A note on transonic flow past a thin airfoil oscillating in a wind tunnel

Journal of Sound and Vibration

A NOTE

(1976) 46(2), 195-207

ON TRANSONIC OSCILLATING

FLOW

PAST A THIN

IN A WIND

AIRFOIL

TUNNEL

S. D. SAVKAR General Electric Company, Corporate Research and Development, Schenectady, New York 12301, U.S.A.

(Received 28 July 1975, and in revised form 13 November 1975)

The problem of a thin airfoil oscillating in a transonic flow duct is examined. Asymptotic solutions valid at high frequency are derived which suggest that the degree of interference from the tunnel walls is weaker than would be thought at lirst. More detailed calculations are then used to deduce the flutter characteristics of such airfoils. It is predicted that the airfoil will suffer a torsional mode instability for a range of parameters.

1. INTRODUCTION

The testing of airfoils oscillating in a tunnel at transonic flows presents two significant difficulties. First is the necessity for the very high physical frequencies required to generate even moderately high reduced frequencies. The second is the effect of the wind tunnel wall interference. In this note the problem of wind tunnel interference is addressed, albeit in a linearized sense, with the specific intent to examine details of the interference at moderate and high reduced frequencies. Also included are calculations related to a phenomenon sometimes referred to as choke flutter. Wind tunnel interference has been examined for the subsonic and supersonic cases in the well-known papers of Woolston and Runyan [l] and Miles [2]. When interpreted in the sense of acoustic wave guide modes, the subsonic solution displays the characteristic feature of cutoff for 206/c < (2n + 1) n.v’m, where o, 2b, c, M and n are, respectively, the circular frequency, tunnel height, blade chord, Mach number and wave guide mode order, the airfoil being on the duct centerline (a list of notation is given in Appendix B). In the transonic limit, M --f 1, however, the cutoff frequency for all the modes goes to zero and all disturbances propagate away from the airfoil. The supersonic interference discussed by Miles is in the nature of the leading edge Mach wave reflecting off the tunnel wall and intercepting the airfoil again. That reflected Mach wave completely misses the airfoil if 2(M2 - 1)‘j2b/ c > 1. This condition cannot be satisfied in the transonic limit due to the receding wavelet whose phase velocity is (M - 1). One is therefore led to expect considerable interference from the tunnel walls in the transonic limit. Nor is the problem of interest from the point of view of tunnel interference alone. The problem also represents an airfoil in a zero stagger cascade whose members oscillate 180”out of phase with each other. Some of the features of such unstaggered cascades a’lsocarry over to staggered cascades. Indeed the growing trend in fan design is for the use of transonic and even supersonic relative tip speeds. Such fans of course operate at a very low incidence angle, under which restriction (of low incidence) at subsonic tip Mach numbers the damping forces in flexure and torsion are generally positive. However, the high tip Mach number operation has uncovered a new mechanism for flutter which comes into play when the blade tip relative 195

