f f i c i e n t s as a f u n c t i o n o f
-113-
1 3 i ) a r e t h e o n l y ones who considered
IWHIHSOA 9 SON&W
the n o n l i n e a r equations i n which no d i s t i n c t i o n i s made between steady and unsteady Flow a r e a b i e t o p r e d i c t t h e d e t a i l s o f the p e r i o d i c a l shock-wave
o
I
\
e x c u r s i o n s - w i t h o u t a d d i t i o n a l measures. T h i s i s
w,
\ 180'
900
276
160"
i l l u s t r a t e d w i t h the r e s u l t s o f some c a l c u l a t i o n s , which e x h i b i t t h e d i f f e r e n t types o f shock-wave motion d e s c r i b e d i n chapter 9 . F i g u r e 13.15 c o n t a i n s the instantaneous p r e s s u r e
U,
0.873
d i s t r i b u t i o n s computed by Magnus and Yoshihara
.m '0
(Ref.
-THEORY
--- EXPERIMENT
II'HARMONIC ONLY)
121) f o r t h e a i r f o i l w i t h f l a p a t M-
=
0.875.
I t i s e v i d e n t t h a t the shock m o t i o n i s o f t y p e A , i.e.
VARIATION IN MOMENT
.cz-
t h e shock moves almost s i n u s o i d a l and remains
present d u r i n g t h e complete c y c l e o f o s c i l l a t i o n .
JCM
VkRIATION IN HINGE MOMENT
.oc2r
Fig. 13.14
Time histories of the unsteady airloads as calculated by Magnus and Yoshihara (Ref. 121).
l i n e a r i z e d transonic small-perturbation equation. T h i s problem can be s o l v e d by a s h o c k - f i t t i n g p r o cedure, f o r which purpose an a d d i t i o n a l r e l a t i o n must be f u i f i l i e d a t t h e mean shock p o s i t i o n . T h i s a d d i t i o n a l r e l a t i o n can be d e r i v e d by matching a m a l i - p e r t u r b a t i o n expansion o f t h e n o n l i n e a r p o t e n t i a l e q u a t i o n w i t h the unsteady shock r e l a t i o n s f o r an o s c i l l a t i n g shock wave.
In principle,
t h i s idea was employed a l r e a d y i n t h e s t u d i e s o f Coupry and P i a z o i l i (Ref. and Landahl (Ref.
l o g ) , Eckhaus (Ref. I I O ) ,
2 4 ) . Rzcent work a l o n g these
/ i n e s i s r e p o r t e d by Hafez, R i z k and Murman (Ref.
F i g . 13.15
177). I n c o n t r a s t h e r e w i t h , t h e methods based on
-115-
Calculated inStBntBIleOUS pressure distributions at the upper surface shoving type-A Shock motion (reproduced from Ref. 121).
A good agreement
i5
o b t a i n e d between the shock
numerical v i s c o s i t y .
t r a j e c t o r y c a l c u l a t e d f o r Mm = 0.875 and the measured t r a j e c t o r y f o r
M_
= 0.90.
the shock has disappeared, probably d i s s i p a t e d by
The small s h i f t
F i n a l l y , a more i l l u s t r a t i v e exampie o f a
i n f r e e - s t r e a m Mach number agrees v e r y w e l l w i t h
type-C shock-wave m o t i o n ( c a l c u l a t e d by Bal ihaus
t h e f a c t t h a t t h e c a l c u l a t e d steady pressure d i s -
and Steger, Ref.
t r i b u t i o n a t Mm = 0.875 upstream o f the shock wave
a pulsating parabolic-arc a i r f o i l . This a i r f o i l
127) i s shown i n f i g u r e 13.18 f o r
15 chord l e n g t h s o f a i r -
shows a s y s t e m a t i c d e v i a t i o n from t h e measured
thickens during the f i r s t
p r e s s u r e s ( s e e f i g u r e 13.1).
f o i l t r a v e l and becomes f l a t a g a i n d u r i n g t h e sub-
which corresponds t o
a s h i f t i n Mach number o f about 0.02.
The sugges-
sequent 15 chord lengths.
During t h e t h i c k e n i n g
t i o n (Refs. 121 and 162) t h a t t h i s small d e f i c i e n -
p a r t o f t h e motion (Fig. 13.18a),
cy may be caused by some w a l l - i n t e r f e r e n c e e f f e c t s
forms and propagates downstream,
seems p l a u s i b l e , a t l e a s t f o r a p a r t . Comparative
maximum downstream l o c a t i o n a t t = 18.25 chord
c a l c u l a t i o n s w i t h t h e method o f Bauer e t a l .
l e n g t h s o f t r a v e l . As t h e a i r f o i l becomes t h i n n e r ,
(Ref.
a shock wave reaching i t s
118), however, show t h a t t h e presence o f t h e bound-
t h e shock wave reverses d i r e c t i o n and propagates
a r y l a y e r upstream o f t h e shock wave, which i s
o f f the f r o n t o f the a i r f o i l
(Fig.
i3.18b).
n e g l e c t e d i n t h e c a l c u l a t i o n s o f r e f e r e n c e s 121 and 162,
i s r e s p o n s i b l e f o r t h e m a j o r p a r t o f the
observed s h i f t i n l o c a l Mach number upstream o f
FLAP POSITION
t h e shock wave. Ballhaus and G o o r j i a n (Refs.
165, 166) NACA 61Aüü6 AIRFOIL WITH FLAP
succeeded i n reproducing a l l t h r e e types o f shock-
U,*O.eS4
wave motion. T h e i r r e s u l t s f o r type-B and type-C m o t i o n a r e shown i n f i g u r e s 13.16 and 13.17,
liPOHz.kiO.179
6 GWRJIAH ____ 8ALLHAUS UAGNUS 6 YOSHIHARA
SHOCK POSITION 's/<
res-
p e c t i v e l y . Since the c a l c u l a t i o n s deal w i t h a frequency o f 90 Hz, whereas t h e corresponding t e s t s were performed a t 120 Hz, no d i r e c t compari s o n between the c a l c u l a t e d and t h e measured shock t r a j e c t o r i e s can be made. A c l o s e r l o o k a t t h e subsequent i n s t a n t a neous p r e s s u r e d i s t r i b u t i o n s o f f i g u r e 1 3 . i 6
re-
v e a l s t h a t t h e shock vanishes d u r i n g a p a r t o f t h e c y c l e , a l t h o u g h the l o c a l f l o w remains supersonic.
In f i g u r e 13.16 a comparison i s made a l s o between t h e r e s u l t s o f B a l l h a u s and G o o r j i a n and t h e r e s u l t s o f Hagnus and Yoshihara.
i t appears
t h a t , f o r t i i s t h i n a i r f o i l and i t s r e l a t i v e l y
.2
.<
.6
small a m p l i t u d e o f o s c i l l a t i o n , a s a t i s f a c t o r y agreement i s obtained. From f i g u r e 1 3 . i 7 ,
which shows a type-C
shock-wave motion, i t can be observed t h a t a shock wave i s formed a t some t i m e between a and b, which s t r e n g t h e n s and propagates upstream. Between c and d t h e embedded supersonic r e g i o n i s c o m p l e t e l y e l i m i n a t e d and, a l t h o u g h t h e f l o w i 5 e n t i r e l y 5ub-
Fig. 13.16
s o n i c , t h e shock wave c o n t i n u e s t o propagate upstream, as shown a t times g , h, and i. A t t i m e j ,
-i16-
C a l c u l a t e d instantaneous p r e s s u r e d i s t r i b u t i o n s a t t h e upper surface, showing t y p e 4 shock m o t i o n ( r e p r o duced from Ref. 165).
e n g i n e e r i n g type o f approach has t o be Followed i n the coming p e r i o d .
I n i h i s r e s p e c t , the ideas
developed by Magnus and Yoshihara (Ref. serve a t t e n t i o n ,
121) de-
s i n c e t h e i r r e l a t i v e l y simple
"viscous-ramp'' model lends i t s e l f f o r easy impiem e n t a t i o n i n i n v i s c i d c a l c u l a t i o n methods. I n r e l a t i o n t o the importance o f t h e boundary l a y e r ,
a l s o t h e i n s i g h t i n t o t h e e f f e c t s o f t h e Reynolds number s h o u l d be f u r t h e r increased. S m a l l - p e r t u r b a t i o n methods a r e v e r y a t t r a c t i v e from a computational p o i n t of view. Therefore, the l i m i t s of such methods s h o u l d be i n v e s t i g a t e d , i n p a r t i c u l a r when they a r e a p p l i e d t o t h i c k s u p e r c r i t i c a l a i r f o i l s o f the t y p e considered i n
L
.iL
this investigation.
I n t h i s respect, i t i s the
o p i n i o n o f the a u t h o r t h a t an improvement of the s m a l l - p e r t u r b a t i o n t h e o r i e s may be o b t a i n e d
-
es-
p e c i a l l y f o r t h e o r i e s i n which a d i s t i n c t i o n i s made between s m a l l unsteady p e r t u r b a t i o n s superimposed
-
upon a non-uniform steady f l o w f i e l d
i f the
-1.2,
F i g . 13.17
13.4
C a l c u l a t e d Instantaneous p r e s s u r e d i s t r i b u t i o n s s t t h e upper surface, showing type-C shock m o t i o n ( r e p r o duced from Ref. 165).
