T Fluid Dynamics Researchoratory Report No. 63-3

SOME RECENT DEVELOPMENTS ININTERFERENCE THEORY FOR

AERONAUTICAL APPLICATIONS*

byHOLT ASHLEY

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JULY 1963

*Supported in part by the U. S. Air Force Office of Scientific Research under Grant

AF-AFOSR-156-63 and by Office of Naval Research under Contract NONR-1841 (80).

MIT Fluid Dynamics ResearchLaboratory Report No. 63-3

SOME RECENT DEVELOPMENTS IN

INTERFERENCE THEORY FOR AERONAUTICAL APPLICATIONS*

by

HOLT ASHLEY**

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JULY 1963

*This work was supported in part by the U. S. Air Forcer

Office of Scientific Research under Grant AF-AFOSR-156-63and by Office of Naval Research under Contract NONR-1841 (81).The author is indebted to North American Aviation, Inc.,and to Aerospace Division, The Boeing Company, forpermission to use certain computed results.

**Department of Aeronautics and Astronautics. This paper is

prepared for presentation at the Sixth Symposium of theDivision of Fluid Mechanics, Polish Academy of Sciences,Zakopane, September, 1963.

TABLE OF CONTENTS

Section 1 Introduction 1

Section 2 Nonplanar Wings in Subsonic Flight 4

Section 3 Slender Configurations at SupersonicSpeed 12

Section 4 Oscillatory Motion in Sonic Flight 21

Section 5 Remarks on Special Topics;Conclusions 24

References 28

Figures 31

ABSTRACT

Except for particular examples which have fundamentalinterest rather than engineering utility, exact solutionsto loading problems on three-dimensional lifting surfaceshave proved unobtainable even by small-perturbatianmethods. Beginning with the eminently successful lifting-line idealization, there has resulted a proliferation oftheories embodying various physical and consistent orinconsistent mathematical approximations. Nearly all ofthese are now rendered obsolete by the general availabilityof digital computation machinery with extraordinary speedand capacity, because the best engineering approach isnow unquestionably through numerical treatment of the exactlinearized integral equation appropriate to the problem underconsideration.

A useful by-product of these developments is thatthe erstwhile restriction to a single, isolated wing whosemean surface lies close to one coordinate plane no longerneed be accepted. Integral representations car, be devisedfor the coupled flow fields due to aggregates of liftingsurfaces, combinations of wings and bodies, or nonplanarmean surfaces. A number of applications of this modifiedapproach to the phenomenon known as "interference" aredescribed, some including numerical results and comparisonswith experiment. Both steady flight and simple harmonicoscillatory motions of small amplitude are included in theexamples. Those flows which involve an incompressible orsubsonic main stream are constructed by superposition ofproperly-oriented doublets of acceleration potential; thesingular behavior of the integrals is handled in the samemanner as for a planar surface. In supersonic cases thetechnique adopted consists of an extension of the conceptof velocity potential aerodynamic influence coefficients.Specialization to sonic flight speed is discussed, buthere the linearization is permissible only when the motionis unsteady.

The paper concludes with a review of related subjects,such as the importance of coupling with flow produced bywing thickness, the influence of a ground-plane, andapproximate means of accounting for second-order nonlinearityin supersonic flight.

I. Introduction

Although typical flight vehicles consist of aggregatesof adjacent or intersecting lifting surfaces and bodies,classical wing theory has dealt very largely with theairflow around individual, almost-plane configurationsin uniform motion. As a consequence, there have beendeveloped various approximate schemes for piecing togetherthe classical solutions - methods that fall under theheading of interference or interaction theory. With thevehicle's motion known, the loading on each element ofthe interfering system is initially estimated as if allother disturbance sources were removed from the field.Two types of correction are then applied. First, theupwash pattern is computed at each lifting surface whichwould be generated in its absence by other elements ofthe system; applying this as an incremental angle of attack,the additional loads are calculated by suitable linearizedtheory. Second, the upwash due to other elements isfound along the centerline of each body. Applying thisas an incremental angle-of-attack distribution to thebody, loads are found, usually by some modification ofthe well-known slender body theory. In steady-stateinterference analysis, both of the foregoing steps can besomewhat refined by extending any large, central body toinfinity and working at the Trefftz plane.

An excellent summary of prior literature and ofinterference procedures along the lines just describedwill be found in Ferrari's article (Ref. 1). Anotheruseful review of both subsonic and supersonic theoryhas been published by Lawrence and Flax (Ref. 2). Theslender-body approach forms the basis of many contributions,notably Refs. 3, 4, 5, 6 and 7.