196

S. D. SAVKAR

speed exceeds Mach 1. This latter mechanism appears to be the same as the unstable oscillations of an isolated, thin airfoil in a supersonic stream first analyzed by Miles [3] using linearized perturbation methods. Miles demonstrated that these oscillations are unstable only in the torsional mode. Such a behavior, predicted by linearized analysis, has been experimentally confirmed [4-61. To be sure, the cascade geometric parameters and phase relations between blades alter the essential details of the stability boundary; in particular, Kurosaka [7] found that they tend to increase the region of instability. Moreover, in both the Miles study and the analysis of Kurosaka, the boundaries of the region of instability broaden as the Mach number is decreased. This latter issue, that of choke flutter or flutter in the limit as M -~. 1, can be motivated from two aspects. First, instances of choke flutter problems have been encountered in very low pressure ratio operations along a given speed line of a compressor. Aerodynamically, the flow through the compressor is choked or transonic. While the precise mechanism is not fully understood, one can plausibly argue the problem of choke flutter as the transonic limit of the supersonic case. Such an argument of course implies a connection between the two modes of flutter. Thus, heuristically extrapolating the above subsonic and supersonic results, one expects a reversal in the sign of the damping coefficient in the transonic region as one proceeds from the high subsonic to the low supersonic Mach number. Moreover, in this argument. the broadening of the supersonic stability boundary, as the Mach number approaches unity, suggests that the origin of the supersonic instability is in the transonic range. At the outset we note that the degree of interference found from the linearized unsteady transonic formulation is somewhat less than initially expected, this is not inconsistent with some experimental observations by Orlik-Riickemann and Olsson [8]. On the other hand severe torsional flutter is predicted. These predictions should be viewed in a qualitative sense since the limits of accuracy of the basic formulation are by no means clear. The solutions themselves, however, do show a consistency with proper limits of other known solutions, such as that of Miles [2], and appear to represent the physical phenomena plausibly. 2. FORMULATION Unlike steady transonic flow, unsteady, inviscid transonic flow can be linearized. In the free field-that is, in the case of an isolated airfoil-the linearization is valid provided that the frequency of the oscillation is high enough. This has been demonstrated most carefully by Landahl [9] and also by Landahl, Mollo-Christensen and Ashley [lo]. The frequency limitation is expressed by the inequality k/lM-

11% 1,

k = m/a,,

(1)

where k is the reduced frequency defined with respect to the blade chord and undisturbed speed of sound. It is particularly important to note that the transonic solution is not the limit of either the supersonic or subsonic solution as M + 1 (see reference [9]). This point is also brought out in Appendix A where the implications of Miles’ solution [2] are examined. Consider an airfoil, of chord c, undergoing harmonic oscillations, of frequency w, in a tunnel, of half height b (see Figure 1). Let the transonic flow Mach number be M and the undisturbed speed of sound a,,, assumed to be a constant. Upon defining the perturbation velocity potential 4, V = UcV4eiwr, U = Mao, for the perturbation velocity V and employing the normalized variables b/c = b, q = y/c and [ = x/c, one obtains the following two-dimensional linearized equation, valid at moderate and high frequencies [9] : c#I,,- (2iwMc/a,)

4c + (o’c’/ai)$

= 0.

(2)

197

OSCILLATING AIRFOIL IN TRANSONIC FLOW

//////////////////////////////N/////////////

Figure 1. Geometry of the airfoil in a flow duct.

At the airfoil surface dy/dt = aUeimt, a < 1. Note that a is the generalized complex upwash. In non-dimensional terms one may then write &=-a(r),

at

q=O*,

&=O,

at

v =+/I.

(3)

Since in the transonic flow the influence of the downstream disturbances can be largely neglected, it is convenient to carry out the calculations in terms of the following Laplace transforms with respect to c : 4 = ie-‘< (b(<, q) d5,

5 = ie-‘r a(l) dc. 0

0

Thus one obtains for the governing equation f$,, - (2iMks - kZ) I$ = 0, where k = m/a, devised as

(4)

is the reduced frequency. A solution to the above system can be readily $+ = [(a coth nfi)/n] [cash 2~ - tanh @Isinh Art], $- = -[(a coth @I)/A][cash 2~ + tanh 1/3sinh 1~1,

q2

o+,

‘I <

o-,

(5)

in which A2= 2iMk.s - k2. To an additive function independent of rl one may express the perturbation pressure, p, as d = -2(pM2 a$/2) (s + ik/M) 6.

(6)

Hence the pressure differential across the airfoil is dp/(pU2/2) = (p- -p+)/(pU2/2)

= [(4Zcoth@?)/n](s+

ik/M).

(7)

Except for the inversion of the above results, this completes the formal solution of the problem posed. The solution is exact in so far as the governing equation is concerned. In what follows, a preliminary discussion of the isolated airfoil is presented, followed by a discussion of the nature of the tunnel interference and finally calculations of flutter.