=
C O N C L U D I N G REMARKS
O
@
From t h e preceding e v a l u a t i o n i t i s apparent t h a t the i n c l u s i o n o f a i r f o i l thickness,
10
SHOCK PROPAGATES 0 0 W S T R E m
in-
cidence, and t r a n s o n i c shock motions i n i n v i s c i d f l o w c a l c u l a t i o n s leads t o an improvement o f the t h e o r e t i c a l p r e d i c t i o n s i n a q u a l i t a t i v e sense. Q u a n t i t a t i v e l y , a l a r g e discrepancy w i t h t h e r e a l flow remains as a r e s u l t o f the boundary l a y e r , which determines the f i n a l l o c a t i o n o f t h e shock t o a l a r g e e x t e n t , and by t h a t t h e o v e r a l l
un2 -I o
steady a i r l o a d s . Because the m o d e l l i n g o f unsteady boundary l a y e r s i s only i n i t s f i r s t s t a g e ( f o r a
-5
o
i
10
@ SHOCK PROPAGATES UPSTREAM
review o f t h e p r e s e n t s t a t u s , t h e reader i s r e f e r r e d t o reference i 7 8 ) ,
F i g . 13.18
i t i s u n l i k e l y t h a t i n the
near f u t u r e s o p h i s t i c a t e d c a l c u l a t i o n methods w i l l become a v a i l a b l e f o r t h i s purpose. T h e r e f o r e , an
-117-
Example o f type-C shock-wave m o t i o n as c a l c u l a t e d b y B a l l b a u s and Steger (Ref. 127) for a p a r a b o l i c a i r f o i l w i t h pulsating thickness.
steady f l o w f i e l d t h a t forms p a r t o f the s m a i i -
suppressed.
p e r t u r b a t i o n s o l u t i o n i s repiaced by a more accu-
F i n a i i y , from the c o n s i d e r a t i o n s o f t h e
r a t e steady f l o w f i e l d . The l a t t e r f l o w f i e i d
quasi-steady r e s u l t s (section 13.2)
should be o b t a i n e d from more advanced steady c a i -
clear that,
c u l a t i o n methods t h a t i n c l u d e boundary-layer e f -
of comparisons between t h e o r y and wind-tunnel
f e c t s , o r from experiment.
data,
I n t h i s way, d i f f e r -
i t has become
i n o r d e r t o improve t h e r e i i a b i i i t y
t h e r e i s an u r g e n t need for methods t o assess
ences i n unsteady a i r l o a d s due t o d i s c r e p a n c i e s
the amount o f w a i l i n t e r f e r e n c e i n unsteady exper-
i n t h e steady f l o w f i e l d (such as d i f f e r e n c e s
iments i n t r a n s o n i c t e s t s e c t i o n s w i t h s l o t t e d o r
in
shock p o s i t i o n and shock s t r e n g t h ) may be
14 IMPACT 14.1
porous w a l l s .
OF THE NLR INVESTIGATIONS AN0 FUTURE PROSPECTS
IMPACT OF THE NLR INVESTIGATIONS
t h i s respect i t i s worth noting t h a t f o r t h i s a i r f o i l c a l c u l a t i o n s a r e planned w i t h t h e a c c u r a t e
As s t a t e d b e f o r e , t h e aim o f t h e NLR inves-
b u t v e r y time-consuming method o f Magnus and I n t h i s way, a b a s i s i s c r e a t e d f o r
t i g a t i o n s was t o e x p l o r e t h e n a t u r e o f two-dimen-
Yoshihara.
s i o n a l unsteady t r a n s o n i c f l o w s and t o support the
comparison w i t h f a s t e r b u t more a p p r o x i m a t i v e
development o f t h e o r e t i c a l and s e m i - e m p i r i c a l pre-
methods. The NLR 7301 a i r f o i l w i l l be a s t r o n g
d i c t i o n methods. i n t h i s r e s p e c t , t h e i n v e s t i g a -
t e s t case f o r s m a l l - p e r t u r b a t i o n methods.since i t
t i o n s have proved t o be successful.
i s r a t h e r t h i c k and has a b l u n t nose, which may
The p h y s i c a l
i n s i g h t gained from t h e experiments d e s c r i b e d here
cause a v i o l a t i o n o f the s m a i i - p e r t u r b a t i o n
and p a r t l y pub1 ished i n p r e v i o u s papers has found
assumption. The v a l u e o f t h e NLR 7301 a i r f o i l a s a
a l r e a d y wide a p p l i c a t i o n i n t h e l i t e r a t u r e t h a t d e s c r i b e s the c h a r a c t e r i s t i c s of unsteady t r a n -
b a s i s f o r comparison w i l l be increased f u r t h e r by
s o n i c f i o w (see, f o r i n s t a n c e , references i78-182).
the t e s t s scheduled f o r a model o f t h i s a i r f o i l
The measured p r e s s u r e d i s t r i b u t i o n s o f t h e NACA
i n the l i - f t . t r a n s o n i c wind tunnel o f NASA Ames
64A006 a i r f o i l served as t h e b a s i s f o r comparison
Research Center. W i t h these t e s t s , n o t o n l y an
i n v a r i o u s t h e o r e t i c a l s t u d i e s and,from d i s c u s -
independent. check on t h e NLR experiments i s ob-
s i o n s w i t h s p e c i a l i s t s i n t h e f i e l d o f unsteady-
t a i n e d , b u t a l s o an e x t e n s i o n o f the range o f t h e
f i o w c a l c u l a t i o n methods, i t t u r n e d o u t t h a t t h e
Reynolds number t o 1 5 m i l l i o n , i n s t e a d o f 2 m i l l i o n
NLR f i n d i n g s concerning the p o s s i b l e types o f
a t NLR.
unsteady shack-wave m o t i o n had a c o n s i d e r a b l e
When t h e p r e s e n t s t a t e o f a f f a i r s
impact on the i n t e r p r e t a t i o n o f c a l c u l a t e d r e s u l t s .
i 5
surveyed,
i t i s the a u t h o r ' s o p i n i o n t h a t a s a t i s f a c t o r y
p r e d i c t i o n o f unsteady a i r l o a d s i n a t t a c h e d twodimensional t r a n s o n i c f l o w w i l i be p o s s i b l e w i t h i n
14.2
a couple of years.
FUTURE PROSPECTS
i n the p e r i o d t o come, f u r t h e r e v a l u a t i o n s The n e x t and more complicated s t e p w i l l be
o f t h e v a r i o u s c a l c u l a t i o n methods a r e needed by means o f mutual comparisons and by means o f com-
t h e development o f p r e d i c t i o n methoas for t h r e e -
p a r i s o n s between t h e o r e t i c a l and experimental
dimensional unsteady t r a n s o n i c f l o w s . From a prac-
r e s u l t s . Furthermore, c a l c u l a t i o n methods have t o
t i c a l p o i n t o f view, t h i s i s o f h i g h i n t e r e s t ,
be deveioped t h a t i n c o r p o r a t e t h e e f f e c t o f v i s -
s i n c e three-dimensional
e f f e c t s a r e known t o be
important i n t r a n s o n i c flow. The f i r s t a t t e m p t s
cosity.
i n t h i s d i r e c t i o n have been p u b l i s h e d r e c e n t l y .
One o f t h e examples t h a t lends i t s e l f f o r f u r t h e r e v a l u a t i o n s i s t h e NLR 7301 a i r f o i l .
In
Ruo and Theisen (Ref. i 8 3 ) , Isogai (Ref. 135) and
- 1 18-
Dowell (Ref.
184) deveioped methods which take the
e f f e c t o f wing t h i c k n e s s i n t o account a t Mm
i
50 t o 100).
1.0,
P r o j e c t e d a g a i n s t the s t a t e o f a f f a i r s far
a l t h o u g h shock waves have n o t been i n c l u d e d . For some r e c t a n g u l a r wings i t was shown ( R e f s .
three-dimens i o n a l steady t r a n s o n i c flows, where
135,
t h e c a l c u l a t i o n s a r e s t i l l very lengthy,
183) t h a t wing t h i c k n e s s has a s i g n i f i c a n t e f f e c t .
i t has
t o be expected t h a t f o r unsteady f l o w adequate
F i n i t e - d i f f e r e n c e methods f o r three-dimensional
three-dimensional f i n i t e - d i f f e r e n c e
flow,
n o t be a v a i l a b l e f o r r o u t i n e c a l c u l a t i o n s i n the
which t a k e shock waves i n t o account, have
been r e p o r t e d by Isom and Garadonna ( R e f s . 168,
methods w i l l
coming years. Therefore, more a p p r o x i m a t i v e b u t
169) f o r a h e l i c o p t e r r o t o r and by W e a t h e r h i l l ,
f a s t e r methods, which may be used t o i n t e r p o l a t e
E h l e r s and Sebastian (Ref.
between a l i m i t e d number of r e s u l t s of more ad-
171) for a p i t c h i n g
r e c t a n g u l a r wing w i t h an NACA 64A006 cross section.
vanced methods, seem t o be a t t r a c t i v e i n the near
Whether methods l i k e these w i l l become successful
future.
