flNational Research Conseil nationalto0 1 Council Canada de recherches Canada

A STUDY OF

TRANSONIC FLUTTEr, OF ATWO-DIMENSIONAL AIRFOIL

SUSING THE U-9 AND p-k METHODS

by

I Natonal B.H..l(. LeeNationa Aeronautical Establishment

k' Reproduced FromOTAA k E CEBest Available Copy OTW

NOVEMBER 1984 AR12 M8

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AERONAUTICAL REPORT

~ ~ 251 04a' NRC NO. 23959oL R -6155

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K 7 .. *

A STUDY OF TRANSONIC FLUTTER OF A TWO-DIMENS!ONAL AIRFOIL

USING THE U-g AND p-k METHODS

UNE ETUDE SUR LES VIBRATIONS AEROELASTIQUES EN REGIME TRANSSONIQUE

D'UN PROFIL AERODYNAMIQUE B[DIMENSIONNEL AU MOYEN DES

METHODES U.g ET p-k

by/par

B.H.K. Lee Accession For

.NTIS GRA&IDTIC TAB 0Unannounced 0Justificatio

By ,,,

Distribution/

Availability Codes

vail and/or

fa E umea* aw"m*• LH. Ohman, Head/Chef* High Speed Aerodynamics Laboratory/ G.M. Lindberg* Laboratoire d'a~rodynamnique i hautes vitesses Director/Directeur

.• SUMMARY

Transonic flutter of a NACA64A00G airfoil undergoing plungingand pitching oscillations is studied using the U-g and p-k methods• The

.: aerodynamic coefficients are calculated using an improved version of anI•. ONERA unsteady transonic aerodynamics code which include the second.' time derivative term of the velocity potential in the governing equation."- ~Comparisons with LTRAN2-NLR show good agreement up to and in some•:" cases exceeding kc = 0.4, except for the pitching moment curves at* the

••. transonic dip Mach number of 0.85. All i~atter results are presented forS~M = 0.85. The p-k method gives flutter speeds identical to those from the

3 U-g method. Subcritical damping ratios using the U-g method with Frueh'sand Miller's damping formula are quite close to those obtained from the p-k

: method, especially for large values of the airfoil-air mass ratio. Response of' ~the airfoil to externally applied forces and moments is studied using the p-k

method and a viscous damping model for coupled motions.

.. ~RESUME•

•: II s'agit de l'dtude des vibrations a6rodlastiques en rdgime-• transsonique d'un profil NACA64A006 soumis • des oscillaticns dans le

Splan vertical au moyen des mdthodes U-g 't p-k. Les coe•.icients adrodyna-: miques sont calcul6s a partir d'une version amdliorde du code des vitesses

adrodynamiques instationnaires en rdgime transsonique de I'ONERA," ~laquelle a recours • Ia seconde ddrivatif du temps du potentiel de la vitesse

:• dans l'6quation principale. Des comparaisons avec LTRAN2-NLR montrent:.: que les valeurs obtenues sont conformes jusqu'•, et quelque'fois surpassant,g kc - 0.4, sauf dans le cas des cotzrbes de moment de tangage pour un nombre

de Mach creux transsonic de 0.65. La mdthode p-k fournit des vitesses de" ~vibrations adro61astiques identiques a celles de la mdthode U-g. Des rapportsS~d'amortissement moins critiques employ6s dans le cadre de la mdthode U-g

avec les formules d'amiortissement de Frueh et de Miller sont tr~s prochesde ceux obtenus avec Ia mdthode p-k, surtout pour les grandes valeurs de

4[ rapport de masse de proqile-d'air. La r&~ponse du profil • des forces et • des• "., moments exte'rieurs est 6tudi~e par la m~thode p-k et un module d'amortis-

CONTENTS

Page

SUM M AR Y ........................................................... (iii)

SYMBOLS ............................................................ (vii)

1.0 INTRODUCTION ...................................................... . 1

"2.0 FLUTTER ANALYSIS OF TWO-DEGREE-OF-FREEDOM AIRFOIL MOTION ...... 2

2.1 The U-g Method .................................................... 32.2 The p-k Method..... ............................................ 4.2.3 Viscous Damping Model for Coupled Motions ............................. 5

3.0 RESULTS AND DISCUSSIONS ...................-........................ 6

3.1 Aerodynamic Coefficients ............................................ 63.2 Flutter Analysis of NACA64AO06 Airfoil ...........-..................... 73.3 Response of Airfoil to External Excitations .............................. 7

4.0 CONCLUSIONS ....................................................... 8

5.0 REFERENCES--------------------------------------------------------- 9

Table 1 Distances from Leading Edge for Cp Plotted in Figure 21 .................... 11!PILLUSTRATIONS

Figure Page

"1 Two-Degree-of-Freedom Airfoil Motion ................................... 13

2 Variation of Lift Coefficient Due to Plunge With Reduced Frequency for a"NACA64AO06 Airfoil at M .= 0.8 ...................................... 14

-- 3 Variation of Moment Coefficient Due to Plunge With Reduced Frequency"for a NACA64AO06 Airfoil at M = 0.8 .................................. 15

4 Variation of Lift Coefficient Due to Pitch With Reduced Frequency for aNACA64AO06 Airfoil at M = 0.8 ...................................... - 16

S5 Variation of Moment Coefficient Due to Pitch With Reduced Frequencyfor a NACA64AO06 Airfoil at M = 0.8 .................................. 17

6 Variation of Lift Coefficient Due to Plunge With Reduced Frequency for aNACA64AO06 Airfoil at M = 0.85 ..................................... 18

7 Variation of Moment Coefficient Due to Plunge With Reduced Frequencyfor a NACA64AO06 Airfoil at M = 0.85 ................................. . 19

(iv)

Io

ILLUSTRATIONS (Cont'd)

Figure Page

8 Variation of Lift Coefficient Due to Pitch With Reduced Frequency for aNACA64AO06 Airfoil at M = 0.85 ..................................... 20

9 Variation of Moment Coefficier,t Due to Pitch With Reduced Frequencyfor a NACA64AO06 Airfoil at M = 0.85 ................................. 21

10 Variation of Lift Coefficient Due to Plunge With Reduced Frequercy for aNACA64AO06 Airfoil at M = 0.875 .................................... 22

11 Variation of Moment Coefficient Due to Plunge With Reduced Frequencyfor a NACA64AO06 Airfoil at M = 0.875 ................................ 23

12 Variation of Lift Coefficient Due to Pitch With Reduced Frk lutncy for aNACA64AO06 Airfoil at M = 0.875 .................................... 24

13 Variation of Moment Coefficient Due to Pitch With Reduced Frequencyfor a NACA64AO06 Airfoil at M = 0.875 ................................ 25

14 Variation of Lift Coefficient Due to Plunge With Mach number for aNACA64A006 Airfoil at kc = 0.1 and 0.2 ............................... 26

15 Variation of Moment Coefficient Due to Plunge With Mach Number for aNACA64AO06 Airfoil at kc = 0.1 and 0.2 ................................ 27

16 Variation o' Lift Coefficient Due to Pitch With Mach Number for aNACA64A006 AKrfoil at kc = 0.1 and 0.2 ............................... 28

17 Variation of Moment Coefficient Due to Pitch With Mach Number for aNACA64AO06 Airfoil at kc = 0.1 and 0.2 ............................... 29

18 Cp Distribution for Plunging Motion on the Upper (--) and Lower (- -

Surfaces of a NACA64AO06 Airfoil at M = 0.85 and kc = 0.5 ................. 30

19 C Distrib. )n for Plunging Motion on the Upper (-) and Lower (- - -)

Surfaces of a NACA64AO06 Airfoil at M = 0.85 and kc = 2 .................. 31

20 Shock Wave Position for Plunging Oscillation of a NACA64AO06 Airfoil atM = 0.85, kc = 0.5 and 2 ............................................. 32

21 Cp Variation With Time on Upper Surface of a NACA64AO06 Airfoil atM = 0.85, kc = 0.5 and 2. (The distances from leading edge where Care computed are given in Table 1) ..................................... 33

22 Variation of U/bw,,, w/cj,, and k. With p for a NACA64AO06 Airfoil atM = 0.85 ......................................................... 34

23(a) Comparison of Damping Ratio and w 1 /w,, With U/bwQ, Between U-g andp-k Methods for a NACA64AO06 Airfoil at M = 0.85 and M = 50 .............. 35

(v)

ILLUSTRATIONS (Cont'd)