3. ASYMPTOTIC LIMIT OF THE FREE FIELD SOLUTION For subsequent reference, the free field calculations result simply, in the limit j? -+ +a~, in the expression 4 = 5 e+/A. Inversion of expression (8) (details of the straightforward

(8) contour integration process are

S.

198 omitted)

D. SAVKAR

yields the results for the potential,

and for the pressure [(ik/M)cc(c - r) + ?<([ - S’)]e-ikM~2’2e-i~e’ZMd5, (9b) d-F These results are in agreement with those of Nelson and Berman [I l] and of Rott [12]. Again, within the limits of the formulation, the results are exact. Nelson and Berman obtained this result by the suppression of the receding wave in the Miles supersonic solution [3]. If now the substitutions s’= Z2 and p(i - 0 = a,([ - 5) are made, the expression (9b) for the pressure becomes

[(ik/M)

a((’

_

22)

+ p((

e-ikM~*/2Z2-~kZ2/2M dZ.

_ z”)]

This form is suitable for the application of the method of stationary phase [13] in the limit as k/M becomes very large. In that limit the asymptotic result for the pressure at the surface of the airfoil is cc(O) eMinj4~~e-iW2M M

+

7

e-in12 +

0

(10)

In essence, expression (10) comprises the piston theory formula, the first term, together with a correction for the bow wave which decays as i-r,“. The fact that here the piston theory contribution has been recovered in the limit of high frequency is not surprising; indeed the validity of that term even in subsonic flows has been discussed by Landahl et al. [IO]. Besides the first two terms, note the term 0(1/k), which does not arise in the case of the supersonic ducted airfoil (see Appendix A). This points up the fact that the supersonic solution will not proceed in the limit, in a simple way, to the transonic solution. However, as with the supersonic airfoil, the transonic solution, in the high frequency limit, does not exhibit any flutter instability.

4. ASYMPTOTIC

BEHAVIOR

OF THE

DUCTED

TRANSONIC

AIRFOIL

One can now return to the ducted airfoil and examine the nature of tunnel interference. From the results in section 2, the solution for the Laplace transformed pressure at the airfoil surface can be expressed as -+ P ~ = -[(olcoth PU2

j.j)/j.] (S + ik/M).

In order to separate the contribution due to the isolated airfoil from that of the tunnel ference, cothi$ may be expanded in a series of exponentials:

(lla) inter-

199

OSCILLATING AIRFOIL IN TRANSONIC FLOW

The first curly bracketed term is recognized at once as the isolated airfoil contribution, while the series sums up the influences of the images in the tunnel walls. Formally inverting the latter term by term yields the solution (12) where A represents the inversion of a(s + i/c/M). One can now repeat the earlier steps of the stationary phase calculations as in section 2, setting r = Z2, and noting, however, that the point of stationary phase falls outside the blade chord (or the interval of integration for lift, 0 to 1) unless /3 is less than 2-l. This process yields solutions, for /? 2 l/2, a(o)

,-@)

e-iWZM+n/4)

l%WWM

+

0..

4’ ei(k/2MNC+4n*

M*t3*/2)

e-in/4

27ckM(4n2 M2 b2 - c2)

(13)

7

and, for 1 > 28 > l/(N + l), i(k12MK+4n~M2

#*IO

e-in/4

2nkM(4n2 M2 Ip’- c2) e2ik17n

,

(14)

a(x) being the unit step function, =l for x b 0’ and =0 for x < O-. Consider first the solution for /? r 2-l : i.e., in the limit of a large ratio of tunnel height to airfoil chord, 2b/c. The major contributions then consist of those of the isolated airfoil and of the leading edge disturbances from the image system in equation (13). Notice that the /

Figure 2. Reflection of the advancing wavelets.