I n t h i s respect, methods l i k e those of
-
f o r p r a c t i c a l a p p l i c a t i o n s w i l l depend l a r g e l y on
Cunningham (Refs.
t h e r e d u c t i o n of computing times t h a t can be
s o n i c and supersonic l i n e a r t h e o r y v i a a compa-
achieved, e s p e c i a l l y s i n c e i n a e r o e l a s t i c inves-
t i b i l i t y r e l a t i o n a t t h e mean shock p o s i t i o n , and
i85
i 8 7 ) , who combines sub-
t i g a t i o n s u s u a l l y a l a r g e number of combinations
t h e semi-empirical method o f Garner ( R e f . 188)
of Mach number, reduced frequency, and v i b r a t i o n
s h o u l d deserve a t t e n t i o n .
modes have t o be c a l c u l a t e d ( i n comnon p r a c t i c e
15 REFERENCES
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Collar " A e r o e l a s t i c problems a t h i g h speed". J o u r n a l o f t h e Royal A e r o n a u t i c a l S o c i e t y , Vol. 51 (Jan. 1947) pp. 1-34.
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"Numerical computation o f t r a n s o n i c f l o w s w i t h shock waves". Symposium Transsonicum l i , G ö t t i n g e n (Eds. K. Oswatitsch and D. Rues; S p r i n g e r V e r l a g , B e r l i n , 1976) pp. 384-422.
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Goetz. R.V. Doaaet i r . "Some e f f e c t s of t i p f i n s on wing f l u t t e r c h a r a c t e r i s t i c s " . NASA TN D-7702 (1974). ~
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Bailey "On t h e computation of two- and three-dimensional steady t r a n s o n i c fiows by r e l a x a t i o n methods". AGARD L e c t u r e s e r i e s 63 "Progress i n Numerical F l u i d Dynamics" (1974).
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Boerstoel "Review o f t h e a p p l i c a t i o n of hodograph t h e o r y t o t r a n s o n i c a e r o f o i l design and t h e o r e t i c a l and experimental a n a l y s i s of shock-free a e r o f o i l s " . Symposium Transsonicum l i , G ö t t i n q e n (Eds. K. D s w a t i t s c h and D. Rues; S p r i n g é r Verlag, B e r l i n , 1976) PP. 109-133.
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Nieuwland "Transonic p o t e n t i a l f l o w around a f a m i l y of q u a s i - e l l i p t i c a l a e r o f o i l sections". D i s s e r t a t i o n U n i v e r s i t y of Groningen, t h e Netherlands (1967). ( A l s o NLR-TR T.172, 1967.)
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17. D.W.
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34. R.J. Zwaan
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35. H. H e r t r i c h
18. H.H.
Holde? "The t r a n s o n i c f l o w p a s t twodimensional a e r o f o i i s " . J o u r n a i o f the Royai A e r o n a u t i c a l S o c i e t y , V o l . 6 8 , No. 644 (Aug. 1964) pp. 501-516.
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Pearcy, A . B . Haines, J. Osborne "The i n t e r a c t i o n between l o c a l e f f e c t s a t t h e shock and r e a r s e p a r a t i o n " . AGARO CP No. 35 "Trans o n i c Aerodynamics" (1968).
36. H. Bergh, H. Tijdeman
" I n v e s t i g a t i o n s on t h e t r a n s o n i c f l o w around a e r o f o i l s " . D i s s e r t a t i o n T e c h n i c a l U n i v e r s i t y D e l f t , the Netherlands (1969). ( A l s o NLR TR 69122 U, 1969.)
R.J. Zwaan "Calculations o f ins t a t i o n a r y p r e s s u r e d i s t r i b u t i o n s and genera l i z e d f o r c e s w i t h t h e d o u b l e t - l a t t i c e method!' NLR TR 72037 U (1972).
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38. J. Becker
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"Unsteady Transonic Flow". (Pergamon Press,New York/London, 1961).
measurements i n compressible f l o w and compari s o n w i t h l i f t i n g s u r f a c e theory". AGARD rep o r t No. 617 (1974). Lodge, H. Schmid "Unsteady pressures due t o c o n t r o l surface r o t a t i o n a t low supersonic Comparison between t h e o r y and experspeeds iment". AGARO r e p o r t No. 647 (1976).
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-
Stark "Numerical l i f t i n g surface theory Problems and progress". A I A A J o u r n a l , V o l . 6, No.ll (Nov. 1968) pp. 2049-2060.
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"Grundlagen der A e r o e l a s t i k " . 27. H.W. F ö r s c h i n g ( S p r i n g e r V e r l a g , B e r l i n , 1974).
-
41. A.L. E r i k s o n , R.C. Robinson "Some p r e l i m i n a r y r e s u l t s i n t h e d e t e r m i n a t i o n o f aerodynamic d e r i v a t i v e s o f c o n t r o l surfaces i n the t r a n s o n i c speed range by means of a f l u s h type e l e c t r i c a l p r e s s u r e c e l l " . NACA RM A8H03 (1948).
4 2 . R.F. Thompson
-
Rodden "A comparison o f methods used i n i n t e r f e r i n g I i f t i n g surface theory". AGARD r e p o r t No. 643 (1976).
-
J.H. Greidanus, A . I . v.d. Vooren, H. Bergh "Experimental d e t e r m i n a t i o n o f t h e aerodynamic c o e f f i c i e n t s o f an o s c i l l a t i n g wing i n incompress i b l e , two-dimens i o n a l flow". NLR-reports F.101, F. 102, F.103 and F. 104 (1952).
31. W . R . L a i d l a w - " T h e o r e t i c a l and e x p e r i m e n t a l p r e s s u r e d i s t r i b u t i o n s on o s c i l l a t i n g low aspect r a t i o wings". MIT A e r o e l a s t i c and S t r u c t u r e s Research Lab. r e p o r t 51-2 (1954).
-
"Unsteady p r e s s u r e measurements on 32. H. Bergh a wing w i t h o s c i l l a t i n g f l a p i n two-dimenslonal flow". Proc. 3 r d European A e r o n a u t i c a l Congress ( B r u s s e i s , 1958). "Die Druck-, A u f t r i e b s - und 33. B. Laschka Momentenverteilungen an einem harmonisch schwingenden P f e i l f l ü g e l k l e i n e r Streckung i m niedrigen Unterschallhereich. Vergleich zwischen T h e o r i e und Messung". ? r o c . 4 t h ICAS Congress ( P a r i s , August 1964). (Ed. R. Dexter; Mac M i l l a n , London, 1965).
-
" I n v e s t i g a t i o n o f a 42.7' swept back wing model t o determine the e f f e c t s o f t r a i ï i n g edge t h i c k n e s s on t h e a i l e r o n hinge-moment and f l u t t e r c h a r a c t e r i s t i c s a t t r a n s o n i c speeds". NACA RM L50J06 (1950).
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"A comparison o f methods used 28. D.L. Woodcock i n l i f t i n g s u r f a c e theory". AGARD r e p o r t No. 583 (1971).
2 9 . W.P.
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40. C.G.
25. R.L. B i s p i i n g h o f f , H. Ashley, R.L. Halfman " A e r o e i a s t i c i ty". (Addison-Wesl ey Publ. Comp., Reading, Mass., 1957).
26. M. Landahl, V.J.E.
-
39. R. Destuynder, H. Tijdeman - "An i n v e s t i g a t i o n on d i f f e r e n t techniques f o r unsteady p r e s s u r e
23. AGARO Manual on A e r o e l a s t i c i t y (1968).
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-
" I n t e r f e r i n g l i f t i n g surfaces i n unsteady subsonic flow Comparison between t h e o r y and experiment". AGARD r e p o r t No. 614 (1974).
' ' T h e o r e t i c a l and exp e r i m e n t a l r e s u l t s f o r the dynamic response o f p r e s s u r e measuring systems". NLR-TR F.238 (1965).
2 4 . bi. Landahl
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37. R . Roos,
21. H. Bergh "A new method f o r measuring t h e p r e s s u r e d i s t r i b u t i o n on h a r m o n i c a l l y o s c i l l a t i n g wings". Proc. 4 t h ICAS-Congress ( P a r i s , 1964). (Ed. R. Dexter; Mac M i l l a n . London, 1965). ( A l s o NLR MP.224, 1964.1
22. H. Bergh, H. Tijdeman
-
#'Binary wing-control surface f l u t t e r c a l c u l a t i o n s w i t h t h e o r e t i c a l and e m p i r i c a l aerodynamic d e r i v a t i v e s f o r twdimens i o n a l incompress i b l e flow". NLR TR 68069 U (1968).
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20. B.H. Spee
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"Zur e x p e r i m e n t e l l e n Prüfung i n s t a t i o n ä r e r d r e i d i m e n s i o n a l e r Tragflächent h e o r i en be i inkompress i b l e r S t römung". M i t t . d. Max Planck I n s t . f u r Strömungsforschung, N r . 40 (1967).
Pearcy "The aerodynamic d e s i g n o f s e c t i o n shapes f o r swept wings". Advances i n A e r o n a u t i c a l Sciences, V o l . 3 (London, 1962).
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"Calculated r e s u i t s f o r o s c i l l a t i n g T - t a i l s i n subsonic f l o w and comparisons w i t h experiment". NLR M P . 2 5 3 (1967).