"Figure Page

M 23(b) Comparison of Damping Ratio and W.21Iw• With U/bwa Betv-ecn U-g andp-k Methods for a NACA64AO06 Airfoil at M = 0.85 and p = 50 .............. 36

24(a) Comparison of Damping Ratio and w I/cw With U/bw• Between U-g andp-k Methods for a NACA64AO06 Airfoil at M = 0.85 and p = 75 .............. 37

24(b) Comparison of Damping Ratio and W2/1W, With U/bwcA Between U-g and,. p-k Methods for a NACA64AO06 Airfoil -t M = 0.85 and p = 75 .............. 38

"25(a) Comparison of Damping Ratio and w I/1w With U/bw, Between U-g and-: p-k Methods for a NACA64AO06 Airfoil at M = 0.85 and p = 100 ............. 39

25(b) Comparison of Damping Ratio and C'02/I" With U/bw, Between U-g andp-k Methods for a NACA64AO06 Airfoil at M = 0.85 andp = 100 ............. 40

26(a) Comparison of Damping Ratio and w I/ o1 With U/bw, Between U-g andp-k Methods for a NACA64AO06 Airfoil at M = 0.85 and p = 150 ............. 41

26(b) Comparison of Damping Ratio and w2 /w, With U/bw, Between U-g andp-k Methods for a NACA64AO06 Airfoil at M = 0.85 and p = 150 .............. 42

27(a) Comparison of Damping Ratio and w /Iw, With U/bw, Between U-g andp-k Methods for a NACA64A006 Airfoil at M = 0.85 and p = 250 ............. 43

* 27(b) Comparison of Damping Ratio and W2/W, With U/bw, Between U-g andp-k Methods for a NACA64AO06 Airfoil at M = 0.85 and p = 250 ............. 44

28 Variation of Amplitude of Displacement Response With U/bco and w/cwafor an Externally Applied Sinusoidal Force .............................. 45

29 Variation of Phase Angle of Displacement Response With U/bwo and w/w.for an Externally Applied Sinusoidal Force .............................. 46

30 Variation of Amplitude of Pitch Response With U/bcoa and wow, for anExternally Applied Sinusoidal Force .................................... 47

"" 31 Varip.tion of Phase Angle of Pitch Response With U/bwo, and wc/c for anExternally Applied Sinusoidal Force .................................... 48

32 Variation of Amplitude of Displacement Response With U/bw, and wlwo/for an Externally Applied Sinusoidal Moment at Elastic Axis .................. 49

33 Variation of Phase Angle of Displacement Response With Ub'bw. and ,for an Externally Applied Sinusoidal Moment at Elastic Axis .................. 50

34 Variation of Amplitude of Pitch Response With U/bw,, and ý)Iw, for anExternally Applied Sinusoidal Mom"nt at Elastic Axis ...................... 51

(vi)

I

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

ILLUSTRATIONS (Cont'd)

* Figure Page

35 Variation of Phase Angle of Pitch Response With Uibw,0 and w•w. for anExternally Applied Sinusoidal Momenit at Elastic Axis ...................... 52

36 Comparison of Amplitude of Displacemert Response Between ViscousDamping and p-k Methods for an Externally Applied Sinusoidal Force ......... 53

37 Comparison of Phase Angle of Displacement Response Between Viscousj Damping and p-k Methods for an Externally Applied Sinusoidal Force .......... 54

38 Comparison of Amplitude of Pitch Response Between Viscous Dampingand p-k Methods for an Externally Applied Sinusoidal Force ................. 55

39 Comparison of Phase Angle of Pitch Response Between Viscous Dampingand p-k Methods for an Externally Applied Sinusoidal Force ................. 56

* 40 Comparison of Amplitude of Displacement Response Between ViscousDamping and p-k Methods for an Externally Applied Sinusoidal Momentat Elastic Axis ..................................................... 57

41 Comparison of Phase Angle of Displacement Response Between ViscousDamping and, p-k Methods for an Externally Applied Sinusoidal Moment

at Elastic Axu ..................................................... 5842 Comparison cf Amplitude of Pitch Response Between Viscous Damping

and p-k Methods for an Fxtemally Applied Sinusoidal Moment atElastic A xis ....................................................... 59

43 Comparison of Phase Angle of Pitch Response Between Viscous Dampingand p-k Methods for an Externally Applied Sinusoidal Moment atElastic A xis ............................ .......................... 60

SYMBOLS

Symbol Definition

ah non-dimensional distance measured from midchord to elastic axis

b semi-chord of airfoil

c chord

CP pressure coefficient

Ch lift coefficient due to plunge

q. lift coeff;cicat due to pitch

Cmh moment coefficient due to plunge

(vii)

S° -: • • . " .* - b °• o . ° - ' , ". - "-T -- •- -* • " * .t. " -. * • ""- , ° *. ' . -° " % o' .

SYMBOLS (Cont'd)

Symbol Defmition

C0n moment coefficient due to pitch

g structural damping coefficient

h plunge displacement

ke reduced frequency wc/U

m mass of airfoil per unit span

M free stream Mach number

p complex eigenvalue

p non-dimen•sional complex eigenvalue

q dynamic pressure

Qh total forces

externally applied force

Q1, total moment

externally applied moment

r. radius of gyration about elastic axis -

t time

U free stream velocity

x, non-dimensional distance measured from elastic axis to center of mass

°defined in Equation 34

17 defined in Equation 35

pitch angle

damping ratio defined in Equation 18

damping ratio defined in Equation 31

°viscous damping for plunging motion

viscous damping for pitching motion

flutter eigenvalue defined in Equation 9

(viii)

7-..

SYMBOLS (Cont'd)

Symbol Definition

p airfoil-air mass ratio, m/irpb 2

non-dimensional displacement h/c

P air density

Co frequency

W1, uncoupled bending frequency

W•a uncoupled torsional frequency

(ix)

. .. . . ..

A STUDY OF TRANSONIC FLUTTER OF A TiWO-DIMENSIONAL AIRFOIL

USING TIlE U.g AND p-k METiHODS

1.0 INTRODUCTION

"Two and three-degree-of-freedom transonic flutter of two-dimensional conventional and,uperzritical airfoils has been the subject of numerous investig-ttions in recent years (Refs. 1-12). Thedrop in flutter iclocity at transonic Mach nmber is an important factor in design of modern airfoils.

1 In the past, before unsteady transonic computation codes were available, the flutter speed in thetransonic range was estimated by extrapolatihg the subsonic and supersonic branches of the fluttercurves. Among the methods for predict ing unsteady transonic flow past oscillating airfoi' (Refs. 13-19),those based on the transonic small disturbance equation are most efficient, and hence are widely used

* in flutter calculations of two-dimensional airfoils.

Two approaches have been employed in the analysis of transonic flutter. The first utilizesthe :onventional U-g method (Ref. 20). The aerodynamic lift and moment coefficients are determinedfrom transonic flow calculations and the structural equations are soi.ed for the flutter velocity(Refs. 1-5). Airfoils that have been studied include NACA64AO06, NACA64AO10 anid MBB A-3 withvariations in parameters Such as mass ratio, center of mass location, frequency ratio, etc.

The second approach determines the aeroelastic responses by coupling aerodynamiccomputation codes with structural eqtations of motion using a variety of numerical integrationtechniques (Refs. 6-12). Such an approach was first taken by Ballhaus and Goorjian (Ref. 6) whocarried out the aeroelastic response of a NACA64AO06 airfoil with a single degree-of-freedom in

* pitch. Extensions of this procedure to two and three-degree-of-freedom and other airfoils such asNACA64A010 and MBB A-3 have bpea reported by Yang et al. (Refs. 8-10).

The original finite-difference scheme for solving the transonic small disturbance equation,called LTRAN2 by Ballhaus and Goorjian (Ref. 16), is accurate only at low reduced frequencies

F (kc < 0.075). Houwink and van der Vooren (Ref. 17) improved this code by retaining the time-derivative terms in the boundary and auxillary conditions. Their modified code called LTRAN2-NLR

* claimed accuracy up to k, = 0.4. Similar modifications have also been carried out by Couston and* Angelini (Ref. 18) at ONERA. Further improvements have been reported by Rizzetta and Chin

"(Ref. 19) who retained the second time-derivative term of the velocity potential in the governingequation and presented results for a NACA64A10 airfoil with flap oscillations up to reducedfrequency of 5. Isogai (Ref. 5) developed a similar computation code, and claimed accuracy up tokc = 0.25 for the entire transonic Mach number range (from subcritical Mach number to above Machnumber 1), Similar improvements to the Couston'! and Angelini's code (Ref. 18) have been carriedout at ONERA, and this code has been made available to NAE. Minor modifications (Ref. 21) havebeen made and these included addition of some graphics for the output data. The use of these variousmodified or improved unsteady t-ansonic aerodynamic codes in flutter analysis can be found inReferences 1-13.

