Supersonic unstalled flutter.[aerodynamic loading of thin airfoils ...

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NASA Technical Memorandum 79001 N79-11000 ( NASA- T M-79001) Y UTTEB (A S A)

S PEBSONIC ONST LLED 01A 24 P HC A02/ F A01 CSCL

unclas G3/02

36983

SUPERSONIC UNSTALLED FLUTTER

J . J. Adamczyk , M. E. Goldstein, and M. J . Hartmann

Lewis Research Center Cle ve land , Ohio

TECHNICAL PAPER to be presented at the Fifty - second Meeting of the Propulsion and Energetics Panel sponso red by AGARD Cleveland , Ohio , October 23 - 27 , 1978

-

.,

Prepar d for Agard P ropul ion and En rg t ics Cl ve land, hio Octob

•

I'

an 1

23-27, 197

lJ!\STALLE D FL T ER

S PE

by J . J . Adamc zyk, M . E . ational A ronautic

oldst in, and M. J . Hartmann

and Spa e Admini tration

L wi Research C nt r Clev lal ld, \

hio 441 35

S MMARY Rec ntly two flutt I' ana ly

n d v loped at

ASA Lewis Res arch

t of s up r onic wl s tall d flutter of a cascad of two-

C nte r to pI' dict th dim ensional airfoils. In

s hay lJ

Th fir:'st of th s

alJalyzes th e ons t of sup rsonic flutt r

at low 1 v Is of aerodynam ic loading (i. ' ., br.::!kpre sur ), whil examin

th

occ urr nc ' of

loading . Both of th s

uper onic Hutt

analys

' 1'

at mod rat

the

c nd

lcve ls of a e rodynamic

are ba s d on the linearized unsteady inviscid

equations of gas dynamic s to model the flow field s urrowldin g th e ca cade. d tails of the deve lopme nt of th

olution to 'a(;h of th s

lished . Th obj ective of Lhe pres 'nt pa p " l"

mod Is hav b

Th

n pub-

utilize th se analys s in a para-

lS L

metric study to show the eff ects of ca c

nt pap r.

Th

obj ect ive of th

r s ult from th is tudya r

pre ' nt pap ' 1' is to utili z

'il 'ct

analy es in a par'ametric s tudy to s how th

Mach num b r, reduce u fr 'quen y , and backpr of th

nd

Ai;

a pp arod in th op n literatu r

d v lop ' d ana l

lit' flutt r, whil

ur " up r onic nutt r.

ba kpr

ha

of th

f this flutt ' 1' mod

NA A Lewi R ) ea r h

to high ba ckp1' ''

up r so nic £lutt r 'an

lightly bolow th op ratLng lin

whic h docum nt th

thes

upe r oni flutt r can

PI'

R gion II , th

\

ur

to high ba kpr

U l' C 011

of ca cad ' g om try , in1 t sll p ' r 'oni flutt e r .

ov ra1

co rr ' lated again t ' xp ' rim ' nta l qua litativ ob s r-

vatio n t o validat the mo de l MODEL Th


' 1111 ' ''1' , 111l ' ,',\ : 1 ,'Ill 111111':11'1,\

fil l\\'

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Tltl :' :1.':4 l1ll1jl l \llli :1\ ' , ' , 1 I'd II ." I II 11 111 ' :1 l' I It"lll) Illll ' llll pk .. Ih, ' 1I 11:4I ' r

' i ll:1ti

:Isc:ld

Th' \':tlu '

s t:lgg ' 1' :lJl"'1 and

slight s tabili z in g- cffL'ct

f

us d pr vi us l, in ca mput in I!: th ' rt' ' lilts in figur

arid ' llii a l to th s Th 'e' r

l', ti

lIr

err 'ets S

d in fan

' 1'\1

b unda)'.\' t

\\' rk pre, 1 f r t r ' i n:l L

in fig'lll' s 11 and 1:..

and 7 .

I'

(il!:. l).

'h

P l':1ting- at

'ss uri zi l1 ::'," ph no m ' l1 n ha s b

l'

Ul"

ado op ratin with a finjt ba h..'Pr ~ s ur

a

t r s i nal fLLlt tel' th :I I1 :1 :1S ':1d'

l r s i nat (luLL

n th

tion

that a

Thi ' b:1 k

fan is rais d (s

t hi )'h ba kpr

s ults for mod rat

l'

ptibL

angl \\'l1

f

t

impli'

S lI S

~.

Ul'

us

'UT

Thu

it

'S

::Illll

13 t

t

th' th o r

of

ur at I w ba k-

17 \ hi h

\V

1'C

bas 'd

' of f re 'd m b ndin " flutter all b'

Ull

an

n Iud 'd that back-

EXpl'l'.Lm ntal ('vid

' 11

, to

uppo rt th i

'

7

COUPLED FLUTTER

Carta (ref.

) showed the existence of coupled flutter in compressor rotors.