43. J.A. Wyss, R . M . Sorenson - "An i n v e s t i g a t i o n of the c o n t r o l s u r f a c e f l u t t e r d e r i v a t i v e s o f an NACA 651-213 a i r f o i l i n the Ames 16-foot h i g h speed wind tunnel". NACA RM A51J10 (1951).
44. A.B.
-
Henning "Results o f a rocket-model i n v e s t i g a t i o n o f c o n t r o l s u r f a c e buzz on a 4 p e r c e n t t h i c k unswept wing and on 6, 9 and 12 p e r c e n t t h i c k swept wings a t t r a n s o n i c speeds". NACA RM L53129 (1953).
-
45. O.J. M a r t i n , R.F. Thompson,
C.W. Martz " E x p l o r a t o r y i n v e s t i g a t i o n o f t h e moments on o s c i l l a t i n g c o n t r o l surfaces a t t r a n s o n i c speeds". NACA RM L55E31b (1955).
46. O.E.
-
Reese, W.C.A. Carlson "An experimental i n v e s t i g a t i o n o f t h e h i n g e moment c h a r a c t e r i s t i c s o f a c o n s t a n t chord c o n t r o l s u r f a c e o s c i l l a t i n g a t h i g h frequency". NACA RM A55J24 (1955).
-
4 7 . R.F. Thompson, W.C.
-
Moseley "Oscillating h i n g e moments and f l u t t e r c h a r a c t e r i s t i c s o f a f l a p t y p e c o n t r o l s u r f a c e on a 4 percent t h i c k unswept wing w i t h low aspect r a t i o a t t r a n s o n i c speeds". NACA RM L55K17 (1956).
,120-
48. S . A . Clevensan
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-
"Some wind tunnel experiments on s i n u l e deoree o f freedom f l u t t e r o f a i l e rens i n the h i g h subsonic speed range". NACA TN 3687 (1956).
-
49. W.C.
" E f f e c t s of Reynolds 63. Y . Nakamura, i. Woodgate number and frequency parameter on c o n t r o i s u r face buzz a t h i g h subsonic speeds". NPL Aero r e p o r t 1312 (1970).
-
-
Hartz "Experimental h i n g e moments on f r e e l y o s c i l l a t i n g f l a p t y p e c o n t r o l surfaces'! NACA RM L56G20 (1956).
64. J.B. B r a t t , A. Chinneck "Measurement o f mid-chord p i t c h i n g moment d e r i v a t i v e s a t h i g h speeds". R E li, No. 2680 (1947).
-
"Aerodynamics 50. R.F. Thompson, S.A. Clevenson o f o s c i l l a t i n g c o n t r o l surfaces a t t r a n s o n i c speeds". NACA RM L57D22b (1957).
65. A.J. Wyss, R. H e r r e r a
-
51. R.F. Thompson, W.C.
Moseley " E f f e c t of h i n g e - l i n e p o s i t i o n on t h e o s c i l l a t i n g h i n g e moments and f l u t t e r c h a r a c t e r i s t i c s o f a f l a p t y p e c o n t r o l a t t r a n s o n i c speeds". NACA RH L 5 7 C l l (1957).
-
66. A.J. Wyss, J.C. M o n t f o r t "Effects o f a i r f o i l p r o f i l e on t h e two-dimensional f l u t t e r d e r i v a t i v e s f o r wings o s c i l l a t i n g i n p i t c h a t h i g h subsonic speeds". NACA RM A54C24 (1954).
-
52. J.B. B r a t t , J.W.C. M i l e s , R.F. Johnson "Measurements o f t h e d i r e c t h i n g e moment d e r , i v a t i v e s a t subsonic and t r a n s o n i c speeds f o r a cropped d e l t a wing w i t h o s c i l l a t i n g flap". R E M, No. 3163 (1957).
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"Effects o f r i g i d 67. J.C. M o n t f o r t , J.A: Wyss s p o i l e r s on t h e two-dimensional f l u t t e r d e r i v atives o f 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 a t h i g h subsonic speeds". NACA RM A54122 (1954).
-
53, J.A. Wyss, R.M.
Raymer, J.E.G. Townsend "Measurements o f the d i r e c t p i t c h i n g moment d e r i v a t i v e s f o r two-dimensional f l o w a t subs o n i c and supersonic speeds and f o r a wing o f aspect r a t i o 4 a t subsonic speeds". R E M, No. 3257 (1959).
-
Moseiey, R.F. Thompson " E f f e c t o f cont r o l t r a i l i n g edge t h i c k n e s s on the o s c i l l a t ing h i n g e moment and f l u t t e r c h a r a c t e r i s t i c s o f a f l a p t y p e c o n t r o l a t t r a n s o n i c speeds". NACA RN L58625 (1958).
-
69. H.C. Lessing, J.L. Troutman, G.P. Meness "Experimental d e t e r m i n a t i o n o f the p r e s s u r e d i s t r i b u t i o n on a r e c t a n g u l a r wing o s c i l l a t i n g i n t h e f i r s t bending mode f o r Mach numbers from 0.24 t o 1.30". NASA TN-0344 (1960).
-
55. W.C.
Moseiey, T.G. Gainer "Effect o f wing t h i c k n e s s and sweep on the o s c i l l a t i n g h i n g e moment and f l u t t e r c h a r a c t e r i s t i c s o f a f l a p type c o n t r o l a t t r a n s o n i c speeds". NASA TM X-123 (1959).
-
70. S.A. L e a d b e t t e r , S.A. Clevenson, W.B. Igoe "Experimental i n v e s t i g a t i o n o f o s c i i l a t o r y aerodynamic f o r c e s , moments and pressures a c t i n g on a tapered wing o s c i l l a t i n g i n p i t c h a t Mach numbers from 0.40 t o 1.07". NASA TN-DI236 (1960).
-
56. J.A. Wyss, R.M. Sorenson, B.J. Gambucci "Measurements o f t r a n s o n i c f i u t t e r d e r i v a t i v e s f o r a mid-span c o n t r o i s u r f a c e on a m o d i f i e d d e l t a wing". NASA TM X - i 5 7 ( l P 6 0 ) .
57. J.A.
71. H. Tijdeman, H. Bergh
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-
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Moseley, T.G. Gainer "Some e f f e c t s o f c o n t r o l p r o f i l e and c o n t r o l t r a i l i n g edge a n g l e on the o s c i l l a t i n g h i n g e moment and f l u t t e r c h a r a c t e r i s t i c s of f l a p t y p e c o n t r o l s a t t r a n s o n i c speeds". NASA TM X-170 (1960).
59. H. Loiseau
-
"Analysis o f pressure d i s t r i b u t i o n s measured on a wing w i t h o s c i i l a t i n g c o n t r o l surface i n two-dimensional h i g h subsonic and t r a n s o n i c flow". NLR-TR F. 253 (1967).
Wyss, R.E. Dannenberg, R.M. Sorenson, B.J. Gambucci " E f f e c t o f boundary l a y e r s u c t i o n and s p o i l e r s on t r a n s o n i c f l u t t e r d e r i v a t i v e s f o r a mid-span c o n t r o l s u r f a c e on an unswept wing". NASA TM X-160 (1960).
58. W . C .
-
68. J.B. B r a t t , W.G.
Sorenson, B.J. Gambucci " E f f e c t s o f m o d i f i c a t i o n s t o a c o n t r o l surface on a 6 oercent t h i c k unsweot winu on the t r a n s o n i c c o n t r o l s u r f a c e d e r i ;at i,ve;tl. NACA RM A58604 (1958).
54. W.C.
-
"Effects o f angle of a t t a c k and a i r f o i l p r o f i l e on the two-dimensional f l u t t e r derivatives f o r 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 a t h i g h subsonic speeds". NACA RM A54H12 (1954).
-
"The i n f l u e n c e o f the 72. H. Tijdeman, H. üergh main f l o w on the t r a n s f e r f u n c t i o n o f tubetransducer systems used f o r unsteady p r e s s u r e measurements". NLR 1IP 76052 U (1976).
73. J. Zwaanevei d - "Aerodynamic c h a r a c t e r i 5 t i c s of t h e s u p e r c r i t i c a l shock-free a i r f o i l sect i o n NLR 7301".
NLR TR 76052 U (1976).
-
"Mesure de c o e f f i c i e n t s aérodynamique i n s t a t i o n n a i r e s de gouvernes en t r a n s sonique". La Recherche Aéronautique, No. 97 (Nov.-Dec. 1963).
"The dynamics and thermody74. A.H. Shapiro namics of c o m o r e s s i b l e f l u i d flow". Vols. I and l l . (The Ronald Press Company, New York, 1953).
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75. W.R.
Sears (ed.) "General theory o f h i g h speed aerodynamics". Vol. V I : "High Speed Aerodynamics and J e t Propulsion". ( P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , 1954).
60. Y. Nakamura, Y. Tanabe "Some experiments on c o n t r o i s u r f a c e buzz". N a t i o n a l A e r o n a u t i c a l L a b o r a t o r y (NAL) TR-72T (1964).
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61. H. Loiseau "Etude e x p é r i m e n t a l e des f l o t t e ments à un degr€ de i i b e r t é en écoulement transsonique". ONERA NT 95 (19661.
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76. G.Y.