The U-g method gives critical flutter speeds in agreement with other methods such as thetraditional British approach with lined-up frequency parameters, but overestimates the relative damp-ing ratio at other speeds (Ref. 22). The use of Frueh's and Miller's (Ref. 23) correction formula fordamping gives useful values of the decay factors, but Woodcock and Lawrence (Ref. 24) found theU-g method is still inferior to the British method. Hassig (Ref. 25) developed a method which givesdamping values in excellent ageement with the British method. The p-k method, as called by Hassig(Ref. 25), iterates for the zeros of the flutter determinant ushig values of the aerodynamics interpo- 2lated from given values at a set of frequency parameters. This method assumes that for sinusoidalmations with slowly varying amplitude, the aerodynamic forces can be approximated by those basedon constant amplitude.

:..,- :. ..:•:- .• .. .. ,. .-.,'.., -,,- ,,.. , ., ,.-. ,, .: . -, .. -.. :..,-..-, . , . .. ,- ,.-. ,--,. .. ...- .. ..: . . .. . .. . . ... . .. .,,. .. .. : . .. .- . .. . 2

-2-S.~

The use of time marching techniques which couple the aerodynamics with the structuralequations of motion should g~ve accurate results for the critical flutter speeds and damping ratios atother speeds. However, this method is time consuming and is not suited for flutter ara.ysis wh.eremany parameters such as mass ratio, frequency ratio, etc. are varied.

The first part of this report presents aerodynamics data of a NACA64AO06 airfoil oscillatingin j. h and plunge using the high frequency version of ONERA unsteady transonic code. The resultsare compared with those obtained by Yang and Chen (Ref. 9) employing LTRAN2-NLR. To the bestof the author's knowledge, data from the ONERA code have not been reported by ONERA in theopen literature. They are, therefore, presented in somewhat detailed manner in this report. From theaerodynamics obtained from the ONERA code, the U-g method is used to compute critical flutterspeeds and damping ratios for a two degree-of-freedom NACA64AO06 airfoil using Frueh's andMiller's formula (Ref. 23). The results using the p-k method are compared with those from. the U-gmethod. Response curves from the p-k method and those using a viscous dampirg model for coupledmotions are also included for forced sinusoidal motion.

2.0 FLUTTER ANALYSIS OF TWO-DEGREE-OF-FREEDOM AIRFOIL MOTION

Figure 1 shows the notations used in the analysis of a two -degree-of-freedom motion of anairfoil oscillating in pitch and in plunge. The bending deflection is denoted by h, positive in thedownward direction. a is thQ pitch angle about the elastic axis, positive with the nose up. The elastic K.axis is located at a distance ahb from the midchord, while the mass center is located at a distance xabfron the elastic axis. Both distances are positive when measured towards the trailing edge of the.irfoil. The aeroelastic equations of -notion have been derived by Fung (Ref. 20) and can bt writtenas:

2 Qh (

mb

Q..

xi+r2+r2W~a (2)mb

2

where • = h/b is the non-dimensional displacement, m is the mass per unit span of ,ne airfoil, CWh, W.are the uncoupled plunging and pitching frequency respectively, r, is the radius of gyration aboutthe elastic axis, and Qh and Q, are the total forces r.nd moments acting about the elastic axisrespectively. In the absence of externally applied forces and moments, Qh and Q, can be writ'en as:

Qh = -qc (( 2 4+C•a (3)

Q, qc2 mh i ' e a) (4)

where q is the dynamic pressure, CQ and Cm are the lift and moment coefficimnts with the subscriptsh and a denoting unit plunging and pitching motion., respectively.

-•..'....."†††††††††.....•............ . .. ............. . ,

-3-

2.1 The U-g Method

In the U-g method, a structural r'amping coefficient g is introduced into Equations (1)and (2) by multiplying the third term of the two equations by the factor (1 + ig). For harmonic

* oscillations, • and a can be written in the following form:

= o eiwt (•)

a = ao eiwt (6)

Equations (1) and (2) can be expressed as:

[ i~ k ( coal)] I[ k-7r a =

A2 [ X. +_ tj + , 2 r2 +- a--•- o = 0 (8)

C ir

where = m/lrb2 p is the airfoil-air mass ratio, kc -c/U is the reduced frequency and

w2 b2

-= (1+ig)- (9)

The complex eigenvalue X can be solved from the following quadratic equation

AX2 + BX + C =0 (10)

where

A )(11)

B= -y--,/d-r2a (12)

C= ad - bc (13)

4* - - - - -

":"1 Ch

a -k--- (14)4 C 21r

b2= -k2x --- (15)

1 - x.- + (16)

2 Cm,d k 12 + (17)4 ca 7

Hassig (Ref. 25), quotiz'- from Frueh and Miller (Ref. 23), gives an expression for thedamping ratio which is written in the present notations as:

go[-(Lg)1((18)

f" 2.2 The p-k Method

Equations (5) and (6) can be written in the following form:

"= eP (19)

of ao ePt (20)

where p = + iw, and j is the damping factor. Substituting into Equations (1) to (4) gives:

p2 +4 (.)( ,o) +-% o o {2 x.+ + }a = 0 (21)

P2 X,- Cmh t + r2 p2 + 4r 2 ) 8 a(, 0 (22)'"-.~v - t U / 7rP a 0(2

,M.1

PI

-5-

Cwhere • = p, and can be solved from the following:

Dr+E + F 0 (23)

where D = r2 - x2 (24)

aa

E = 1+ir2 -x. (+b) (25)

F = ad-bc (26)

Qh2 8bW (27)

- 4 (28)

4= rA - Ch (29)

4r bwa\2 8S4r2 V(- C,. (30)

The damping ratio " is given as:

Vp2 + W2 (31)

2.3 Viscous Damping M el for Coupled Motions

4 In the study of t e dynamic response of a -laig to random loads or buffeting, a viscous"damping force is often usq in the structural equations of motion in place of the aerodynamic"damping force, and the daping ratio is obtained from flutter calculations (see, for example, Ref. 26).Introducing a viscous dampng term in Equations (1) and (2), and assuming harmonic motion of theform given in Equations (5) and (6), the structural equations of motion are:

QhZh(W) to -W 2 X a0 mb = (32)

qa

.6.

"2 Q.

""- W x0 to + r- Z (w) ao = (33)mb 2

where Zh(Lw) - - w2 + i 2•" hw (34)

Z0 (w) = 2 _ + i 2tc.c (35)

,h and • are the damping ratios for plunging and pitching motions. Equations (32) and (33) can beused to study the response of the airfoil to external loads.

"* 3.0 RESULTS AND DISCUSSIONS

3.L Aerodynamic Coefficients

In this study only two-degree-of-freedom motions in pitch and plunge are considered. Theaerodynamic coefficients presented are those for lift and moment due to unit plunge displacement

S and pitch rotation respectively, namely, C~h, Cmh, CQ, and Cmo,,. The computations are performedusing the ONERA improved version of Couston's and Angelini's (Ref. 18) unsteady transonic aero-dynamics code. The iesults for a NACA64A006 airfoil are presented herein in a somewhat detailedmanner since it does not appear that computations using this computer code have been compared

*. with other transonic codes and reported in the open literature.

In Figures 2 to 13, C~h, Cmh, C,, and Cm are plotted versus kc for Mach numbersM 0.8, 0.85 and 0.875. The amplitudes for the plunging and pitching motions are h/c = 0.04 andS= 0.1 deg. respectively, and the pitch axis is located at the 1/4 chord point. The results are computedup to a value of kc = 2. Shown also in the figure., are computations by Yang and Chen (Ref. 9) usingLTRAN2-NLR which is claimed to be accurate up to kc = 0.4. The results from the ONERA codeagree quite well with LTRAN2-NLR up to and in some cases exceeding kc = 0.4, except for the Ccurves (especially the imaginary part) at M = 0.85. At higher frequencies, LTRAN2-NLR is notaccurate since, unlike the ONERA code, the second time derivative of the velocity potential term hasnot been included in the governing equation.