Because of its author's interest in dynamic loadingand aeroelastic stability, the present paper is especiallydevoted to streamlined interfering systems performingsmall simple harmonic oscillations. Steady motion isthen included as a low-frequency limit. The literatureon such time-dependent flows over multiple surfaces andwing-body combinations is surprisingly sparse. Anapproximate study of the case of incompressible fluidappears in Ref. 8, pp. 63-75. Statler and Easterbrook(Refs. (9-10) have proposed methods for subsonic flight,

-2-

and more recent second-order transonic and supersonictheories for slender midwing-body combinations are describedby landahl (Ref. 1I) and others (e.g., Refs. 12 Roddenand Revell (Ref. 23) have also discussed certain interferenceproblems.

Except for particular examples, such as supersonicconical flow and the slender-body limit, exact three-dimensional solutions have proved unobtainable even forplanar lifting surfaces in steady flight. This deficiencyhas led, over the years, to the emergence of numerousanalyses embodying various physical and consistent orinconsistent mathematical approximations, which aredetailed in the cited references and in several excellentbooks now available on aeronautical aerodynamics. Inthis connection, the present paper hopes to emphasizetwo points. First, the bulk of these approximate methodshave been rendered obsolete, save for rough preliminaryestimation, by the wide availability of digital computationmachinery with extraordinary speed and capacity. Second,the restriction to an isolated, planar wing no longerneed be accepted, because systematic integral representationscan be devised for the coupled flow fields due to verygeneral interfering systems. Of these observations theformer is not particularly original, but the latter hasnot received the recognition which perhaps it deserves.All of the discussion and numerical applications thatfollow are designed to illustrate how coupled-flow problemsare set up and solved. It is hoped that the exampleswill be adequate to demonstrate the feasibility of practicalrealization.

The general framework for constructing mathematicalstatements is by superposition of doublet and source-typesingular solutions of the appropriate differential equationsfor small perturbations upon a uniform stream U parallelto the positive x-direction. Except for occasional commentsabout nonlinearities, therefore, linearized ideal-fluidtheory is employed throughout. In view of numerousexcellent published treatments (e.g. Chapter 1 of Miles,Ref. 7), it hardly seems necessary to give extensivedetails of the problem setup.

The basic dependent variables are the accelerationor pressure potential

"V(~,=_ __- (1)

-3-

and the disturbance potential 4 (x, y, z, t), definedby stating that the total velocity potential of theflow is

j5U + (2)

Here p is static pressure, ,o is density, and subscript ocidentifies properties of the remote stream. In termsof Cartesian coordinates x, y, z (Fig. 1) and time t,either qD or / is a solution of

;L -/ 1 o */< / (3)

M being the free-stream Mach number. The normal derivativeof V is specified over all wing and body surfaces, theformer being replaceable by the two sides of the wing'smean surface. Physical variables must be continuousexternal to the flight vehicle and its wake, and disturbancesmust die out appropriately at large distances. The unique-ness of solutions usually has to be assured by invokingan auxiliary condition, such as the Kutta hypothesis ofsmooth flow-off (continuous )V or 79 ) at all subsonictrailing edges.

The information of practical interest, once theproblem is solved, usually consists of the pressuredistribution over all wing and body surfaces or someweighted integral thereof, although occasionally othercharacteristics of the field are required. Pressure canbe obtained from Y/ through Eq. (1) or from T , in mostcases, through the linearized Bernoulli equation

__-__. Z__ P (4)

In seady flow the dependence on t disappears. When theperturbations are simple harmonic, all dependent variablesare replaced with their complex amplitudes ( 7 ) byoperations such as (R.P. "Real Part")

(x, y, z, t) = R. P. (x, Y, z) ) e

Also I±41is substituted for / t. The circular frequencyW)provides a measure of "unsteadiness" in a given problemwhen converted to a reduced frequency

U.7 (6)1 being a representative streamwise dimension such aswing semichord or body length.

-4-

II. Nonplanar Wings in Subsonic Plight

Consider small-amplitude vibrations of a liftingsurface S like the one illustrated in Fig. 1, lettingthe free surface recede to infinity for the moment. Inthe examples it is assumed that S has zero thickness.In fact, when S happens to be almost plane, thicknesseffects may be treated separately and independentlyfrom those of angle of attack, camberetc. The samedecoupling can be accomplished in the unsteady loadsproblem even for nonplanar wings. But it is not generallypossible in steady flow, and non-zero thickness must berepresented 'y distributing sources over S, which interactwith the doublets describing the lifting part of thefield. The procedure is straightforward and adds nothingessential to the present dibcussion.