Nieuwland, E.M. Spee "Transonic shockf r e e flow, f a c t o r f i c t i o n ? " . AGARD CP No.35 "Transonic Aerodynamics1' (1968).
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62. Y. Nakamura "Some c o n t r i b u t i o n s on a c o n t r o i s u r f a c e buzz a t h i g h subsonic speeds". J o u r n a l o f A i r c r a f t , V o l . 5, No. 2 (March-April 1968) pp. 118-125.
77. J. Z l e r e p
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"Theorie der SchalInahen und der Hyperschal Iströmungen". ( V e r l a g G. Braun, K a r l s r u h e , 1966).
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7 8 . J.W. B o e r s r o e l . J.A.
van Eamond "Desian o f basic aerofoils f o r a supercritNLR TR 75059 C (1975).
shack-free, i c a i wing".
9 4 . F.R. Goldschmied
-
"On the frequency response viscous compressible f l u i d s as a f u n c t i o n O f Stokes number". J o u r n a l o f Basic Engin e e r i n g , T r a n s a c t i o n s o f ASME, V o l . 9 2 (1970) Of
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79. H. V . Helmholtz "Verhandlungen der N a t u r h i s t o r i s c h - M e d i z i n i s c h e n V e r e i n s zu H e i d e l b e r g vom J a h r e 1863". Ed. I i I (1863) p. 16.
PP. 333-347.
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154. H. Ashley, W.S.
Park " C a l c u l a t i o n af aerodynamic der i v a t i v e s i n unsteady two-dimensional t r a n s o n i c Flaw u s i n g O o w e l i ' s l i n e a r i z a t i o n method". P r i n c e t o n U n i v e r s i t y , AM8 r e p o r t No. 1238-T (1975). ( S e e a l s o AIAA J o u r n a i , V o l . 1 4 , No. I O (Oct. 1976) pp. 1345-1346,) P.H.
-
167. N.J. Yu, A.R.
-
Seebass, 1I.F. Ballhaus "An i m p l i c i t s h o c k - f i t t i n g scheme for unsteady t r a n s o n i c f l o w computations". A I A A paper 77-633. 3d Comp. F l u i d Dynamics Conference (Albuquerque, New Mexico, June 1977).
Rodden "A d o u b l e t - l a t t i c e method f o r c a l c u l a t i n g l i f t d i s t r i b u t i o n s on o s c i l l a t i n g s u r f a c e s i n subsonic fiows". A i A A J o u r n a l , Vol. 7, No. 2 (1965) pp. 279-285.
-124-
-
"Unsteady subsonic and t r a n s o n i c 163. H.P. isom p o t e n t i a i flaw over h e l i c o p t e r blades". NASA CR-2463 (1974).
Engineering, V o i . 99, No. PP. 8-39.
-
-
180. M.T. Landahl "Some developments i n unsteady t r a n s o n i c flow research('. Symposium TransSonicum I I , Göttingen. (eds. K. Oswatitsch and D. Rues; S p r i n g e r Verlag, B e r l i n , i976).
-
Tijdeman "On the m o t i o n o f shock waves on an a i r f o i l w i t h o s c i l l a t i n g f l a p " . Symposium Transsonicum I I , Göttingen. (eds. K. O s w a t i t s c h and D. Rues; S p r i n g e r V e r l a g , B e r l i n 1976).
Ballhaus "Some r e c e n t progress i n t r a n s o n i c f l o w computations". L e c t u r e s e r i e s on Computational F l u i d Dynamics (Von Karman I n s t i t u t e , Rhode-St.Genese. Belgium, March 1976).
"On t h e unsteady aerodynamic 172. H. Tijdeman characteristics o f o s c i l l a t i n g a i r f o i l s in two-dimens iona I t r a n s o n i c flow". Paper presented a t the ONR T r a n s o n i c Flow Conference (Univ. o f C a l i f o r n i a . Los Angeles, March 1976).
-
182. J.H. Wu, T.H. Houiden "A survey o f t r a n s o n i c aerodynamics". A i A A paper 76-326; A I A A 9 t h F l u i d and Plasma Dynamics Conference (San Diego, J u l y 1976)'
Persoon "Unsteady a i r l o a d s on an o s c i l l a t i n g superc r i t i c a l a i r f o i l " . G A R D S p e c i a l i s t s Meeting on "Unsteady A i r l o a d s i n Separated and Trans o n i c flow" ( L i s b o n , Porbugai, A p r i l 1977).
Ruo, J.G. Theisen " C a l c u i a t i o n o f unsteady t r a n s o n i c aerodynamics f o r o s c i l l a t i n g winqs w i t h thickness". NASA CR-2259 (1973). (See a l s o A I A A paper 73-316; Dynamics Speciali s t s Conference ( W i l l i a m s b u r g , March l 9 7 3 ) . )
184. E.H.
Murman " A n a l y s i s o f embedded shock waves c a l c u l a t e d by r e l a x a t i o n methods". AIAA J o u r n a l , Voi. 12 (May 1974) pp. 626-633.
175. J. Smith
-
"Values o f w a l i i n t e r f e r e n c e c o r r e c t i o n f o r the NLR P i l o t t u n n e l w i t h 10 % open s l o t t e d t e s t s e c t i o n " . EIL% i n t . r e p o r t AC-74-01 (1974).
176. J.I.1.
-
Kooi " I n v e s t i g a t i o n on t h e a p p l i c a b i l i t y o f the Bauer, Garabedian, Korn and Janeson program f o r t h e c a l c u l a t i o n o f the viscous p r e s s u r e d i s t r i b u t i o n s on a i r f o i l s " . NLR i n t . r e p o r t AC-77-026 ( i n Dutch) (1977).
177. H . i l . Hafez,
-
Rizk, E.M. Murman "Numericai B o l u t i o n o f t h e unsteady smalld i s t u r b a n c e equation". AGAR0 S p e c i a l i s t s f l e e t i n g on "Unsteady A i r i o a d s i n Separated and Transonic fiow" ( L i s b o n , P o r t u q a l , A p r i i 1977).
178. W.J.
M.H.
-
Dowell "A s i m p l i f i e d t h e o r y l a t i n g a i r f o i l s i n transonic flow: and extension". A i A A paper 77-445; Special i s t s Conference (San Diego,
-
174. E.M.
-
183. S . Y .
-
173. H. Tijdeman, P. Schippers, A.J.
-
181. W.F.
171. W.H. W e a t h e r i l l , F.E. E h l e r s , J.D. Sebastian "Comoutation o f t h e t r a n s o n i c o e r t u r b a t i o n f l o w ' f i e l d s around two- and thkee-dimensional o s c i l l a t i n g wings". NACA CR-2599 (1975).
-
-
179. J.R. S p r e i t e r , S. Stahara "Unsteady trans o n i c aerodynamics an a e r o n a u t i c a l c h a l lenge". Symposium on "Unsteady Aerodynamics" (ed. R.B. Kinney; Univ. o f Tucson, Arizona, 1975).
-
"Numerical c a l c u 169. F.X.Caradonna, M.P. Isom l a t i o n o f unsteady t r a n s o n i c p o t e n t i a l f i o w over h e l i c o p t e r biades". AIAA paper 75-168, 1 3 t h Aerospace Sciences Meeting (Pasadena, Jan. 1975). 170. H.
1 (narch 1977)
of oscilReview Dynamics March 1977).
-
185. A.M.
Cunningham j r . "The a p p l i c a t i o n o f qeneral dvnamic l i f t i n q s u r f a c e elements t o problems i n unsteady t r a n s o n i c flori". NASA CR-112264 (1973).
-
-
186. A.M. Cunningham j r . "An o s c i l l a t o r y Kernel f u n c t i o n method f o r l i f t i n g surfaces i n mixed t r a n s o n i c flow". AIAA paper 74-359, AIAA/ASME/ SAE 1 5 t h S t r u c t u r e s , S t r u c t u r a l Dynamics and M a t e r i a l s Conference (Las Vegas, A p r i l 1974).
-
187. A.ll.
Cunningham j r . " F u r t h e r developments i n t h e p r e d i c t i o n o f o s c i i l a t o r y aerodynamics i n mixed t r a n s o n i c f l o w " . A I M paper 75-93; 1 3 t h Aerospace Sciences Meeting (Pasadena, Jan. 1975).
188. H.G.
-
Garner "A p r a c t i c a l approach t o the p r e d i c t i o n o f o s c i l l a t o r y pressure d i s t r i b u t i o n s on wings i n s u p e r c r i t i c a l flow". RAE TR 74181 (1975).
