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Technical Notes

Transonic Unsteady Potential Flow:

Scaling Analysis of Linear and

Nonlinear Dynamics

Earl H. Dowell

Duke University, Durham, North Carolina 27708

DOI: 10.2514/1.40426

I. Introduction

B Y ORDER-OF-MAGNITUDE estimates, called scalinganalysis [14], several interesting questions about unsteadytransonic potential flow may be addressed. Similarity laws basedupon dimensional analysis have been developed previously forpotential-flow models, largely within the context of steady flows[5,6], but also see the interesting discussion of Williams [7,8] onsimilarity laws in unsteady flows. Such similarity laws are veryvaluable and tellus how one may express the results ofanalysisin themost compact dimensionless form. But they do not address thequestion posed here, which is how large must the airfoil motion befor nonlinear aerodynamic affects to be important, as predicted bypotential-flowmodels? To answerthis question,scaling analysis maybe used.

Both similarity laws and scaling analysis seek to extract as muchinformation as possible by analysis without resorting to numericalcomputations. As such, they are useful complements to compu-tations and experiments. As a particular example and a primarymotivation for the present work, the limit cycle oscillations of the F-

16 aircraft are considered.Bejan [1] calls the approach employed herescaling analysis and

he has used this approach very effectively for a wide range of topicsin thermodynamics and heat transfer. The present author hasindependently developed a similar approach for topics in aero-elasticity [2,4].

II. Scaling Analysis

It is well established that for small perturbations, e.g., thin air-foils and/or airfoils undergoing small motions, the governing fieldequation for the aerodynamicvelocity potential in transonic unsteadypotential nonlinearflowis as follows:x, y,and z are the conventionalCartesian coordinates in the chordwise, spanwise, and transversedirections,

xx1 M2

L yy zz 1=a212U1xt tt 0 (1)

M2L M21

1 1

xU1

(2)

Here, M1 is the freestream Mach number and ML is the local Machnumber that varies with x, y, z, and t through its dependence on x.

Subscripts x, y, and t on denote derivatives with respect to one ofthe spatial variables x, y, and z or time t.

To make order-of-magnitude estimates we consider oscillatoryflows of frequency ! with a wing of chord c and span .

The velocity potential is related to airfoil shape and motionthrough the small-perturbation boundary condition: that is,

@

@z

z0

w (3)

where w is the downwash. Here, the primary interest is with airfoilmotion, so for a typical oscillatory angle of attack, the order-of-magnitude estimate ofw may be taken as follows:

wU1 (4)

Equation (4) will be manipulated as thoughit werean equation,butit is in fact only (but crucially) an order-of-magnitude estimate. Note

that there is another term of the order of the reduced frequency thatformally appears in Eq. (4) that has been neglected here.Subsequently, it will be shown that reduced frequencies of the orderof unity or less are of interest, and hence this term may be omittedwhen making order-of-magnitude estimates. Also note that weassume that the motion of the airfoil (say, dynamic angle of attack)is much larger than the thickness ratio of the airfoil, , which is truefor the results presented here. It would be of interest to consider thecase in which and are of the same order of magnitude, but this isleft to future work.

III. Linear Aerodynamics

To betterfix ideas as a foundation for the discussion of nonlineardynamics and aerodynamics to follow, first consider the simplerlinear theory. For the linear theory, the local Mach number ML issimply set equal to the freestream Mach numberM1, which in turn,for our purposes, will be set to 1.

Consider now the order of magnitude of the several terms inEq. (1). They are as follows:

xx =c2 (5)

yy =2 (6)

zz =2 (7)

(where is a length scale yet to be estimated)

U1a21

xt U1a21

!

c(8)

Note that in estimating the order of magnitude of the various terms,numerical factors of order 1, such as ( 1), may be ignored:

1

a21tt

!2

a21(9)

By taking the ratio of Eqs. (69) to Eq. (5) one may estimate therelative size of the several terms. The results are as follows.

Ratio of Eq. (6) to Eq. (5):

c=2 (10)

Ratio of Eq. (7) to Eq. ,(5):

c=2 (11)

Received 26 August 2008; revision received 1 October 2009; accepted forpublication 31 January 2010. Copyright 2010 by 2010 by Earl Dowell.Published by the American Institute of Aeronautics and Astronautics, Inc.,with permission. Copies of this paper may be made for personal or internaluse, on conditionthat thecopier paythe $10.00per-copyfee to the Copyright

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0001-1452/10 and $10.00 in correspondence with the CCC.Professor, Departmentof MechanicalEngineering and Materials Science;

dowell@ee.duke.edu. Honorary Fellow AIAA.

AIAA JOURNALVol. 48, No. 5, May 2010

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Ratio of Eq. (8) to Eq. (5):

M1

!c

a1

(12)

Ratio of Eq. (9) to Eq. (5):

!c

a1

2

(13)

Again recall that forour purposes,M1 istakentobeoforder1andset

to 1 since our interest is in transonic flow, but the cases M1 1 orM1 1 could be considered by similar methods. In the generalcase, Eqs. (1013) are all of unit order of magnitude. When one ormore of these ratios are substantially less than unity, certainsimplifications may be made, as will be discussed next.