The four aerodynamic coefficients are plotted versus Mach number in Figures 14 to 17 forreduced frequency kc = 0.1 and 0.2. It is seen that at M = 0.85 there are large changes in the aero-

dynamic coefficients, especially the moment coefficients. This Mach number coincides with thetransonic dip observed for the NACA64AO06 airfoil by Yang et al. (Refs. 1, 2). At M = 0.7 and 0.8the flow past the airfoil contains no shock waves. At M = 0.85, Fig. 18 shows that a shock wave ispresent for part of a plunge cycle at kc = 0.5. The results are typical up to a value of kc approximately1.4, above which a shock wave is present for the full cycle as shown in Figure 19 for kc = 2. Theshock positions* for the third and fourth cycles of computations are shown in Figure 20 for k' = 0.5and 2.0. It is seen that at low frequencies, the shock movements are much larger than those at higherfrequencies.

For plunging motions, Cp plots for the upper surface of the airfoil at M = 0.85 for the thirdand fourth cycles of computations are shown in Figure 21 for k. = 0.5 and 2.0. The numbers from 3to 29 marked on the curves represent computation grid points, and their distances from the leadingedge non-dimensionalized w.r.t. the chord are given in Table 1. Because of the interrupted shock wave

$The shock positions are determined, according to Reference 21, from grid points where ACp/Ax is a maximum, while also satisfyingthe criterion t6Cp/Ax > 2,

4. . .• . . . . ° . • • ,

-7-

motions at low frequ, ncies, shown for example in Figure 21 (kc 0.5), the pressures in the regionbehind the shock sho%,, irregular oscillations. At higher frequencies, when shock waves are presentedat all times in a plunge cycle (Fig. 21, kc 2.0), the pressures behind the shock oscillates sinusoidally.

Results for pitch oscillations similar to those presented in Figures 18 to 21 are not reportedin this report. For a NACA64AO10 airfoil, Isogai (Ref. 5) gives some quite detailed results for pitching"motion.

3.2 Flutter Analysis of NACA64AO06 Airfoil

Flutter calculations using the ONERA unsteady transonic aerodynamics code are compared"in Figure 22 with those obtained by Yang et al. (Ref. 1) using an implicit scheme (Ref. 6) and arelaxation approach (Refs. 14, 15) for the aerodynamic coefficients. In this figure, the U-g method isused and the flutter speed U/bw., reduced frequency kc and frequency ratio /wlwc, are plotted againstairfoil.air mass ratio p. The Mach number is 0.85, and the following parameters are used: x. = 0.25,-_r,, = 0.5, Wh /Cj = 0.2, pitch axis = 0.25 (ah = -0.5), which are the ones used by Yang et al. (Ref. 1).This figure shows lower flutter speeds using ONERA code than those from Reference 6, except at amass ratio of 50. The present results can be taken to be more accurate than Reference 1 since the"aerodynamics are improved by taking the second time derivative of the velocity potential in the

-- governing equation. The aerodynamic coefficients are in good agreement with LTRAN2-NLR (seeFigs. 6 to 9) in the range of values for kc between 0.1 and 0.2 where flutter speeds are calculated.Comparisons of the aerodynamics from the indicial and relaxation methods given in Reference 2 with

* either the LTRAN2-NLR or the present results show tWe relaxation approach yields closer agreementthan the .ndicial method which gives much poorer results especially for the CQ and Cm coefficients.

The effect of p on the damping and frequency ratios are shown in Figures 23 to 27 forM = 0.85. The results for " and -7 of the two modes are computed using Equations (18) and (31). Thecritical flutter speeds and frequencies obtained from both methods are identical. For the bendingmode, the differences in subcritical damping and frequency ratios between the p-k and U-g methodsdecrease for increasing p. At u = 250, these two methods give almost identical results. For the torsionmode, the differences between " and -f increase with p while the differences in W 2 /1. decrease. Thefairly good agreement between the two methods may be a special case for a two-degree-of-freedomtwo-dimensional airfoil motion. Results for subcritical damping from Reference 24 show poor com-parison between the U-g and the British method with lined-up frequency parameter, which has in turnbeen demonstrated by Hassig (Ref. 25) to be in good agreement with the p-k method.

3.3 Response of Airfoil to External Excitations

To investigate the response of the airfoil as the flutter speed is approached, • and a are- solved from Equations (1) and (2) for two cases: Q' = 0 and Qh = 0, where Q, and Qh are the

externally applied sinusoidal excitation moment and force at the airfoil's elastic axis respectively.

mbw2 mbw2

Using the p-k method, the amplitude and phase of to and a. forQ'=0 and:•'•Qh Qh

mb 2W2 mb 2 2

to and ao , for Q' = 0 are plotted against - and O./bw• for M = 0.85 and p = 100,Q.(A)

in Figures 28 to 35. For a purely translational excitation force, the displacement amplitude (Fig. 28)decreases with increasing values of U/bwQ until a minimum is reached around U/bw, = 2.5 to 3.0.Further increase in U/bw, results in a rapid growth of the amplitude. Tl.e pitch amplitude (Fig. 30)vshows a similar behaviour for w1w/, in the vicinity of the uncoupled bending frequency. Theamplitude near the uncoupled torsional frequency decays as the flutter speed is approached.Figures 29 and 31 show the corresponding phase angles for displacement and pitch rotationrespectively.

.8-

When a moment is applied at the elastic axis, the displacement increases rather gradually"initially until U/bcWQ approaches tne flutter speed (Fig. 32). Similar behaviour of the pitch amplitude

.. °0. near the uncoupled bending frequency is shown in Figure 34. The peak response near = 1decays from large values at small U/bw. to very small values as U/bcw, approaches the critical value.The phase angles for the displacement and pitch response are shown in Figures 33 and 35.

. "In Section 2.3 a viscous damping model far the airfoil motion is given. In the study of" dynamic response of wing to external loads, such as buffeting, often the aerodynamic damrping is

combined with the structural damping to yield a total damping proportional to the ve!ocity (see, forexample, Ref. 27). The damping values can be determined either experimentally or from fluttercalculations, such as the p-k method. Figures 36 to 43 show the displacement and pitch response ofa NACA64AO06 airfoil at M = 0.85 and p = 100. The values of ýh, ý' and Wh/W. are obtained for

. U/bw)o = 4 using the p-k method. Curres for other values of U/bw. are quite similar to those shownin these figures. For a translational excitation force (Q, = 0), Figures 36 to 39 show the viscous

, damping and p-k methods give quite similar response curves. However, when the external excitationis a moment applied at the elastic axis (Qh = 0), Figures 40 and 42 show the large response obtainednear the uncoupled bending frequency from the p-k method carinot be duplicated usirg the viscousdamping model, even though the response for pitching motion near the uncoupled torsional frequencyagrees fairly well. The phase curves for the displacement show a difference of nearly 180 degreesbetween the two methods, while those for the pitch rotation are reasonably good except near theuncoupled bending frequency.

* 4.0 CONCLUSIONS

The improved version of ONERA unsteady transonic aerodynamics code, which include

-. the second time derivative term of the velocity potential in the governing equation, has been used to"compute aerodynamic coefficients for plurging and pitching motions of a NACA64AO06 airfoil atvarious Mach numbers. The results agree quite well with LTRAN2-NLR up to and in some cases

* exceeding kc = 0.4, exce it for the Cma curves (especially the imagi.'ary part) at M = 0.85. ThisMach number coincides with the transonic dip and large changes in the aerodynamic coefficients,

especially the moment coefficients, are observed. Results of shock motions are given for the airfoiloscillating in plunge. At M 0.85 and kc below 1.4, computations show that a shock wave is present"only for part of a plunge cycle. The shock motions are fairly large and the pressures in the regionbehind the shock show irregular oscillations.

Flutter speeds for a NACA64AO06 airfoil using the U-g mehod and aerodynamiccoefficients from ONERA code are lower than those reported by Yang et al. using LTRAN2. Thecomparisons are made at M = 0.85 for various airfoil-air mass ratios. The p-k method gives flutterspeeds and k¢ values identical to those from the U-g method. The computed subcritical dampingratios at M = 0.85 for different t show the U-g method using Frueh's and Miller's formula gives

* results quite close to the p-k method, especially for large values of p. The comparison is always veryclose for small values of U/bw. irrespect of p. The fairly good agreement between these two methodsmay bc a special case for a two-degree-of-freedom two-dimensional airfoil performing plunging andpitching oscillations. Results for more complicated cases show that subcritical damping ratios obtainedfrom the U-g method agree poorly with the British method with lined-up frequency parameters, whichhas in turn been demonstrated to be in good agreement with the p-k method.