-
McCroskey "Some c u r r e n t research i n unsteady f l u i d dynamics". J o u r n a l o f F l u i d s
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( b ) no steady f l o w ( c ) tube long enough, so t h a t end e f f e c t s a r e neg1 i g i b l e
-
I n t h e equations(B.9) s = R (P~W/U)' I
=
-
k =
(pCp/A)* wR/ao
(6.12) the f o l l o w i n g f o u r parameters can be d i s t i n g u i s h e d :
,
t h e shear wave number
,
t h e square r o o t o f t h e P r a n d t l number
,
t h e reduced frequency, b e i n g p r o p o r t i o n a l t o t h e r a t i o o f tube r a d i u s t o wave l e n g t h
y
=
,
Cp/Cy
the r a t i o of s p e c i f i c heats
A s f o r a g i v e n gas I and y o f t e n can be considered as c o n s t a n t s , the two i m p o r t a n t parameters a r e the
-
shear wave number, 5, and t h e reduced frequency, k . To o b t a i n a s o l u t i o n o f an i n f i n i t e l y long, e q u a t i o n s (8.8)
-
r i g i d tube o f constant c i r c u l a r c r o s s - s e c t i o n ,
(6.12) have t o s a t i s f y t h e f o l l o w i n g boundary c o n d i t i o n s and assumptions:
( a ) a t t h e tube w a l l the r a d i a l and the a x i a l v e l o c i t y must be zero: at
n = l ,
u = O
and
i.e.,
v = O ;
(6.13)
( b ) the r a d i a l v e l o c i t y must be z ero a t the tube a x i s due t o t h e axisymmetry o f t h e problem: at
and u , p. o , (c)
" = O ,
i.e.,
" = o ;
(8.14)
and T have t o remain f i n i t e
t h e heat c o n d u c t i v i t y o f t h e tube w a l l i s l a r g e i n comparison w i t h t h e h e a t c o n d u c t i v i t y o f the fluid:
i.e., at
0
= 1
,
T = O
(isothermal walls)
(6.15)
6.1.2 B r i e f review o f s o l u t i o n s g i v e n i n t h e l i t e r a t u r e
(%.a) -
The problem o f s o l v i n g equations r e l a t e d l i k e Helmholtz (Ref.
(8.12)
i s a c l a s s i c a l one,
t o which famous names a r e
79), K i r c h h o f f (Ref. S O ) , and Rayleigh (Ref. S i ) . Since then many papers
have been w r i t t e n on t h e s u b j e c t , o f t e n i n r e l a t i o n t o s t u d i e s d e a l i n g w i t h t h e dynamic behaviour o f p r e s s u r e t r a n s m i s s i o n l i n e s . R e c e n t l y t h e present a u t h o r gave a review of t h e s o l u t i o n s g i v e n i n the l i t e r a t u r e (Ref. 8 1 ) .
I t appears t h a t t h e s o l u t i o n f o r t h e pressure p e r t u r b a t i o n p i n general can be
p u t i n the form
In t h i s expression, two types o f p r e s s u r e waves can be d i s t i n g u i s h e d , namely waves propagating i n the p o s i t i v e 6 - d i r e c t i o n and waves p r o p a g a t i n g i n t h e o p p o s i t e d i r e c t i o n . An i m p o r t a n t q u a n t i t y i n t h i s process i 5 t h e p r o p a g a t i o n c o n s t a n t
r
=
Ï'
+ ir",
which c o n s i s t s o f a r e a l p a r t
a t t e n u a t i o n o f the waves o v e r a u n i t d i s t a n c e 6. and an imaginary p a r t
Ï",
r",
r e p r e s e n t i n g the
b e i n g t h e phase s h i f t over
t h e same d i s t a n c e . Once i t was recognized t h a t t h e problem i s governed by t h e parameters 5 and t o r e w r i t e t h e most s i g n i f i c a n t a n a l y t i c a l s o l u t i o n s f o r t h e propagation c o n s t a n t parameters (Table 6.1). The s o l u t i o n s c o l l e c t e d i n t a b l e 8.1
k , i t appeared p o s s i b l e r i n terms o f these two
can be d i v i d e d r o u g h l y i n t o two groups. The
f i r s t group comprises s o l u t i o n s , o b t a i n e d as a n a l y t i c a l approximations o f t h e f u l l s o l u t i o n o f equations
-133-
1896
R a y l e i g h (Ref. 81) ("narrow tube")
t
Karam and Franke (Ref. 88)
Rohmann and Grogan (Ref. 87)
Transition "very wide"
Transit ion "wide/very wide"
'hide/nar row"
Transition
Transition "narrow/w ide"
Weston (Ref. 83)
1953
Zwikker and Kosten 1949 (Ref. 85) (independent I y a l 50 I bera I 1.1950, Ref. 86)
I939
1868
K i r c h h o f f (Ref. 80) ("wide tube")
84)
I863
Helmholtz (Ref. 79)
K e r r i s (Ref.
Year
Author
r = 1 + 77-5
I+i I
Formula f o r the p r o p a g a t i o n c o n s t a n t
r =
r i + ir" TABLE
B.l
t
I n reference 83, t h e s o l u t i o n f o r i s a f a c t o r JZ i n e r r o r .
f o r the p r o p a g a t i o n c o n s t a n t
r'
Review o f s i g n i f i c a n t a n a l y t i c a l 5 o I u t i o n s
-
(8.8)
( 8 . 1 2 ) , g i v e n by K i r c h h o f f i n t h e form o f a transcendental e q u a t i o n ( R e f s . 8 0 , 8 1 ) . The s o l u t i o n s
i n t h e second group were d e r i v e d d i r e c t l y from t h e b a s i c equations by t h e i n t r o d u c t i o n o f one o r more s i m p l i f y i n g assumptions. The f i r s t a p p r o x i m a t i o n s o f t h e f u l l tubes and by R a y l e i g h f o r "narrow"
83),
tubes.
K i r c h h o f f s o l u t i o n were produced by K i r c h h o f f h i m s e l f for "wide" L a t e r , h i g h e r - o r d e r a p p r o x i m a t i o n s were g i v e n by Weston ( R e f .
who d e r i v e d formulae f o r the t r a n s i t i o n s Ilnarrow/wide",
"very-wide''
"wide/narrow",
"wide/very-wide"
and f o r
tubes.
A n a l y t i c a l s o l u t i o n s i n t h e second group, o b t a i n e d d i r e c t l y frm m i r e o r i e s s s i m p l i f i e d b a s i c equations were p r e s e n t e d by, among o t h e r s , K e r r i s (Ref. 8 4 ) , Z w i k k e r and Kosten (Ref.
85). I b e r a l l (Ref.
86), Rohmann and Grogan (Ref. 87), and by Karam and Franke (Ref. 88). A c l o s e r examination o f t h e formulae i n t a b l e 8 . 1 r e v e a l s t h a t t h e expressions do n o t c o n t a i n the reduced frequency,
k,
except f o r t h e "wide/very-wide"
and "very-wide"
In. t h e l a s t two cases t h e expressions f o r t h e a t t e n u a t i o n ,
approximations o f Weston (Ref. 83).
among o t h e r t h i n g s , c o n t a i n the t e r m
which equals t h e a t t e n u a t i o n o f p i a n e waves i n f r e e a i r ( R e f .
8 i ) , and which i s independent o f t h e tube
radius. I n t h e "wide"-tube
s o l u t i o n of K i r c h h o f f ,
t h e parameters s and o a r e present, which i n d i c a t e s t h a t
b o t h v i s c o s i t y e f f e c t s and h e a t c o n d u c t i v i t y have been accounted f o r . P u t t i n g y = 1 ( i s o t h e r m a l c o n d i t i o n s ) reduces t h e K i r c h h o f f s o l u t i o n t o t h a t o f Helmholtz. For b o t h s o l u t i o n s t h e r e s u l t I i m holds,
r
= i
5-
t h i s being. t h e s o l u t i o n f o r p l a n e waves i n f r e e a i r w i t h o u t d i s s i p a t i o n .
i n which t h e diameter i s assumed t o be so smail t h a t
The s o l u t i o n by R a y l e i g h f o r "narrow tubes",
heat i s t r a n s f e r r e d f r e e i y from t h e c e n t e r o f t h e tube t o t h e w a l l , does n o t c o n t a i n the parameter
I,
which i n d i c a t e s t h a t o n l y v i s c o s i t y e f f e c t s a r e i n v o l v e d . A s mentioned above, Weston's formulae a r e h i g h e r - o r d e r approximations o f t h e f u l l K i r c h h o f f solut i o n , and t h e r e f o r e i t i s n o t s u r p r i s i n g t h a t t h e f i r s t terms o f t h e l'narrow/wide'' a p p r o x i m a t i o n equal t h e R a y l e i g h s o l u t i o n , whereas t h e f i r s t term o f t h e "wide/narrow" t h e c h a r a c t e r i s t i c s o f t h e "wide"
and "wide/very-wide"
t r a n s i t i o n s show
tube o f K i r c h h o f f .
The a n a l y t i c a l s o l u t i o n s b e l o n g i n g t o t h e second group, o b t a i n e d d i r e c t l y from s i m p l i f i e d b a s i c e q u a t i o n s , a r e o f a d i f f e r e n t type, w i t h t h e e x c e p t i o n o f t h e "high-frequency"
s o l u t i o n of Karam and
Franke (Ref. 8 8 ) . which l o o k s v e r y s i m i l a r t o t h a t o f K i r c h h o f f . The mutual r e l a t i o n s h i p o f t h e m a j o r i t y o f t h e s o l u t i o n s t o one a n o t h e r , except Weston's "wide/ very-wide'' and "very-wide'(
approximation,
i s shown i n f i g u r e 8 . 2 ,
where t h e a t t e n u a t i o n , Y',
and t h e
phase s h i f t , S ' , a r e p l . o t t e d as f u n c t i o n s o f t h e shear wave number s . A p o i n t o f i n t e r e s t r e v e a l e d by f i g u r e 8.2
i s t h a t t h e S o l u t i o n g i v e n f o r t h e f i r s t t i m e by Zwikker
and Kosten and d e s i g n a t e d h e r e as t h e "low-reduced-frequency
s o l u t i o n " passes c o n t i n u o u s l y from R a y l e i g h ' s
s o l u t i o n i n t o t h e s o l u t i o n o f K i r c h h o f f . T h i s "iow-reduced-frequency n e x t s e c t i o n , can be shown t o be v a l i d for
k
t a b l e 8. 1, except f o r t h e "wide/very-wide"
and "very-wide"
CE
solution",
1 and k / s c< I. The s o l u t i o n covers a l l the s o l u t i o n s i n a p p r o x i m a t i o n s o f Weston.