Sowhat may one conclude from Eqs. (1013)? A number of well-known and perhaps not-so-well-known results may be obtained.

First, if

!c

a1 1 (14)

then the flow is significantly unsteady and the terms involving timederivatives (9) must be retained, which is a reassuring but hardlysurprising result. If !c=a1 1, then unsteadiness may beneglected or at least is of modest importance.

Second, if

c= 1 (15)

then the flow is two-dimensional and the yy term (6) may beneglected, which is a well-known result.

Third, if

c= 1 (16)

then the xx term may be neglected and if

!

a1 1 (17)

then

!ca1

!a1

c

1 (18)

and the time derivative terms (8) and (9) may be neglected, whichreduces Eq. (1) to

yy zz 0 (19)

This is the well-known slender-body, or slender-wing, theory seenfrom a new perspective based upon scaling analysis.

Finally, consider

!c

a1 1

Then Eq. (1) reduces to the following:

zz 1

a21tt 0

which is the well-known linear piston theory result.Now what about Eq. (11)? This allows one to estimate , the scale

of the flow in the z direction. To see this most clearly, return toEqs. (59) and now construct the ratios of Eqs. (5), (6), (8),and(9) toEq. (7), as follows.

Ratio of Eq. (5) to Eq. (7):

=c2 (20)

Ratio of Eq. (6) to Eq. (7):

=2 (21)

Ratio of Eq. (8) to Eq. (7):

M1

!

a1

=c (22)

Ratio of Eq. (9) to Eq. (7):

!

a1

2

(23)

The largest of these ratiosdetermines ,forthe zz termmust balancethe largest of the other terms. Thus, for example, assuming!c=a1 1, then if

c= 1 (24)

it follows that , and if

c= 1 (25)

it follows that c.On the other hand, if

c= 1 and

!c

a1

1 (26)

then it follows that

a1=!

IV. Nonlinear Dynamics and Aerodynamics

Itis the estimateof that is a keyto considering thenonlinearcase,with ML, the local Mach number, now given by Eq. (2).

Consider now the nonlinear term in Eq. (1) and again set thefreestream Mach number, M1 1, for our purposes. An order-of-magnitude estimate for this nonlinear term is as follows: Note againthat 1 is of order 1 for the purposes of scaling analysis:

xxx

U1 xx

xa1

2

a1c3

(27)

Now compare this term to zz; recallfrom Eq. (6) thatzz =2.

IfEq.(27)isofthesameorderasEq.(6) or greater, thenonlinear term

must be retained; conversely, if it is much smaller, it may be safelyneglected. Thus, construct the ratio of Eq. (27) to Eq. (6):

2

a1c3

=2

2

a1c3

(28)

Thus, the critical order of magnitude of is determined as follows:

cr a1 c3=2 (29)

For < cr, linear theory is sufficient, and if cr, a nonlineartheory is required. But how do we determine ?

Again, we require zz to balance the other linear terms. Inparticular, we ask that it balance the remaining dominantlinear term.So consider each remaining linear term in turn:

zz yy )

2 =2 ) (30)

or

zz U1a21

xt )

2

!

a1c)

a1c

!

r(31)

or

zz 11

a21tt )

2

!2

a21)

a1!

(32)

Now Eq. (30) will be appropriate to the slender-body limit (31)tothetwo-dimensional flow limit at low frequencies, and Eq. (32) will beappropriate to two-dimensional flows at high frequencies.

Using Eqs. (3032) in turn in Eq. (29), one can deduce thefollowing.

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From Eqs. (29) and (30),

cr a1c3=2 a1cc= (33)

which tells us that cr is far beyond the range of small-perturbationtheory itself for small-aspect-ratio wings 1c, or slender-wingtheory. Recall that for small-perturbation theory to be applicable,one requires that c= 1. Hence, the slender-body limit of small-perturbation potentialflow cannot model aerodynamicnonlinearitiesaccurately.

From Eqs. (29) and (31),

cr a1cc2

a1c=! a1c

!c

a1

(34)

so that only for small !c=a1 will cr be much less than a1c andwithin the range of small-perturbation theory, i.e., cr 1c.

From Eqs. (29) and (32),

cr a1c3=2 a1c

3!2

a21

or

cr 1

c!c1

2

(35)

and thus for such a cr, small-perturbation theory will again beviolated, since Eq. (32) assumes !c=a1 1, and hence fromEq. (35),

cr 1c

Thus, of the three cases, Eqs. (3032), only the case of Eq. (31)(i.e., !c=a1 of order one or less) is of remaining interest, asanticipated in the earlier discussion.