The p-k method can be used to study the response of an airfoil to externally appliedexcitation forces and moments. Two examples are given for either a translational force or a momentap,' ied at the elastic axis. Using the computed values of damping ratios ýh and r,, for the bendinga. A torsion modes together with the ratio w/,, the response using a viscous damping model hasbeen determined and compared with the p-k method. It is shown that for translational excitation

* force, the response curves for the two methods are in good agreement. However, when the externalexcitation is a moment applied at the elastic axis, the large response obtained near the uncoupledbending frequency from the p-k method cannot be duplicated using the viscous damping model, eventhough the response for pitching motion near the uncoupled torsional frequercy agrees fairly well.

"4..

-9.

5.0 REFERENCES

1. Yang, T.Y. Flutter Analysis of a NACA64AO06 Airfoil in Small DisturbanceGuruswamy, P. Transonic Flow.Striz, A.G. J. Aircraft, Vol. 17, No. 4, April 1980, pp. 225-232.Olsen, J.J.

2. Yang, T.Y. Flutter Analysis of Two-Dimensional and Two-Degree-of-FreedomStriz, A.G. Airfoils in Small-Disturbance Unsteady Transonic Flow.Guruswamy, P. Flight Dynamics Laboratory, WPAFB, Ohio, AFFDL-TR-78.202,

December 1978.

3. Traci, R.M. Small Disturbance Transonic Flows About Oscillating Airfoils andAlbano, E.D. Planar Wings.Farr, J.L. Flight Dynamics Laboratory, WPAFB, Ohio, AFFDL-TR-75-100.

June 1975.

4. Rizzetta, D.P. Transonic Flutter Analysis of a Two-Dimensional Airfoil.Flight Dynamics Laboratory, WPAFB, Ohio, AFFDL-TM-77-64-FBR, July 1977.

5. Isogai, K. Numerical Study of Transonic Flutter of a Two-DimensionalAirfoil.National Aerospace Laboratory, Ch5fu, Tokyo, Japan, NAL-TR-617T, July 1980.

6. Ballhaus, W.F. Computation of Unsteady Transonic Flows by the IndicialGoorjian, P.M. Method.

AIAA Journal, Vol. 16, Feb. 1978, pp. 117-124.

7. Rizzetta, D.P. Time-Dependent Response o0 Two-Dimensional Airfoil inTransonic Flow.AIAA Journal, Vol. 17, Jan. 1979, pp. 26-32.

8. Yang, T.Y. Transonic Flutter and Response Analyses of Two 3-Degree-of-Chen, C.H. Freedom Airfoils.

J. Aircraft, Vol. 19, Oct. 1982, pp. 875-884.

9. Yang, T.Y. Flutter and Time Response Analyses of Three Degrees of FreedomChen, C.H. Airfoils in Transonic Flow.

Flight E inamics Laboratory, WPAFB, Ohio, AFWAL-TR-81-3103, A g. 1981.

10. Yang, T.Y. Transonic Time-Response Analysis of 3-Degree-of-FreedomBatina, J.T. Conventional and Supercriticcl Airfoils.

J. Aircraft, Vol. 20, Aug. 1983, pp. 703-710.

11. Carretta, C. Simultaneous Resolution of Aerodynamic and AeroelasticCouston, M. Equations of Motion for Transonic Two-Dimensional Airfoils.Angelini, J.J. International Conf.-on Numerical Methods for Coupled Problems,

Swansea, 7-11 Sept. 1981.

12. Edwards, J.W. Time-Marching Transonic Flutter Solutions Including Angle-of-Bennett, R.M. Attack Effects.Whitlow, Jr., W. J. Aircraft, Vol '0, Nov. 1983, pp. 899-906.Seidel, D.A.

.- :'~......'.--:•. ".'.. ••-',.'..,....... ..-.. - '.." '. -.-- .....-. .........-... .- -....-.-....-.-..... . .

"-10-

" 13. Magnus, R. Unsteady Transonic Flows Over an Airfoil.Yoshihara, H. AIAA Journal, Vol. 13, Dec. 1975, pp. 1622-1628.

14. Farr, J.L. Computer Programs for Calculating Small Disturbance TransonicTraci, R.M. Flows About Oscillating Airfoils.Albano, E.D Flight Dynamics Laboratory, WPAFB, Ohio, AFFDL-TR-74-135,

Nov. 1974.

. 15. Traci, R.M. Perturbation Method for Transonic Flows About OscillatingAlbano, E.D. Airfoils.Farr, J.L. AIAA Journal, Vol. 14, Sept. 1976, pp. 1258-1265.

16. Ballhaus, W.F. Implicit Finite-Difference Computations of Unsteady TransonicGoorjian, P.M. Flow About Airfoils.

AIAA Journal, Vol. 15. Dec. 1977, pp. 1728.1735.

17. Houwink, R. Improved Version of LTRAN2 for Unsteady Transonic Flow

van der Vooren, J. Computations.AIAA Journal, Vol. 18, Aug. 1980, pp. 1008-1010.

18. Couston, M. Numerical Solutions of Nonsteady Two-Dimensional TransonicAngelini, J.J. Flows.

J. Fluids Engineering, Vol. 101, Sept. 1979, pp. 341-347.

19. Rizzetta, D.P. Effect of Frequency in Unsteady Transonic Flow.* Chin, W.C. AIAA Journal, Vol. 17, July 1979, pp. 779-781.

20. Fung, Y.C. An Introduction to The Theory of Aeroelasticity, John Wiley &Sons, N.Y., 1955.Ia

21. Jones, D.J. Private Communication.

22. Lawrence, A.J. Comparison of Different Methods of Assessing the Free OscillatoryJackson, P. Characteristics of Aeroelastic Systems.

RAE Technical Report 68296 (ARC CP 1084), 1968.

23. Frueh, F.J. Prediction of Damping Response from Flutter Analysis Solutions.Miller, J.M. Air Force Office of Scientific Research, Arlington, Va., Rept.

65-0952, June 1965.

24. Woodcock, D.L. Further Comparisons of Different Methods of Assessing the FreeLawrence, A.J. Oscillatory Characteristics of Aeroelastic Systems.

RAE Technical Rept. 72188, Dec. 1972.

25. Hassig, H.J. An Approximate True Damping Solution of the Flutter Equationby Determinant Iteration.J. Aircraft, Vol. 8, Nov. 1971, pp. 885-889.

26. Mullans, R.E. Buffet Dynamic Loads During Transonic Maneuvers.Lemley, C.E. Flight Dynamics Laboratory, WPAFB, Ohio, AFFDL-TR-72-46,

Sept. 1972.

27. Jones, J.G. A Survey of the Dynamic Analysis of Buffeting and RelatedPhenomena.Royal Aircraft Establishment, RAE TR72197, Feb. 1973.

I/

S. *9

:- . TABLE 1

DISTANCES FROM LEADING EDGE FOR Cp PLOTTED IN FIGURE 21

Position x/c Position x/c

3 0.04029 17 0.5769

5 0.08516 19 0.6712

7 0.1410 21 0.7546

U 9 0.2079 23 0.8269

11 0.2857 25 0.8883

13 0.3745 27 0.9386

15 0.4744 29 0.9780

0f

0.

S

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

.13-

p-

C r

MEAN POSITION

MID-CHORD

ELASTIC A '

CENTRE OF MASS

FIG. 1: TWO-DEGREE-OF-FREEDOM AIRFOIL MOTION

•4

-14-

NACA 64A006m~ 0. 8A

6.0

S ~YANG 8 CHEN I

5.0A

*C A

3.0-

2.0-

1 . 0 0'

0'

-1.0 I0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

*kFIG. 2: VARIATION OF LIFT COEFFICIENT DUE TO PLUNGE W~IT REDUCED

FREQUENCY FOR A NACA64AO06 AIRFOIL AT M =0.8

-15-

1.4

1.2 NACA64ACO06M= 0.8 -

1.00 YANG B CHEN

0.8-

0.6- Re

0.400.4°

Cm0

-0.2_

-1.40

-1.6

. .. . . . . .. ..

h 0 0. .4 0 6 0. .0 12 1. .6 18 2. .

FREQENC FO "AA4A""IFILA .

12

10- NACA 64A006M= 0.8

8-oYANG a CHEN

YAN

6-

Re

-2

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0. 2.2

FIG. 4: VARIATION OF LIFT COEFFICIENT DUE TO0 PITCH WITH REDUCED -FREQUENCY FOR A NACA64AO06 AIRFOIL AT M =0.8

-17-

0

NACA64AO06A M- 0.8

00 00

0 YANG I CHENA :

-o.- 0.