Superimposed on the mean-surface shape zo(y) in Fig. 1is the small displacement

(x , , = 6 2 (X, Y) e(7)

the "real part" operation from Eq. (5) being henceforthdropped. This gives rise to a normal displacement

7yi(x y)e t , (Xy)e sec0(y) (8)

and to a normal velocity of fluid particles in contactwith S 6 C

(X +L /X (9)

vn must equal the normal derivative & for all pointsx, y or x, s on S (s being a single-valued curvilinearspanwise coordinate).

For subsonic flow, Y is chosen as the primary unknown,since pressure discontinuities occur only through S andthe disturbance can therefore be represented by normallyoriented doublets of )v over the bounded area of thewing. From the well-known relation between -Y andit is easily shown that the principal boundary conditionreads x Z , . . , ,''

o(10)for(x, y, z) on S'

-5-

If a layer of p-doublets is distributed over S,classical methods can be used to establish that the layer'sstrength is locally proportional to the discontinuityof 'Y , or of p. In incompressible flow, for instance,Eq. (3) reduces to Laplace's equation, the doubletsingularity is simply a normal derivative of l/r, andthe solution becomes

- 7 Z_11

Here. , , -- are dummy variables replacing x, y, sin the wing surface, while nI is the normal direction atpoint _, , Zo (37) . Subscripts U and L denote "upper"

and "lower sides of S relative to the n direction (cf.Fig. 1). Combining Eqs. (10) and (11) ylelds theincompressible-flow integral equation

Cj~ _ e K/ (12)(J

with the singular kernel function

- ~~ /at~ (yYt A- -J (13)

The double integral of Eq. (12) is to be evaluatedin the sense of Mangler (Ref. 13), the principal singularitybeing associated with a factor (y - )-c. After somemanipulation, which generalizes the work of Watkins,Runyan and Woolston (Ref. 14), a suitable working formof the kernel function is found to be

-6-

[e)X ( )4 YA X, [3 02- +

L- a_ -X

2-*)X Xk0 1

+(7 of -

____ - LU~ll

e+ ljT~ ) Ll ()~

w2.

(1 *Lx (A 0 + 6)

0'~ J.cx --3

2.Y.zC -ZwJ)

-7-

Here Ii, Ki and Li are modified Bessel and Lommelfunctions in standard notation. The auxiliary symbolsare as follows:

X0 X> - Y Y 2 2Y

A generalization of the kernel function to includethe influence of Mach number in subsonic compressibleflow has been derived, but space considerations preventreproducing it here. Several reformulations of Eq. (12)for special physical situations are discussed below:

a.) Pair or collection of non-intersecting liftingsurfaces. In this case a boundary condition likeEq. (12) would be written for each surface and wouldcontain integral terms on the right equal to thetotal number of surfaces, only the term representingthe influence of a particular surface on itselfinvolving a singular integration. One is thusrequired to solve a system of coupled integralequations for[pL - Pu ] over all the interferingelements. K will have essentially the same formas Eq. (14) in all terms. The surfaces do not haveto be plane or parallel.

b.) Intersecting lifting surfaces. This problem diff:,,sfrom the one described under a.) only because pLand pu on a surface may be discontinuous throughthe station where it is intersected by another.The solution procedure described below must bemodified to permit these Jumps but to assure continuityof pressure as the corner is turned from one surfaceonto another.

c.) Effect of a ground plane. By the well-known imageprinciple, a ground plane (cf. Fig. 1) is accountedfor by adding to the field an image which isloaded in a sense symmetrical to the original wing.The effect can be introduced as an additive correctionto K in the single integral of Eq. (12).

-8-

d.) Effect of a free water surface on an oscillatinghydrofoil running at high Froude number F = U/if .The boundary condition of constant pressure atthe free surface z = d (Fig. 1) is known toreduce to 4D = 0 when F>>1. This condition ismet by loading the image in a sense antisymmetricalto the original wing, that is, by makin f Ff ithe same at corresponding points. The uppersurface is the positive z-side in each case. Again,an additive correction to K appears in Eq. (12).

The steady-flow simplification of the kernelfunction is substantial:

2sKo /4[=(Y)+ 0),jJ S f*7 -O x°03k LX 0 ]

-Co (o s9()Y 2021l- X. [Z +x Y.k-- 4 +3 ZZ (16)

The influence of Mach number in steady motion can behandled most simply by means of the Prandtl-Glauertcompressibility correction.

Regardless of the physical circumstances, the numericalprocedure which has been found most effective for solvingintegral equations like Eq. (12) is the same, a directoutgrowth of Watkins' development (Ref. 15) for planarwings. The key idea is to approximate LPL - Pu witha rapidly-convergent series of functions which give theright leading-edge singularity, fulfill the Kuttahypothesis along the trailing edge, and also drop to zerowith the correct infinite slope along side edges. Ifthe wingtips are located at x = * B/2, one introducesthe following variables to transform S onto a rectanglebetween S- -1, +1 and - = 0, 7r:

(17)2

-9-

Here xT and xL are streamwise coordinates of the leadingand trailing edges, respectively. The pressure seriesreads

8 /1_ _S2

LXr -XJ1 8C-- Ot-I- as 5'

(Several series are, of course, needed for a system ofinterfering surfaces.)