200

S. D. SAVKAR

influence of the reflected advancing wave weakens very rapidly and, more interestingly, that near the leading edge the solution is essentially the same as that for the isolated airfoil. Indeed. the point at which the reflected wave has its maximum influence is some distance behind the leading edge. These observations are depicted in Figure 2. from which one can note that the reflected wave corresponds to a disturbance wavelet ahich has progressed and weakened over a considerable time span. Not surprisingly then. no disturbance from the adjacent image5 reaches the leading edge. This behavior is analogous to that displayed by the Miles supersonic solution, which is similarly examined in Appendix A. For narrower spacing, such that p < 2-i, additional terms, corresponding to the geometric acoustic contributions of the image system, must be taken into account. These are included in equation (14). While the Mach cone is essentially a plane centered at the leading edge, the geometric acoustics contribution, or the piston theory term, has its predominant influence confined to a cone of 45”, since the rates of transverse propagation and downstream convection are essentially the same. Consequently, reflections of these disturbances, or the influence of the image blades, intercept the airfoil only when p < 2-l, or c > 26. However, within the geometric acoustics approximation that disturbance is undiminished in strength. Practically speaking, such narrow spacings, even in compressor applications, are not of interest. In staggered cascades, however, the influence of the geometric acoustics contribution can be significant. While the geometric acoustics interpretation is restricted to high frequencies, the singular behavior at the leading edge is an inherent feature of the solution. This may be illustrated by expanding expression (11 a) near the leading edge. Now, as i -+ 0, s + x ; thus, if c1is represented by a truncated power series,

the pressure at the leading edge of the airfoil, in the transform b’lplJ2 which on inversion

-+ (-l/dZEJ@

[a,/z/s

plane, becomes,

in the limit,

+ a,/s312 + . . .I,

tends to

(15) Such subsonic behavior can be traced to the elimination of the receding wave in the solution for the supersonic airfoil. Hence note that the limit s + ~0 of Miles’ solution [2] in the which is in essence the geometric acoustic transform plane is ~5’/pU~ + -a&‘~s, solution applied at the leading edge for large M. The nature of the transonic solution further differs from that of the supersonic solution by the presence of the phase lag 7r/4. Nevertheless, the leading edge solution shows no influence of the tunnel walls. In other words, the solution at high frequencies, and at the leading edge for arbitrary frequencies, is dominated by the isolated airfoil behavior. This conclusion is in fact borne out experimentally : note the observations of Orlik-Riickemann and Olsson [8] as quoted by Landahl [9]. This then suggests a simple way to treat the influence of a staggered cascade in the limit of geometric acoustics. With reference to Figure 3, solution (13) may be rewritten for r = 0+ of the subject airfoil as

mQtLJWLe

i(k/2M)(i,,+4n2

z:

“=I

=(4(4n2

M2 8’li

”) e-in/4

M2 p2 - p.‘)

’

OSCILLATING AIRFOIL IN TRANSONIC FLOW

201

Figure 3. Influence of stagger on the mutual interference

where [, = [ - [,, 5. being the axial shift of the nth cascade member influencing the upper surface. The test blade is n = 0, along which surface the co-ordinate 5 is measured. In the region (c - cl) < 0, the pressure along the upper surface approximates to that of the isolated airfoil. One may similarly modify equation (14), the expression including the influence of the geometric acoustic contribution, and approximate the pressure along the lower surface of the blade. 5. SOLUTION FOR ARBITRARY FREQUENCY (k/]M - 119 1) Consider now, in the sense of the formulation (k/ IM - 11$ l), the solution (7) in the case of pure flexure, for which 2 = g/s so that @/@W/2)

= 4Z[(s + ik/M)/si] coth (A#?),

(16)

or, upon expanding coth(x), (17) There are no branch points and all the contributions come from simple poles. Hence summing up the residues gives the series {(n’ .rr2/k2 /j2 +

AP/(P~~/~) = -4%

l)el(n2n'-k2 82)c12kML@ _ 2)

n2n2-k2fi2

I.