V. Estimate of Critical Magnitude of Motion forPotential-Flow Nonlinearities to be Important

For the case of Eq. (31), what is the critical size of the motion orangle of attackcr corresponding to cr?From Eqs. (3) and (4),

@

@z

z0

w; wU1

or

U1 (36)

From Eqs. (31), (34), and (36),

cr cr

U1

a1c

U1

!c=a1

a1c=!p

!c

a1

3=2

(37)

Butrecallthat!c=a1 1 forEqs.(31), (34),and(37)tobevalidandthat M1 U1=a1 is again set to 1. Note that if ! 0, thennonlinearities are important for anyangle of attack, i.e., cr 0. Thisis also a well-known result from classical steady flow transonicaerodynamics [5,6].

Consider now a numerical example representative of a high-performance aircraft such as the F-16. Choose the following param-eter values: !=2 10 Hz, c 10 ft, and U1 a1 1000 ft=s.Thus, !c=a1 0:6. From Eq. (37),

cr

!c

a1

3=2

0:5 rad 30

This angle of attack for pitch oscillations is much larger than themotions described as limit cycle oscillations(LCOs)that are found inflight tests forfighter aircraft and also in computational models [9].

Hence, this resultis consistent with the conclusions drawn by Dowellet al. [9] that an inviscid potential-flow model without somemodeling of the viscous boundary layer is not capable of describingLCOs for current aircraft of interest. A NavierStokes model thatincludes the effects of separated flow predicts LCO amplitudes thatare typically an order of magnitude smaller than the cr determinedby an inviscid potential-flow model, for example [9].

Note, by the way, that the thickness ratio of the F-16 is of theorder of 0.05, i.e., an order of magnitude less than cr for this

numerical example, consistent with the assumptions of the presentscaling analysis.

VI. Conclusions

The scaling analysis reported here supports the conclusionsreached by numerical computations andflight experimentsregardingthe inability of inviscid potential-flow models to predict LCOmotions perse. Of course, inviscid potential-flowmodels maystill beuseful in predicting the conditions for the onset of LCOandflutter asLandahl suggested many years ago in his classic book [10]. Finally,note that although the governing equations of transonic small-disturbance potential-flow theory are usually derived by othermethods, they may be deduced from full potential-flow theory byscaling analysis [11].

An interesting and open question remains, however. Can one usescaling analysis to predict the order of magnitude of observed LCOmotions based upon the Euler or NavierStokes equations?

Finally, the author notes that the essence of scaling analysishas been developed and used by the author to develop order-of-magnitude estimates of LCO aeroelastic response forflat and curvedplates and also for high-altitude long-endurance (HALE) wings [24]. Bejan [1], the authors colleague, has independently developedsimilar ideas and used them in thescaling analysis of thermodynamicsystems.

References

[1] Bejan, A., Convection Heat Transfer, Wiley, New York, 1984.[2] Dowell, E. H., Nonlinear Flutter of Curved Plates, AIAA Journal,

Vol. 7, No. 3, 1969, pp. 424

431.doi:10.2514/3.5124[3] Dowell, E. H., Nonlinear Flutter of Curved Plates, Part II, AIAA

Journal, Vol. 8, No. 2, 1970, pp. 259261.doi:10.2514/3.5653

[4] Tang, D. M., and Dowell, E. H., Effects of Geometric StructuralNonlinearity on Flutter and Limit Cycle Oscillations of High-Aspect-Ratio Wings, Journal of Fluids and Structures, Vol. 19, No. 3, 2004,pp. 291306.doi:10.1016/j.jfluidstructs.2003.10.007

[5] Heaslet, M. A., and Lomas, H., Small Perturbation Theories ofTransonic and Hypersonic Flow, General Theory of High Speed

Aerodynamics, Vol. 6, High Speed Aerodynamics and Jet Propulsion,Princeton Univ. Press, Princeton, NJ, 1954, Chap. 8.

[6] Cole,J. D.,and Cook,L. P., Transonic Aerodynamics, Seriesin AppliedMathematics and Mechanics, NorthHolland, Amsterdam, 1986.

[7] Williams, M. H., Linearization of Unsteady Transonic FlowsContaining Shocks, AIAA Journal, Vol. 17, No. 4, 1979, pp. 394397.doi:10.2514/3.61137

[8] Williams, M. H., Unsteady Thin Airfoil Theory for Transonic Flowswith Embedded Shocks,AIAA Journal, Vol.18, No. 6, 1980, pp. 615624.doi:10.2514/3.50797

[9] Dowell, E. H., Edwards, J., and Strganac, T., NonlinearAeroelasticity, Journal of Aircraft, Vol. 40, No. 5, Sept.Oct. 2003,pp. 857874.doi:10.2514/2.6876

[10] Landahl, M. T., Unsteady TransonicFlow, Pergamon,New York, 1961.[11] Dowell, E. H., Clark, R., Cox, D., Curtiss, H., Edwards, J., Hall, K.,

et al., A Modern Course in Aeroelasticity, 4th ed., Kluwer Academic,Norwell, MA, 2004.

P. BeranAssociate Editor

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