Another p o i n t o f i n t e r e s t r e v e a l e d by f i g u r e 8 . 2 i s t h a t "narrow"-tube values o f s and "wide"-tube
t o be d e r i v e d i n t h e
s o l u t i o n s f o r h i g h v a l u e s of t h i s parameter.
-135-
s o l u t i o n s a r e v a l i d f o r low
ATTENUATION
i
PHASESHIFT
r
SHEAR WAVE NUMBER S
Fig. 8.2
Propagation constant
r
= ?'+i?" as a function of shear wave number.
8.1.3 D e r i v a t i o n o f the "low-reduced-frequency
solution"
The formuia f o r the p r o p a g a t i o n c o n s t a n t , ?, a c c o r d i n g t o the l'low-reduced-frequency
s o l u t i o n " was
g i v e n f o r the f i r s t t i m e by Zwikker and Kosten (Ref. 851, who o b t a i n e d t h e i r s o l u t i o n i n an i n t u i t i v e way and w i t h o u t f i r m p r o o f . L a t e r , t h e s o l u t i o n was g i v e n more r i g o r o u s l y i n r e f e r e n c e s 22, 86, 89 t h i s section,
a d e r i v a t i o n o f t h e "low-reduced-frequency
solution"
-
91.
In
i s g i v e n , which i n c l u d e s n o t o n l y the
complete s e t o f a c o u s t i c v a r i a b l e s , b u t e x p l a i n s a l s o the c o n d i t i o n s f o r which t h e s o l u t i o n i s v a l i d .
When t h e i n t e r n a l tube r a d i u s i s small i n comparison w i t h t h e wave l e n g t h and t h e r a d i a l v e l o c i t y component, v,
i s small w i t h r e s p e c t t o t h e a x i a l v e l o c i t y , u ( i . e .
e q u a t i o n s (8.8)
-
i<
s
wR/ao c < 1 and v/u < c I ) t h e b a s i c
( 8 . 1 2 ) can be reduced t o :
(8.18)
(8.20) P
= P + T ,
(8.21)
(8.22)
-136-
%IH
3 1 -
m
m
D
-
3
7
_.
(LI
VI
VI
c
n 3
m a
7 '
_. _.
yl
n
_.
Y
VI
m
o-
O
n
m
01
s
o
N
m
-
-<
'D
N
m
-
.
O
n
3
c
3
_. n
N
-_.
m
-
-n
O
O 3
n
_.
(I
_.
n n
m
VI
m
c
"7 Y)
o
_.
n
O 3
n
o,
Y
3
(LI
VI
_. "
0 5
'o
N
m
+
c . (
7
a
_.
u
-
->i-
(LI
v
N
-
0
7
VI
_.
c
i
m
m
N
-
I,
I
n 3 m
o 3
- Y
-.
-n
3 -. n
c
-n
c
o
m
n
"
O
II
3
n
Y
m
N
m
m
O
n
YI
(LI
s
N
o
I,
o
3 - 3
n
c
" a i
o m o O 0 ui: n
7'3-
-. n
-
3
20"
-n
O
II
r
o s 3 -
n
-.
_. m
a
o 3
m o
n = a i
3
(LI
_. 23 :-. 7
I
3
O
+ D
!3
N
m
-
N
3
5 _. o 3
3- :O
i
A f t e r i n t e g r a t i o n w i t h respect t o ii, one has
Jl(i3’2uns))l i3’2as
From t h e boundary c o n d i t i o n v = O a t
-
F(S) = i p
i
1
I d2p
+--
~~(i3’20s)
I)
Y dEz
(
-n J i 3’2s
J o ( i3’2s)
From t h e e q u a t i o n (6.341,
.
(8.32)
= i, i t f o l l o w s t h a t
(8.33)
I)
= O,
t h i s requirement i s f u l f i l l e d i f F ( 6 ) = O , o r
= o .
+
+ F(S)
+-
Due t o t h e a x i a l symmetry, v = O a t
p = A erg
1 1
(i3’2~s)
(8.34)
one can s o l v e f o r p:
with
8 e-“,
r
=
[
~ ~ ( i 3 ’ ~ ys )
-1
i (8.35)
~ ~ ( i “ ~ ns )
Y
J~ ( i 3’20s)
The s o l u t i o n f o r t h e o t h e r a c o u s t i c v a r i a b l e s becomes
(8.38)
(8.39)
The c o n s t a n t s A and B should be determined by s p e c i f y i n g a d d i t i o n a l c o n s t r a i n t s a t b o t h ends o f the tube. From t h e s o l u t i o n f o r the r a d i a l v e l o c i t y , v/u
if
k
v,
i t can be v e r i f i e d a p o s t e r i o r i t h a t t h e c o n d i t i o n
<< 1 and k/s c c I .
I t appears t h a t a s o l u t i o n i d e n t i c a l t o (8.35)
can be o b t a i n e d i f , i n t h e o r i g i n a l s e t o f equations,
t h e e q u a t i o n o f s t a t e and t h e energy e q u a t i o n ( e q u a t i o n 8.11 tropic relation
-” = constant. P/P
-
and 8 . 1 2 )
a r e r e p l a c e d by t h e s i n g l e p o l y -
E v i d e n t l y , t h e thermodynamic process i n s i d e t h e tube can be considered t o
o c c u r p o l y t r o p i c a l l y , w i t h t h e (complex) p o l y t r o p i c c o n s t a n t , n, g i v e n by t h e e x p r e s s i o n (8.35). T h i s p o l y t r o p i c c o n s t a n t i s a f u n c t i o n o f t h e p r o d u c t as, which means t h a t t h e c o n s t a n t does n o t depend on the
-138-
uAGNITUOE n
I'=
1.1,
1.1
1.4
-
Fig. 8.3
" P o l y t r o p i c c o n s t a n t " n for t h e p r o p a e a t i o n of p r e s s u r e waves in c y l i n d r i c a l tubes as a f u n c t i o n O f as.
Ik
-lfL PHASE ANGLE n
v i s c o s i t y and o n l y accounts f o r the e f f e c t o f heat conduction. The development o f t h e p o l y t r o p i c c o n s t a n t with
IS
i s shown i n f i g u r e 8 . 3 . O f i n t e r e s t a r e the a s y m p t o t i c ' v a l u e s o f n:
iim n = 1 as+o
and
Iim
n = y ,
LTS-
which imply t h a t t h e p r o p a g a t i o n of t h e p r e s s u r e waves a t v e r y small values o f as ( f o r instance a t very l o w frequency) happens i s o t h e r m a l l y , whereas a t h i g h v a l u e s of as ( f o r i n s t a n c e a t very h i g h frequency)
the p r o p a g a t i o n takes p l a c e almost a d i a b a t i c a l l y .
8.1.4
E f f e c t o f t h e reduced frequency
The f u l l s o l u t i o n o f t h e b a s i c equations ( 8 . 8 ) "low-reduced-frequency
solution",
-
(8.12),
w i t h o u t t h e assumptions p e r t a i n i n g t o the
i s g i v e n i n t h e o r i g i n a l paper o f K i r c h h o f f
(Ref.
80). T h i s s o l u t i o n i s
o b t a i n e d i n t h e form o f a t r a n s c e n d e n t a l equation, which does n o t l e n d i t s e l f t o f u r t h e r a n a l y t i c a l t r e a t ment. Using t h e knowledge t h a t t h e problem i s governed by t h e four b a s i c parameters s,
k,
a , and y,
the
p r e s e n t a u t h o r r e w r o t e t h e o r i g i n a l K i r c h h o f f s o l u t i o n i n t o t h e more a t t r a c t i v e form (Ref. 8 2 ) :
F(r,s,k,o,y)
= O
.
(8.40)
-139-
1.6!
I
SHEAR 'WAVE XUUBER I
Fig. 8.4
Exact Solution of
Ï
= Ï'+iÏ"
8s a function of shear save number and reduced frequency.
T h i s e q u a t i o n i s s o l v e d n u m e r i c a l l y w i t h t h e Newton-Raphson procedure. As a f i r s t e s t i m a t e , the value o f ? r e s u l t i n g from t h e ')low-reduced-frequency s o l u t i o n " has been used. Some c h a r a c t e r i s t i c r e s u l t s , showing t h e i n f l u e n c e o f t h e reduced frequency, c o n s t a n t , Ï , a r e g i v e n i n f i g u r e 8.4.