(18)

The series has a radius of convergence of unity and is oscillatory. The trouble may be also

S. D.

202 illustrated by using the expansion solution in terms of theta functions

in terms

SAVKAR

of the cxponentials.

of the third

which formal]!

yields a

kind (141:

where <* = k
while oscillatory, has a Cesaro sum for .X= n : S(n) r --J (Cesaro limit). Hence term by term integration can be used and indeed yields an absolutely convergent series [15] for the lift and moment. Thus one can define the lift amplitude as L = ,I‘,”Apdx and obtain, upon term by term integration. f,/(pu/‘c/2)

= -4iE

(l/koM){l

+ iM/k -. (iM/k)ek”““}

c

kP *

{(n’ =2/k’ p’ +

1)

+

2kMp2(ei(n2 n2-k2@Ji2kMBZ_

+Ti II;1

1)

_

2i(n2

,$

_ k2 p2))

i(n’ TI’- k2 p’)’

Consider now the stability of the above system in pure fiexure: i.e., mji + ky = Leiwf, giving -w2mj + lj = RL -t i9L, or o z w,{ 1 - ( l/2)( RL/mjo$

- (i/2)(XL/mjwf)}.

Then, for stability, exp (iof) z exp (io,[ 1 - ( I /2) (RL/rn~%4)] t + ( 1/2)(4Lt/m~h3}, one must have 9L < 0. For the above defined system, then, numerical computations show that 9;L < 0 for all k, M and jj’of interest. Consider now the case of pure torsion: then, with 0 as the pitch angle, ctU = lJB + c(< - &,)d. Upon letting 0 = Ee’“‘, and noting that co is the normalized hinge point, one has e iwf = E{ 1 + (ik/M)(<

63 With v = ik/M,

the Laplace transformed

- lo)} e’“‘.

results are

1 = L%{(l/s)+ V[(l/S2) - (;,/s,]}) Ajj/(pU2/2)

= 4E[(l/s)(l

which readily can be inverted Ap/(pu2/2)

- vi,,) -I- v/s”] [(coth@)/i](s

+ ik/M),

to yield

= -4ic([( l/kpM){v(i + (kfi/M)

- Co) - (v2 - v&/2 + l/2) ec”” + (v’ + 1)) +

5 ((v&~/ip; Ill,

- (0$&M

+

- v&,pZ + p;)e’“i’/”

- 2v([ - Co) -

2)I/pRl,

where &=n2n2/k2P2+

I.

p(n = (nfrc2/k2 /I’ - I ) k2 p2,

CT= 2kMjY2.

OSCILLATING

AIRFOIL IN TRANSONIC

203

FLOW

It is now convenient to define the following coefficients : B=v2+1=i2k2/M2+1=l-k2/M2,

A = v2 - v[,/2 + l/2,

E = vA/ip; - vL P; + cl;,

F=p(:cn$p.+2=2[(n2~2+k2/?2)/(n2~2-k2~2)+

11.

The lift and moment can be defined in terms of the pressure as U(P~’ c/2) = j W(p~2,2)]

d5,

0

=/Cl -

~XP~~C~/~)

io)kfplW2/2)1dL

0

Thus one finds L/(pU2c/2) = -4iZ

[

(l/kjM){v(+ - To) - 2Av(e1’2’- 1) + B) $ (e ‘r,‘u- 1)/K, - 2v(3 -

+ (kalM)~~IW~li&J

lo)/&

-

F/p,}

1,

T/(pU’ c2/2) = -4iE[(l/kfiM){v(+ - Co+ Cg)- 2Av[e”‘” - (2v + lo) (e112”- l)] + B(+ - co)} + (k/?/M) $ {(Ea/i~;)[eia~‘a - (a/& + Co)(eip~“-’ - I)]//.G It=1 -

2v(+

-

Co +

5@lc(,

-

F(3

-

To>/~n)l.