Cm0

Re

0, ..

-1.5

-2.0 , "-

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2kc"---

FIG. 5: VARIATION OF MOMENT COEFFICIENT DUE TO PITCH WITH REDUCEDFREQUENCY FOR A NACA64AO06 AIRFOKIL AT M =0.

:"o..%- *..... ~ e .-. . T ~ . . < t

-18-

7.0- NACA64AO0O

"M 0.85

6.0- YANG a CHEN

- Im

5.0-

cA

4.C

3.0 A

2.0

1.0-

7--Oo_

Ree

0

;| 0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

FIG. 6: VARIATION OF LIFT COEFFICIENT DUE TO PLUNGE WITH REDUCEDFREQUENCY FOR A NACA64AO06 AIRFOIL AT M =0.85

4

-19-

1.4

1.2 - NACA64AO06

1.0 = 0.85

h. 1.0 -

0.80 YANG a CHEN

0.6-

0.4

0.2

0.0-

_ -0.2

-1.0

-1.2A

-. -0.4

-1.6

0A

CI •

'..- -... J

"0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

6k

FIG. 7: VARIATION OF MOMENT COEFFICIENT DUE TO PLUNGE WITH REDUCED

,. FREQUENCY FOR A NACA64AO06 AIRFOIL AT M = 0.85

Is. . . - . 4

,.. '.

-20-

12 _0 NACA64AO06

M 0.85

0YANG aCHEN

8

6-

CLa Re

4

0-

-2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2kc

FIG. 8: VARIATION OF LIFT COEFFICIENT DUE TO PITCH WITH REDUCEDFREQUENCY FOR A NACA64AO06 AIRFOIL AT M = 0.85

,0.e

-21-

0

NACA64AO06

M a 0.85

O YANG 8 CHEN

S-0.5 A

0

SCmR

. , A

00

-1.050

r, A m

II

-' 0 I I I I I I I I

"2"00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2k¢

4 FIG. 9: VARIATION OF MOMENT COEFFICIENT DUE TO PITCH WITH REDUCED .FREQUENCY FOR A NACA64AO06 AIRFOIL AT M =0.85

Lo" /.

-22-

NACA64AO06 /

M= 0.875

7-

0 YANG & CHEN

•6 /-'Im

5-

Cih

4 .

-I3-

,4 0 Ree00

0 0,2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2:." kc

, FIG. 10: VARIATION OF LIFT COEFFICIENT DUE TO PLUNGE WITH REDUCED

FREQUENCY FOR A NACA64AO06 AIRFOIL AT M 0.875

.23.

1.2-

i1.0 NACA64AO06

M m 0.8750.8

0.6- YANG 8 CHEN,,,F.• ORe

0.4-

0.2

0Cmh

"'" " -0.6 -

-1.2

-2.O 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

* FIG. 11: VARIATION OF MOMENT COEFFICIENT DUE TO PLUNGE WITH REDUC~flFREQUENCY FOR A NACA64AO06 AIRFOIL AT M =0.875

,0oor"

-24-

"q I0

NACA64AO06

M= 0.875

8-

0 YANG a CHEN

6-

i

4[

Cl

0-

"'2-

-4-

-6

- 10 1 I I I I I I I i i

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

FIG. 12: VARIATION OF LIFT COEFFICIENT DUE TO PITCH WITH REDUCED

FREQUENCY FOR A NACA64AO06 AIRFOIL AT M = 0.875

UI

25.-

2.0

1.6 NACA64AO06 ,.SM 0.875

1.2 YANG B CHEN

0.8.

0.4'

Cma

0.0'

-0.4-

-0.8

00-1.2=

Im

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

FIG. 13: VARIATION OF MOMENT COEFFICIENT DUE TO PITCH WITH REDUCEDFREQUENCY FOR A NACA64AO06 AIRFOIL AT M =0.875

-- .6 - 0 --.

-26-

.rr6

NACA64AO06

* REAL I kr 0.1x IMAGINARY

1.4- 0 REAL k- 0.2+ IMAGINARY I c

5

1.2

C/h

:: 0.8

,p-0.6-

0.4-

p.

0.2

.% /

I

060.6 0.7 0.8 0.9

M

a- FIG. 14: VARIATION OF LIFT COEFFICIENT DUE TO PLUNGE WITH MACH NUMBER

FOR A NACA64AO06 AIRFOIL AT k= 0.1 AND 0.2

4=' °-". . . -. • o . . . . . . *-.%.. . ...-..- , '.... . . . . . . . . . . . .

- 27 -

0.15

NACA64AOOI

*REAL0.0.10- x IMAGINARY c0.

o REALk 0.+ IMAGINAR'l

O-otI

Cmh

-0.05

0.10-

-0.15-

-0.2C

-0.2. 540.6 0.7 0.8 0.9

M

FIG. 15: VARIATION OF MOMENT COEFFICIENT DUE TO PLUNGE WITH MACHNUMBER FOR A NACA64AOO6 AIRFOIL AT =c 0.1 AND 0.2

-28-

16.0

NACA 64A006

a REAL k 01.-X IMAGINARY k .

o REAL0.+ IMAGINARY

8.0-

4.0-

Ca

0

-4.0-

-8.0-

~I.06 0.7 0.8 0.9M

FIG. 16: VARIATION OF LIFT COEFFICIENT DUE TO PITCH WITH MACH NUMBERFOR A NACA64AO06 AIRFOIL AT k= 0.1 AND 0.2

-29-

1.6

NACA64 A006

0 REAL k~~ *

1.2- x IMAGINARY C

0 REAL+ IMAGINARY kc0.2

0.4-

Cma

0

-0.4-

-0.8-

0.6 0.7 0.8 0.9M

FIG. 17: VARIATION OF MOMENT COEFFICIENT DUE TO PITCH WITH MACHNUMBER FOR A NACA64AO06 AIRFOIL AT =c 0.1 AND 0.2

-30-

-0.8 * 0.-0..CP pC

-0.4*0.4 7~~~ - 00 ---- -

0.0,0

0.4 0.4

0.8 0.8-

1. 20" 1.2 I * I * I0.0 0.2 0.4 0.6 0.8 1.3 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

X/c K/c

- Cp -0.4 -

-080.0~i i i.I

-0. -- -e- -

0.4-

080.8 1121.2 I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 .0 0.2 0.4 0.6 0.8 1.0 1.2K/c K/c

cp C;- 0.44

- - - - -

0.040.4 vI 3w 0.4

0.8- 0.8

1.21. F0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

K/c K/C.

FIG.18: -0,8,IO O LNGN OIN NTEUPE NLOE --- )S'FAEO N CA6AO IFIPTM=08 N .

-0 .4 - -0 .4

- - - - - -- c

0.0

-31-

-0.8 --cp c. - p-0.4- -0- - ----

0.. .

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 CA 0.6 0.8 1.0 1.2X/c X/C

-0. 0.08 ,

0.8 0.8

1.2F . I I I I 1 .2l I I 1 I1 1 10.0 0.2 0.4 0.6x 0. 8 1.0 1.2 0.0 02 0.4 0.6x/0.8 1.0 1.2

cp cp-0.4--- -- *-,------- c po

0c0.0'5

0.-0.4 3

0.8 0.8-

0.0 0.2 0.4 0.6 0.8 1.0 1.2 d0.0 0.'2 0.4 0.6 0.8 1.0' 1.2x /C x /c

cp cp-

p~ c*

0. 0.8

1.2 1.20. 0 0. 2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1012-

K/C K/c

FIG. 19: C p DISTRIBUTION FOR PLUNGING MOTION ON THE UPPER (- )AND

LOWE~R (---)SURFACES OF A NACA64AO06 AIRFOIL AT M 0.85 AND kc2

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

-32 -

0.7- kc= 0.5.

0.6 -X xxxXXX

Z 0. XXNO SH IOCK XX xZ NO SHOCK xX

k :2

0.6-

KXXXXXX xxxxxxxxx~xx xxxxxxxxxxxxxx

X 0.5- xxxxxxxxx XXXXXXXXXXXXXX

0.4-

0 27r 2rT 4 rwt

FIG. 20: SHOCK WAVE POSITION FOR PLUNGING OSCILLATION OF A NACA64AO06AIRFOIL AT M 0.85, kc 0O.5 AND 2

Cp p

I0.