-
k, on t h e propagation
T h i s f i g u r e r e v e a l s a l a r g e e f f e c t o f the reduced frequency on both
a t t e n u a t i o n and phase s h i f t , e s p e c i a l l y i n t h e range o f r e l a t i v e l y low values o f s ( s vaiues o f
k,
t h e exact v a l u e approaches t h e "low-reduced-frequency
For the range o f i n t e r e s t f o r t h e p r e s e n t i n v e s t i g a t i o n ( O 5
k
c 4).
For smaii
solution".
5 0.007),
t h e e f f e c t o f t h i s parameter can
be d i s c a r d e d .
8.2
DERIVATION OF THE FORMULAE FOR THE DYNAMIC RESPONSE OF TUBE-TRANSDUCER SYSTEMS
To a r T i v e a t the s o l u t i o n o f tube-transducer s y ~ t e m s , use solution".
i 5
made o f t h e "low-reduced-frequency
From t h i s S o l u t i o n , b o t h the e x p r e s s i o n f o r t h e p r e s s u r e p e r t u r b a t i o n ( 8 . 3 5 )
f o r the p e r t u r b a t i o n i n a x i a l v e l o c i t y (8.36)
and t h e expression
a r e needed. The l a t t e r expression i s used t o c a l c u l a t e the
mean v a l u e of t h e v e l o c i t y p e r t u r b a t i o n :
-140-
z
w
N
,I
L.
7
5 O
L.
OU
v
c
m
-
1
-
IF
3
yl
' ; i O
o
: s
L.
D
<
7
I1
n
L.
n
D
8
"7
m
n
c
a
7
m
m
3
o
n
_.
n
yl
0
-t
n
.
m
7 u
s
(
n
-. o y i Log
n
_. E
-
-. E
m
u?
c
-
7
<
I
n
ID
I
-
m
L.
n
+
_.
c
r
m
-
-n
O
L.
u
-+
i
P
w
c
m
-
L.N
. -
c
7
-<
01
a
3
u o c
O
(r
n
O
n 5
E. Y
N
c
m
-
-
L.
C
7
_.
I, O
-
X L.
75
L.
O
,I
x
L.
--c
m
1
m
D
m
m
a
m
-
a a a
._
+
II
._
a
a
O
I,
I,
o
O
a
I1
x
-
o
-
x
U
o m 3
Y
L o
o
c ._ c al m
m
m O
-
Ln
z u
Y
I
k
4
.3
M
m
Y)
O
3
O
C
0::
L
m
n 0 3 - c
L
m c
a
0 0
3 - Y )
._
L
u !. - L - a u a .m m w 5 .-m* Lm Eo a
OL 4
r. m
m":
0
m
r
o
.
.-u
<-
o
c o u
Y
L
3
m z
*
Y
o
o
-
-3o
.-
m
L
o
W E
-
" n a
Y
m
.-
O
..
m
m
> ._
Y
=
Q J L
r
X
>
"
i r e
m - u ui
7
m
U
m
m
m
4 m .c m 3
Y
x u 3 m c 3 > w o u n n u c m o m r -
a
: ? "
o m
Y
o
o Y) c m o n
L
a
m e m - o m -
L
0 - 0L
"
.c
a
a
N
-w
m
u OL 4
a u
+
.N
a
-
+
-n
u
a
OL 4
.-
+
-
I1 N
OL
a
3
a
VI
O
$ Y
ZI m r
o
I1
L
a u
I1
NI4
a
ui
O
3
I1
a
O 3
a,
2, .+
n/
I
.
7 ,a
,I ,//,/, _ *,,-,,_ _ - 1-
I
t P
-,L---_(IL
/"" -t ' ...
I
Lx
I
r
Y)
m
n
, /
O1
Y
3
>
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<: U1
= al M,
(C.4)
,
.
(c.5) (C.6)
I f t h e s i m p l i f i e d f l o w model, discussed i n s e c t i o n 9.1
( F i g . 9.7).
is accepted and i f
it is
Here index 2 r e f e r s t o t h e q u a n t i t i e s j u s t b e h i n d
assumed char t h e shock wave moves r e l a t i v e l y t o
t h e shock wave, and index 1 t o the q u a n t i t i e s
t h e a i r f o i l w i t h a v e l o c i t y V ( V < < a ) , then r e l a -
j u s t ahead o f t h e shock wave.
t i o n (C.1)
- 143-
becomes:
m
-
a
-,x 2-1
+
.
0
C
>
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
Y
r
-
N
Y)
a
N
1
x
-+
U
a
\
+
U
a
*
u . .,
O
-
n
N
I,
Y
.-m
a ._
3 <-
.3
X
a U a -
Y
.-
Y
a
3
n
Lo
a
c Y
u. O
c
O ._
w
-E m
U .-
2
3
.-ac m
a
c
C
3
*o
3 Y > >
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 <-
X
a
-
x
a
c
m
, IO . .,
O
3
-+
<-
X
m
m
m \
-rY
N
II
U
3
x a
X
3 .
APPENDIX D
:
A
constant
d e f i n e d i n eq.
(5.1)
A
constant
d e f i n e d i n eq.
(9.ib)
a
v e l o c i t y o f sound
L
lift
u n d i s t u r b e d v e l o c i t y o f sound i n s i d e
L
waveiength
a
o
LIST OF SYMBOLS
k = k'+ik"
a p r e s s u r e tube
a. J B
c o e f f i c i e n t s d e f i n e d i n eq.
C
chord l e n g t h ( c = Z E )
c o n s t a n t d e f i n e d i n eq.
(12.10)
(5.1)
L
l e n g t h o f p r e s s u r e tube
AL
variation in l i f t
2
semi chord
I1
steady moment a bout qua r t e r - c h o r d point
-
steady n o r m a l - f o r c e c o e f f i c i e n t
'k
unsteady n o r m a l - f o r c e c o e f f i c i e n t due .to f l a p o s c i l i a t i o n
e
steady l i f t c o e f f i c i e n t ( f o r small incidences,c2 = ck)
M
unsteady moment about quarter-chord point
steady moment c o e f f i c i e n t about quarter-chord p o i n t
M
Mach number
m
M*
c r i t i c a l ilach number
steady hinge-moment c o e f f i c i e n t
m = m'+im" c x a a
unsteady moment c o e f f i c i e n t about q u a r t e r - c h o r d p o i n t due t o p i t c h oscillation
m = m'+im"
unsteady moment c o e f f i c i e n t about q u a r t e r - c h o r d p o i n t due t o f l a p oscillation
C
C
C
n
steady p r e s s u r e c o e f f i c i e n t
CP
C*
steady p r e s s u r e c o e f f i c i e n t f o r which
C
Mloc = s p e c i f i c heat a t c o n s t a n t p r e s s u r e
P P
cv
AC
P
C
C
C
-
s p e c i f i c heat a t c o n s t a n t volume
N
steady h i n g e moment
unsteady p r e s s u r e c o e f f i c i e n t s , d e f i n e d i n Appendix A
N
unsteady h i n g e moment
N
number o f p r e s s u r e cubes i n a series connection p o l y t r o p i c c o n s t a n t For pressure tube, d e f i n e d i n eq. (5.2)
(5.4)
C
c o n s t a n t d e f i n e d i n eq.
C
wall correction c o e f f i c i e n t , defined i n eq. ( 1 3 . 1 )
n
aerodynamic i n f l u e n c e c o e f f i c i e n t , d e f i n e d i n eq. ( 1 2 . 1 3 )
n = n'+in"
c
c
c
n
energy per u n i t volume
unsteady hinge-moment c o e f f i c i e n t p o l y t r o p i c c o n s t a n t f o r transducer ume
YO I
frequency (Hz)
P
a i r f o i l thickness d i s t r i b u t i o n PO
mode o f v i b r a t i o n
pressure stagnation pressure
p-
s t a t i c p r e s s u r e o f the free-stream flow
pressure loading functions,defined i n eq. (12.10)
Pi
pressure p e r t u r b a t i o n a t the entrance o f a p r e s s u r e tube
Hankei f u n c t i o n o f second k i n d o f order n
PU
p r e s s u r e p e r t u r b a t i o n i n t h e scanning va I ve
imaginary u n i t
PS
time-dependent d e f o r m a t i o n
Ap=Ap'+iAp"
Bessel f u n c t i o n o f f i r s t k i n d o f order n
mean p r e s s u r e Dressure o e r t u r b a t i o n a t the a i r f o i I surface
steady normal f o r c e
dynamic p r e s s u r e ( q = ;PU , :)
unsteady normal f o r c e
tube r a d i u s
Kernel f u n c t i o n f o r subsonic l i f t i n g surface theory
relaxation factor, eq. (10.3)
unsteady aerodynamic i n f l u e n c e c o e f f i c i e n t , defined i n eq. (12.10)
Reynolds number based on chord i e n g t h (Re = o_cU,/ii)
defined i n
reduced frequency ( k = wZ/U_)
r e a l p a r t o f a complex number
k
reduced frequency ( p r e s s u r e tube) ( k = wR/ao)
gas c o n s t a n t
k = k'+ik''
unsteady norma I - f o r c e c o e f f i c i e n t due to oitch o s c i l l a t i o n
k
e
a
u
coordinate i n radial d i r e c t i o n (pressure tube) a i r f o i l contour
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shear wave number ( s = R I i i S w / ~ > ' )
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
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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-
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I n verband met deze