For pure torsional motion one then has 10 + kf9 = Teiwf, or (-10~ + k)e = RT+ ST, from which w x w,[ 1 - _%T/28Zwz- ifT/28Zo: 1. For stability, then, since eta’ x exp [iw.(l - YT/28Zw;f) r + (9T/2&:)

t],

Figure 4. Torsional instability (ordinate imaginary component of moment, co is the hinge point). (a) Function of frequency, M = 1, /?= 1; (b) function of tunnel wall, k = 2.1, M = 1; (c) function of Mach number,k=2+1,B=l.

204

S. D.

SAVKH

must have $7‘~. 0. Whereas the results in pure Aexure do not inhibit instability, as in the one supersonic case, flutter is exhibited in torsional vibrations (see the sample calculations in Figure 4). Hence the governing parameter appears to be k/j’. Recalling the asymptotic result (14), one notices that the oscillations are wholly unbounded for X-p < 2. This enhanced instability appears to be caused by the reflected acoustic disturbances.

6. DISCUSSION The problem of transonic tunnel interference is not a simple one. What has been argued in this paper is that, at least in the limit of near zero thickness airfoil, one can examine this question using the linearized theory. In that limit, the interference, although significant, does not appear to be quite as severe as one might be led to believe at first from the limits of the supersonic or subsonic solutions. Indeed, in the limit of high frequency, the behavior, to the lowest significant orders, is remarkably supersonic-like. This is further borne out by the transonic flutter calculations which strongly suggest that, at least in the qualitative sense, the calculations of Kurosaka [7] will yield plausible representations for near sonic Mach numbers. There are two obvious weaknesses in these calculations. Neither includes the effects of the airfoil shape or thickness. Retaining the assumption of a shockless but axially varying flow does not alter the essential conclusions of this paper. One may illustrate this rather easily by using the formulation of Ruo et al. [16], based on the local axially varying Mach number. Indeed, their governing equation, obtained by using a “modified Prandtl-Glauert transformation”, essentially reduces to ours (see equations (3.2)-(3.4) of reference [16]). So the crux of the unsteady transonic flow issue is not the effect of axially varying Mach numbers (so long as one argues that the Kutta condition does not apply even if the Mach number is very slightly subsonic and the receding wave can be ignored). The real issue appears to be the influence of the finite (but weak) shock and three-dimensional effects. One notes that the weak shock strength is proportional to (M’ - 1). Then, since IM - 1 j is O($,,), where, in kandahl’s notation, @rXis the base flow axial velocity perturbation, if one retains the variable axial base flow, one must account for the finite shocks. In the interim, it can be noted that the dependence of flutter on the reduced frequency k and so!idity p, as predicted by the simple linearized theory, has been borne out experimentally.

7. CONCLUSIONS It has been shown here, in the case of transonic flow past a thin airfoil oscillating at high frequency in a tunnel, that the degree of interference due to the flow duct is relatively less than that usually supposed. The behavior in the limit of high frequency shows some distinct similarities and some marked dissimilarities with that of the supersonic ducted airfoil. On the other hand, at lower frequencies, unstable torsional flutter is predicted. The severity of the flutter is found to increase as the flow duct was narrowed down. Indeed the parameter governing the flutter oscillations is found to be k/l, the product of the reduced frequency and non-dimensional tunnel half-height.

REFERENCES 1. D. S. W~~LSTON and H. L. RUNVAN 1955 Jourrzal of Aerospace Sciences 22, 41-50. Some considerations on air forces on a wing oscillating between two walls for subsonic compressible flow. 2. J. W. MILES 1956 Journal of Aerospace Sciences 23, 671-678. The compressible flow past an oscillating airfoil in a wind tunnel.