404

cpo

FIG. 21: C~ VARIATION WITH TIME ON UPPER SURFACE GF A NACA64AO06AIRFOIL AT M =0.85, kc = 0.5 AND 2. (THE DISTANCES FROM LEADING EDGE

WHERE C~ ARE COMPUTED ARE GIVEN IN TABLE 1.)

-34 -

8NACA64AO06

7- m 0.85, wt/w@ z 0.2X

X INDICIAL +

6 + RELAXATION YAGEALX

u x

4+

3+

0 50 100 15020 250

0.32-

iL0.30

0.28-

0 50 100 15.0 200 250

0.3

0.1

~0.

0 50 100 150 200 250

FIG. 22: VARIATION OF U/bw,,, wiw, AND cW ITH iFOR A NACA64AO06AIRFOIL AT M =0.85

. . . . . . ... . .

- 35 -

0.15 NACA64Ao06

""~ z= 0.85, /x.=50

- p-k METHOD

U-g METHOD

0.10

005-, I

|\"--= _

030

i- 0.0

025

0.2

I 2 3 4 5

4m

-0.05

30.30 •/

WI!

WaI

0.20

0 I 2 345U/ bin

FIG. 23(a): COMPARISON OF DAMPING RATIO AND wil/co WITH U/bc' BETWEENU-g AND p-k METHODS FOR A NACA64AO06 AIRFOIL AT M 0.85 ANDp = 50

I.,,: .•.• " .i. .• . ." . - ' . ' .. . ,., -. " .. ' " . . ,•...: ''' " "" ' .,

//

-36-

"NACA64A006"•"~ "aM 0. 85,p. 5 0/

* 0.o25 - C2.,:| - I

- p-k METHOD

- -•- U-g METHOD/

0.20L/

L" < 0.15 -

I/

0.05W2

* -

0 2 3 4 5 6//w

- 0 i M F A 5 0.0

I..2-3 4 b 6

4.i FI.621) OPRSNO APN AI N 2• IHUb EWE

" UgAN -kMTHD FRA AA6A0 AROI T 108.4D• -5

•... .... . .•''. , .. . .. . . . . ...... .. .,.. . .... ." '.','... , . :.... .. 2

1.2

..37-

0.15NACA64AO06

Mz 0.85, /.L= 7 5

p-k METHOD0-0. -0

---- U-g METHOD

0.0-0.

0'00

-0.05-

z

0.30 '

3/ 4

wo

0.250

0.20-

0 I 2 3 4 5 6

U/bw,

FIG. 24(a): COMPARISON OF DAMPING RATIO AND wil/w. WITH U/bw " BETWEENU-g AND p-k METHODS FOR A N,ýCA64A006 AIRFOIL AT M =0.85 AND/ 1A 75

•" : '""" " " "" " ' " ""i.• .:-' " " . . . . . . . . .. '''""I' .,; =- . +.• . •.. . .. , -. +.".. .o . . * . . oo ...Wa. °° . o.

- 38.

NACA64AO06

=O0.8 5, 750.25 2

p-k METHOD

- U-g METHOD

0.20-

0.15

ZI

S0.15

* 0.05

0 1 2 3 45 6

U/bwaJ

FIG. 24(b): COMPARISON OF DAMPING RATIO AND 2/.WITH U/bw. BETWEENU-g AND p-k METHODS FOR A NACA64AO06 AIRFOIL AT M -0.85 AND 75

- 39

0.15NACA64AOOG

M= 0.85, p =100

p-k METHOD

0.10 ~ U-g METHOD

0 - -2

4, 0.05

.0.0 0I2 3 45 6

-0.0.

\ ."

0.35

0.30-0.05-

0.23 -

Wa

0.20 -

I I I III0 I 2 3 4 5 6 •

U/bw0 .

FIG. 25(a): COMPARISON OF DAMPING RATIO AND wl/lw WITH U/bca BETWEENU-g AND p-k METHODS FOR A NACA64AO06 AIRFOIL AT M = 0.85 AND p = 100

.o

. . . .*,;•.•..•.:• .. '." ,,-.,-, ... ,..... ... ,... . ...- ,..-...,.,.. .. ... . , .-,.,,.'...... . '- -, . *... ., • , .,.. .-. ...... ,

/'

-40 -

NACA64AOO6- ~ ~~M z 0 .8 5 , /J = I0 I "

0.25 -- 0. / "

p-k METHOD

----- U-g METHOD

0.20

o

S0.15z

0.

0.10

0.05 -

I I I I I I'

0I 2 3 4 5 6

1.6-

1.4-Wh

2

Wa I:", ilI I.O * IIU b I I 6

•0 I2 54 5 6

U/bwa

FIG. 25(b): COMPARISON OF DAMPING RATIO AND W2/c.• WITH U/bw BETWEENU-g AND p-k METHODS FOR A NACA64AO06 AIRFOIL AT M = 0.85 AND p = 100

-41.

J 0.15NACA6 4A006

M --O.85, p.: 150

p-k METHOD0.10

U-g METHOD

S0.05-

0z

0.

0.0I2 3 4 5 6

-0.05-

0.30 -

UPI

0.25

0.20-

0 I2 3 4 5 6Ulbwa

FIG. 26(a): COMPARISON OF DAMPING RATIO AND w.Ilw. WITH U/bw,, BETWEENU-g AND p-k METHODS FOR A NACA64AO06 AIRFOIL AT M 0.85 AND y 150

-42-

NACA64AO06 J.,

M= 0.85, 150 I0.25 -- /, /

p-k METHOD Yp/

U-g METHOD

0.20 t2

SI~~ii i'

/ I,

C' 0.15 -IzIC -

0.10

0.05-

I , I , I I I i•.•0 2 3 4 5 6

1.6-

1.4-

1.2 . .. . . . . . ...

0"',%

0 I 2 3 4 56

U//bw

FIG. 26(b): COMPARISON OF DAMPING RATIO AND WJ2/1w WITH U/bw. BETWEEN V

U-g AND p-k METHODS FOR A NACA64AO06 AIRFOIL AT M = 0.85 AND u = 150

":: :..... .......... .,.. .... .. .. :• .:.,,.:.,:..,,... . . . .. ... .. . . . .. ,-.": . . ,, . ..... ..... .*

-43-

O °'5

0 .15 .. , .

NACA64AO06

M= 0,85q,.L 250

- p-k METHOD

0.10- U-g METHOD

I-

S0.05

0.0

0.0 0 I 2 3 4 5 67 7

\ I g-

-0.05-I

0.30 - 77

0.25-

0.20 -

0 2 3 4 5 .6 7

-- \. . .'

Uibw,

0.. 0 .--

FIG.1 2 : CN O

U-. AN oMEHOSFO,.-CA4O0 IROI T .8"ADp 5

-44 -

I "

NACA64A006

0.25 M : 0.35,/ =2 5 0

I_ ~ ~p - k M E T H O D//: :

U-g METHOD

0.20

/I "C 2

_o / I

:i- -/0.15 ,

o /r

J, . -

0.10 S-

0.05-

I I I i I i I i I""0 2 3 4 5 6 7

1.6 -

1.4 -

-.

Wa

, .o , I I i I i I I Ii"0 I 2 3 4 5 6 7

U/bw.

FIG. 27(b): COMPARISON OF DAMPING RATIO AND W2/Iw WITH U/bw BETWEENU-g AND p-k METHODS FOR A NACA64AO06 AIRFOIL AT M = 0.85 AND p 250

-~ -. / .-

-45-

0

CD

-o NACA64AO06 .

M= 0. 8 5 , =100.ClC,

°.0

V3

~Io°

0,

0.N

FIG. 28: VARIATION OF AMPLITUDE OF DISPLACEMENT RESPONSE WITH U/bw.AND w~.FOR AN EXTERNALLY APPLIED SINUSOIDAL FORCE

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

*.M~**.~'.Uo *. . p

.46 -

NACA64AO06

S0. M: o.851•0:Ioo. o cy0

-J6

CC

w i

FIG. 29: VARIATION OF PHASE ANGLE OF DISPLACEMENT RESPONSE WITH U /•AND o:•,FOR AN EXTERNALLY APPLIED SINUSOIDAL FORCE

a-i

;:::.. . -. , '-v '. . .. . .,.- .- . .v .'.'- ... ., - .. -'.... .'-. - . ...- .,.. - ... ,. . .v '.. . . .0 ....

-47.

.40 -

CiC\w{

LnL

C:

C-

CDLo

1D

v

m- NACA64AO06

o M= 0.85, : =I00

c

0.o oE

0

1"6• 2.o

0.5-

FIG. 30: VARIATION OF AMPLITUDE OF PITCH RESPONSE WITH U/bw,, AND ci/wFOR AN EXTERNALLY APPLIED SINUSOIDAL FORCE

.