Unlik coupled flutter in fixed wing aircraft where the coupling between the b nding and torsional mod is due to th aerodynamic forces, the coupling in comp r essor rotors is caused by mechanical restraining forces asso ciat ed with part span shrouds and flexible disks. Carta in his analysis assumed the motion of the roto r disk to be

S = Ae

-i (wt- 27m \~ N

!\

pJ

whe re A is the amplitude of the motion, t is time, w is the circular frequency, n is an integer, N is the number of rotor blades , p is the pitch chord ratio, and the peripherial distance around the wheel. U the rotor blades are rigidly fixed

r

to the defo rming disk the plunging and pitching amplitude of a blade section at a g iven radial location are ho = A cos 0

a

o

f

27mA i1T/2 =-e

Np

r espectively, where 0 is the local stagger angle of the blade section s. The nondimensional aerodynamic work per cycle associated with this motion is W = A1T cos ofrm C

L

L

-

27m pN cos



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\

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V'I V'I

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, L-

CHOKE SOU DARY

WEIGHT FLOW

Figu re 1. - Compressor performance and stability map .

.--L-L l ...

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L-J.

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n

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FLO\ CONFIGIJRAf lO Af \00 RAn ro HIGH BAC PRES UR E .

(al

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1

1

1

LEADI G EDGE ACH AVE I

n 2

fRAILI G EDGE 'ACH \ A ES

I

1 ' - -_ _ _ 1_ __

n•I (b) FLO V CO FlGURAfiO AT LO

SAC PRES URES .

Figure 2.

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l'

2

n

ci

fli

J

,,

,, '~ 1

REGIO I

IG 'AL P (){I .4I . l'

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REDUCED rREOUENCY .

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1.0

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0

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u

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-3 71/2 71 3"/2 I r RBLADE PHASl A GI E

271

rlgu re 3. - Wo rk per cyc le for pi tchi ng motIOn 3bou t mld chord at madera e backpressure (Marh '10. 1. 2l.

o

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REDUCED FREOUENCY. k 0. 25

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.SO 1.0 UNSTABLE REGION

.... -'

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u

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0:: .... D-

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-3 71/2 71 371/2 INlERBLADE PHAS E ANGLE

271

Figure 4. - Work per cycle for pitching motion about mldcho rd at moderate backpres ure (Mach no. 1. 41.

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UNSTABLE REG ION

0. 10

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/::;

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Figu re 15. - Work per cycle for pl unging mol IOn at moder ate backpressu re (Mach no. 1. 61.

SOLID ITY

o o

u.J

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1.2 1.4

o ~------------------~~~--

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STABLE REGlO

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-3L -______L -____--..l_ _ _ _ _---'_ _ _ _ _- - ' 71'/2 71' 371'/2 271' INTERBLADE PHAS E ANGLE Figu re 16. - rhe effec t of solidity on Ihe work per cyc le fo r plungi ng motion at mod era te backpressu re.

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TA GGER A Gl . U STABL[ REG IO

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70

o 50 01--- - - - - -- -

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ffi a.. o

TABl [ RlGIO

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I rr/2 rr rr/2 I TERBLAD E PHA EAGLE

I 2rr

rlg llr 17. - Th e eff ect of tagqer angle on th e \', ~ r per cycle at moderate backpre lire

R DUCED fREOUl CY. k 0. 25

o

. 50

(:;

1.00

U TABLE REGIO

\0

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w a..

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TABL[ R GIO

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o

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4 HARMO IC U 1B[R

Figure I S. - \ or p 'r cycle for coup led moti on at moder at e backpressur e I 1ach 11 0. I. ;:1 .

.1

.L..--J

j

J

' ~.

ORlGINAL PAGE 1~ OF POOR QUAJ,ITY

REDUCED FREOUE CY. k

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U STABLE REGIO

...,

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-10 STABLE I REGIO

-20

'--- - - ' -_ _ _1...-_-'-_

o

J. _ _-'--_--'1

t, HAR MO IC UMBr;,

12

Figu re 19. - Work per cycl e for {;oupled flutter at mod rat e backpressure IMach no. I. 41.

REDUCED FRE(JUE CY.

o

o

~

0. 25 . 50 1.00

U STABL REGIO

...,

10

-'

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0:: ..., Cl.

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0::

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-10 STABLE REG ION

-20

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-4 0 4 HAR 10NIC NU ~ \B ER

--..J 12

Figu re 20. - \lo rk per cyc le fo r coupled motion at moder at e ba ckpressur e IMach no. I. 61.

'-1~L ,L 1 1

'

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LJ

, I "

.1

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1

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REDUCED FREOUE, CY, k

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U STABLE REGIa

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-30

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·4 0 4 HARMON IC NUMBER

( 12

Figure 21. - Work per cyc le for coupled motion at low backpressure (Mach no. 1. 21.

I .1.

. 1..... J . . . ___

I:

ORIGINAL PAGE II OF POOR QU A LfTY REDUCED FREOUENCY, k

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0.25

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UNSTABLE REG ION

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Or-.IC Figure 22. - Work oer cycle tor coupled motion at low back press ure IMach no. 1. 41.

RED UCED FREOUE CY, k

o o

0.25

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20

u.J

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HARMONIC NUMBER Figu re 23. - Work per cycle pressure IMach no. 1. 6).

--LJ ,. L

tor coupled motion at low back-

-

-