OSCILLATING

AIRFOIL IN TRANSONIC FLOW

205

3. J. W. MILES 1959 The Potential Theory of Unsteady Supersonic Flow. Cambridge University Press. (Also see Journal of Aerospace Sciences, July 1956.) 4. Y. NAKAMURAand Y. TANABE1966 Journal of Aircraft 3,405-410. Some contributions on single degree of freedom flutter in two-dimensional low subsonic flow. 5. C. SCRUTON,L. WOODGATE,K. C. LAPWORTH and J. MAYBURRY1962 Aeronautical Research Council Reports and Memoranda No. 3234. Measurements of pitching-moment derivatives for airfoils oscillating in two dimensional supersonic flow. 6. J. B. BRATT,W. G. RAYNER and J. E. G. TOWNSEND 1962 Aeronautical Research Council Report No. 3257. Measurements of the direct pitching-moment derivatives for two-dimensional flow at subsonic and supersonic speeds and for a wing of aspect ratio 4 at subsonic speeds. 7. M. KUROSAKA1974 Transactions of the American Society of Mechanical Engineers Journal of Engineering for Power 96, 13-31. On the unsteady supersonic cascade with a subsonic leading edge-an exact first order theory-Parts 1 and 2. 8. K. ORLIK-R~~CKEMANN and C. 0. OLSSON1956 Aeronautical Research Institute of Sweden (FAA), Report 62. A method for the determination of the damping in pitch of semi-span models in highspeed wind tunnels. 9. M. T. LANDAHL1961 Unsteady Transonic Flow. New York: Pergamon Press. 10. M. T. LANDAHL,E. L. MOLLO-CHRISTENSEN and H. ASHLEY1955 United States Airforce Ofice of Scientific Research TR No. 55-l 3. Parametric studies of viscous and nonviscous unsteady flows. 11. H. C. NELsoN and J. H. BERMAN1953 NACA Report 1128. Calculations on the forces and moments for an oscillating wing-aileron combination in two-dimensional potential flow at sonic speed. 12. N. Rorr 1949 Journal of Aerospace Sciences 16,380-381. Oscillating airfoils at Mach number 1. 13. A. ERDBLYI 1956 Asymptotic Expansions. New York: Dover Publications. See pp. 51-56. 14. M. ABRAMOVITZ and I. STEGUN 1964 Handbook of Mathematical Functions. Washington, D.C.: National Bureau of Standards (NBS AMS 55). 15. R. C. BUCK 1956 Advanced Calculus. New York: McGraw-Hill Book Company, Inc. 16. S. Y. Ruo, E. C. YATES and J. G. THEISEN 1974 Journal of Aircraft 11, 601-608. Calculation of unsteady transonic aerodynamics for oscillating wings with thickness.

APPENDIX

A: OSCILLATIONS

OF AN AIRFOIL

IN A SUPERSONIC

FLOW DUCT

In support of the conclusions of the main text, the asymptotic behavior of an airfoil oscillating in a supersonic flow duct [2] is examined here : P+/PU’ = -WC

+ WMI

j g(L 0) a([ - 0 cK, 0

where “!I E, Jo[K(12 - n2 A2)1/2/M] ll({ - nA),

g(t) = (e-*“‘/m Eo =

1,

En= 2

for

n#O,

K= k/(M2 - l),

A = (26/c) (M2 - 1)“2.

Expressing J,(X) in its integral form and reversing the order of integration, one finds $M2

- 1)1’2p+/pU2 = -[a/a3 + ik/M]n:o

7

j a(( - ~)eiKh(C)d~de,

-n/2 0

with h(g) = II~-‘(<~ - n2 A2)‘/2 sin 8 - 5. For the inner integral, the point of stationary phase for n # 0 corresponds to (o = MnA/2/M2 - sin28. For n = 0, the isolated airfoil, there is no

With thedefinition stationary point, and the argument of the integral is a(C - 5) e iK~M-‘sine-l)C.