". . . .. . , . . . • . . . . .

Co

- 48-

0E

NN0

3 1-0W'S 0

I~clFOR ANETRAL0PLEDSNSIAOC

-49 -

0

C3

0 NACA64AO06N c~ M= O.8 5qL= 100

1.

0.

FIG. 32: VARIATION OF AMPLITUJDE OF DISPLACEMENT RESPONSE WITH U/bwj.AND wlw/w FOR AN EXTERNALLY APPLIED SINUSOIDAL MOMENT AT ELASTIC AXIS

-50 -

0

0

CL

.0.

FI.3:VRAINOEHS NL FDSLCMN EPNEWT /wAN 0O NETRAL PLE IUODLMMN TEATCAI

-51.

0.

03

•" NACA64AO06

O MA O. 85,= SLAIOC

O o

.o O .E I0 " !

4-

0"

Io,

FIG. 34: VARIATION OF AMPLITUDE OF PITCH RESPONSE WITH U/bc, AND c/(,wFOR AN EXTERNALLY APPLIED SINUSO~bAL MOMENT AT ELASTIC AXIS

- .S•'•"•""•'.,,.w' • '."'.," ." . ." "" ." " " "" "" ... ' * .* ." " .- " . .•* S -'- ." -" " " " " " . " . . "" ,

- .* J. ..

- 52 -

0 S .

0

E cý

~~0.0

FIG.3:VRAINOHS NL O IC EPNEWT / N

z~.FRA XENLYAPIDSIUODLMMN TEATCAI

.53-

Io

C•

NACA64AO06o M 0.8 5 ,b a 100

.N U/blbwa: 4p-k METHOD

----- VISCOUS DAMPINGU,

C)

0

U,)

Lrv

Lo

Lo

7

i !:C,'

O.? 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.CQwI w.

FIG. 36: COMPARISON OF AMPLITUDE OF DISPLACEMENT RESPONSE BETWEEN VISCOUSDAMPING AND p-k METHODS FOR AN EXTERNALLY APPLIED SINUSOIDAL FORCE

S. .. ..

.°.

-o - .. -'

-54 -

Co

ci NACA64AOC6

M= 0.85, .L z 100-- U/bwo 2 4

- p-k METHOD

----- VISCOUS DAMPING

C

0'

w

C

N.J

0

A.D

•o .

U)J cý -

7.

o

CD

"Id I(n

4 ° iII

W/.w

FiG 37: COPRSNO HS NL FDIPAEETRSOS EWE

CD

I

VISCOUS DAMPING AND p-k METHODS FOR AN EXTERNALLY APPLIEDSINUSOIDAL FORCE

Ct

- 55 -

Cl

a;

C)

°.

NACA64AO06"-r M: O.= 50/.8 t00

U/bwa6 4o p-k METHODT..VISCOUS DAMPING

ci-

o

,o.,

o 7

o~o 0.25 0.50 0.79 .0 1:.25 1 .50 1. 7E 2.00-

I IWI W"

w/wa

FIG. 38: COMPARISON OF AMPLITUDE OF PITCH RESPONSE BETWEEN VISCOUS DAMPINGAND p-k METHODS FOR AN EXTERNALLY APPLIED SINUSOIDAL FORCE

I-,b

-56-

c':,

CD-

I

(o

to. M= 0.85,/J,= 100 "0 0 U/bwa = 4.'

. 3 - p-k METHOD ,z ", ---- VISCOUS DAMPING •

S0,

CD

I M:.SIL:O

w WDMIo AN -ETEDS AI

(.0 I ,

0.0 02C.0 07 ,0 .5 15 .5 20

,n/%:RI'

FI.3:C OMAIO FPAEANL FPTHRSOSEBTENVSOS4

3; /_ I.I

-57 -

U,)

ooC

E CDCICU APN

L1')

0

o -.

C=)

01

S_ NACA64A0O6, \

0 lu 0.2 0.0 07500 .25,p 1.50 1.7 2.0

FI.4:CMAISNO-MLTD OF DIPLCEEN RESPONS BEWENISOU

-l•: U/bw 6= 4

AMN p-k METHOD " ALE ---- VISCOUS DAMPINGA

0 N" "o- C%

:Ci

CI•°

DAPN N -k MEHDIO NETRAL PLE IUODLMMN T"-

E LASTIC AXIS ..

-........

c¾*

CD

o -

o ,.o

e -

w c- -

co

0 00157

FIG 4OFRV DP A M

MMN A.

0

0.o

O o'-=

.. w .. >K i 2 : . .. . .. . . 7.:,.<r0 M2 050 0.5 .00.850 1.75 2.00,

VICOSDAPNGAD MTOD ORA EENAL APPLIMETHD SIUOIA. . . . .

*o.%.

N "

uý NACA64AO06Ms- 0 .85 , ý: '41O

o U'b: :4

0 p-k METHODoVISCOUS DAMPING

U, °

0- M7 -. 0 . 25 ' 1. S 5 0 -

DAPN AN p-k METHODS FO.N XENALAPID.IUSIA

A I A

1 I I I 1'%0.00 0.25 0.50 0.75 1.030 1.25 1.50 1 .75 2.00 -z

FIG. 42: COMPARISON OF AMPLITUDE OF PITCH RESPONSE BETWEEN VISCOUS"'"DAMPING AND p-k METHODS FOR AN EXTERNALLY APPLIED SINUSOIDAL "'

MOMENT AT ELASTIC AXIS

.. .. . . . . . . .. . •. . . . , .. . . . . . ... ., , .,- ,,. ... ,

-60-

CD

0)

•.,, ~ c - 0

(o

NACA64AO06

M= 0.85, 1 100CDj - U/bwa 4

El p- k METHODSu VISCOUS DAMPING

. .. ° \ C

* 05

0

*C z

* CD

(I

*I IIICD

co

1•70.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

I•; FIG. 43: COMPARISON OF PHASE ANGLE OF PITCH RESPONSE BETWEEN VISCOUS

, DAMPING AND p-k METHODS FOR AN EXTERNALLY APPLIED SINUSOIDALMOMENT AT ELASTIC AXIS

C€

' 6• /i) C

REPORT DOCUMENTATION PAGE / PAGE DE DOCUMENTATION DE RAPPORT

REPORT/RAPPORT REPORT/RAPPORT

LR-615 NRC No. 23959

la lb

REPORT SECURITY CLASSIFICATION DISTRIBUTION (LIMITATIONS)

CLASSIFICATION DE SECURITE DE RAPPORT I.

Unclassified Unlimited2 3

TITLE/SUBTITLE/TITRE/SOUS-TITRE

A Study of Transonic Flutter of a Two-Dimerisional Airfoil Using the U-g and p-k Methods

4

AUTHOR (S)/AUTEUR(S)

B.H.K. Lee5

SERIES/S ERIE

Aeronautical Report

CORPORATE AUTHOR/PERFORMING AGENCY/AUTEUR D'ENTREPRISE/AGENCE D'EXECUTION

National Research Council CanadaNational Aeronautical Establishment High Speed Laboratory

SPONSORING AGENCY/AGENCE DE SUBVENTION

8

DATE FILE/DOSSIER LAB. ORDER PAGES FIGS/DIAGRAMMES

84-11 COMMANDE DU LAO. 69 43

9 10 11 12a 12b

NOTES

13

DESCRIPTORS (KEY WORDS)/MOTS-CLES

1. Transonic Aerodynamics2. Aerofoils (Transonic)

14 3. Transonic FlutterSUMMARY/SOMMAIRE

Transonic flutter of a NACA64AO06 airfoil undergoing plungingand pitching oscillations is studied using the U-g and p-k methods. Theaerodynamic coefficients are calculated using an improved version of anONERA unsteady transonic aerodynamics code which include the secondtime derivative term of the velocity potential in the governing equation.Comparisons with LTRAN2-NLR show good agreement up to and in somecases exceeding kc = 0.4, except for the pitching moment curves at thetransonic dip Mach number of 0.85. All flutter results are presented forM = 0.85. The p-k method gives flutter speeds identical to those from theU-g method. Subcritical damping ratios using the U-g method with Frueh's pand Miller's damping formula are quite close to those obtained from the p-kmethod, especially for large values of the airfoil-air mass ratio. Response ofthe airfoil to externally applied forces and moments is studied using the p-kmethod and a viscous damping model for coupled motions.

15 5

S4-ý85

DT-