S. D. SAVKAK

206 -D z ajag + ik/M,

one obtains n/l

n(M2

-

1)“2_--

P+

=

D

PU2

ni2

*

-Lx([)dG

-‘_n,21

[( _r;,r c K(1 - M-‘sin@

+2$

Z(0)eiK(M~*~inB~I)idH

K(M-‘sine-

I)

I

pI~~~(~~~~i~~,)l~1/2~ “=I -x,2

x c(([ - &J eiKhCeo)e-‘“l” ll( to -- nA) + i e iK[M-I

Q(i - nA)NO) ’ K[([ sin O/M)/([’ - n2A2)‘j2 - l]

do

((Z--n2

1,

A2)*12

sin 8-U

X

with 6(&J = +zAM-~(M~ - sin20). The first term in the series only contributes for nA < to < 1. This is in fact the contribution of the piston theory, or of the geometric acoustic disturbance of the images. For <,, > 1 the point of stationary phase falls outside the blade chord. The first curly bracketed quantities are the isolated airfoil terms, the first and second of which represent, respectively, the piston theory and the bow wave contributions. This can be made explicit by applying stationary phase argument to the second integral to obtain rkCl(M+O-in/4

_

e-ikCi(M-l)+in;41

, +

O( 1/[kc]““) +

..

Thus the first term is immediately recognized to be the piston theory contribution and the second the influence of the advancing (phase velocity M + 1) and receding (phase velocity M - 1) wavelets generated by the leading edge. Note also that the solution for the pressure does not reduce to the transonic solution in the limit M 4 1; hence note the absence of a term 0( l/k). The same type of interpretation results from the second set of terms in the series corresponding to the reflected wavelets : icr(O)Q(c - nA) K

e-iKC+i(K/Mh,Z*-n2

[[

-

2rtM i Kdc2

AZ-in/4

-n2A2 e-iKi-i(KiM)\

II2

Md<2-n2A2~
A2+in,l

M(c2 - n2 A2)“2] + [[ + M(cz - n2 AZ)““]

1

However, it is immediately obvious in view of the step function ‘I([ - nA) that the reflections will influence the pressure on the airfoil only if (at least for n = 1) 1 - (26/c)dm > 0, or c > 2dmb, or when the reflected Mach cone intercepts the blade. The remaining term corresponds to the reflections of the disturbance of piston theory term; however, the cone of influence of this term is even narrower than the Mach cone, being governed by the requirement that x - &,, the argument of a(x), always be such that 0 G x < 1, or that, since then 1 - &, > 0,l > MA/2/M2 - sin* 0, for n = 1. Since the remaining stationarypointsare +rc/2 this requires that (MBb/c)/dm) < 1, or c > 6M, or c/U > 6/a,. This requirement supersedes the influence of step function, I(&, - nA), since lo - nA = nA[(M - dM2 - sin28)/dM2 - sin’01 is generally > 0. Thus the condition is that the time taken for the acoustic disturbance to be convected downstream, c/U, be greater than the time for the disturbance to propagate normal to the plate, strike the tunnel wall and impinge on the plate again, 26/a,.

OSCILLATING

AIRFOIL IN TRANSONIC FLOW

207

These results are generally common, qualitatively, to both the supersonic as well as transonic cases. The transonic case, which differs in that the reflected shock always strikes the plate, however weakly, does not result as a simple limit of the supersonic case, the influence of the receding wave causing increasingly rapid oscillations in the solution as M + 1. Moreover, the influence cone for the geometric acoustics disturbance is now n/4. APPENDIX a0 b c Z k L M p s T U

undisturbed speed of sound tunnel half height airfoil chord moment of inertia reduced frequency, cm/a0 lift undisturbed flow Mach number pressure Laplace transform parameter moment undisturbed flow speed

B : NOMENCLATURE a

generalized complex (pitch and flexure) upwash jl tunnel half height normalized by blade chord, b/c q normalized transverse co-ordinate, y/c [ normalized axial co-ordinate, x/c Co hinge point in pitching oscillations 0 instantaneous pitch angle 0 circular frequency of oscillation p density of air 4 velocity potential