When Eq. (18) is inserted into Eq. (12), with dummyintegration variables, the integrals can be evaluatednumerically at a set of stations Cx, s)on the half-span.For the assumed symmetrical planform shape, the loadingis divided into symmetrical and antisynmetrical portionsin a well-known way. Itegrations must be performed withcare because of the yo - singularity; the work of Ref. 15has proved very useful in this connection. The resultsmay be cast in the matric form

Ea- /21,j {L -r uI* h (19)Here f..j are column matrices, whereas EK] is asquare or rectangular matrix, whose complex elements areintegrals of the kernel weighted by individual terms inthe series, Eq. (18). One solves for the column {am}of unknown coefficients by direct inversion or some %st-squares technique. The load distribution comes fromEq. (18), and generalized forces like lift and pitchingmoment can often be expressed in terms of relatively fewof the anm. Clearly, all of these steps would beunthinkable without high-speed digital computing machinery.The IBM 7090 at Massachusetts Institute of Technologyis able, however, to solve one case of steady loadingon a nonplanar wing or of unsteady loading on a planarwing in less than five minutes, nine to sixteen collocationpoints on the half-span being employed.

It should be mentioned, in passing, that moresophisticated schemes for solving Eq. (12) have beenproposed and deserve further examination in the caseof nonplanar surfaces. For instance, Hsu (Ref. 16 andantecedents) adopts the series (18) but simplifies the

-10-

integrations with a "natural" choice of collocation points.Stark (Ref. 17) has been able to avoid certain difficultieswith rapidity of convergence by focussing on generalized-force computation, using reverse-flow theorems, andintroducing least squares when applying the flow-tangencyboundary condition.

Illustrative solutions of Eq. (12) for a varietyof practical problems are presented by Figs. 2 through7. In each instance the fluid is of constant density,since most applications to date have been carried outin connection with hydrofoil design. Many similar resultsfor plan surfaces in both compressible and incompressibleflow w~ll-e found in Refs. 18, 16, 17 etc.

Figure 2, adapted from a recent study of groundeffect by Saunders (Ref. 19) demonstrates the excellentaccuracy that can often bV obtained at low incidenceeven on quite thick wings. The reader is directed toRef. 19 for sources and interpretation of the data.Shown plotted vs. height above ground in chords, theordinate is dimensionless lift per radian of angle ofattack for three rectangular surfaces:

L _ds _(20)

The dimensionless nose-up pitching moments, associatedwith these lifts, about a spanwise axis 25% of the wayback from the leading edge are graphed in Fig. 3. Theground plane's influence is seen to cause an increasein lift while displacing its center of action aft. Asan example of combined ground and non-planar effects,Fig. 4 plots lift-curve slope for a V-wing at differentheights and dihedral angles.

Figures 5 - 7 concern oscillatory motion in thepresence of a free surface, 5 and 6 being taken from arecent paper by the author and two colleagues (Ref. 20).In Fig. 5, the vibration consists of a verticaltranslation

Sz (xAy - h e ,(21)

-11-

whilVig. 6 refers to nose-up pitch-angle displacementZe about the quarter-chord axis. The abscissa isreduced frequency, Eq. (6), with length 1 chosen toequal the semichord b. Being complex numbers, thedimensionless aerodynamic loads are shown in magnitudeand phase-angle form, e.g.,

C4 C46/CL'- R#LC (22)

Froude number in Figs. 5 and 6 is assumed large enoughto permit use of the antisymmetrical image wing.Geometrical characteristics are listed on the figures.

The calculations in Fig. 7, previously unpublished,refer to the two-dimensional hydrofoil running at finiteFroude number. The integral of Eq. (12) is here replacedby a single integration along the chord, but the kernelfunction (Ref. 20) must account for the various trainsof surface waves set up by the motion. The importantparameter of this problem is

J(23)

One sees the singular behavior near kF2 = 0.25 that wasdiscuesed by Crimi and Statler (Ref. 21). F = 10 isevidently large enough, however, to permit the infinite-Froude-number approximation for practical reducedfrequencies.

No subsonic interacting systems containing bodieshave been successfully worked through by the methods ofthis report. It is expected that recourse will have tobe taken to fundamental representations, in terms ofsource and doublet solutions of the governing differentialequation distributed over all surfaces bounding theflow. The circumstances where complete linearizationis permissible, especially for pressure calculation, areby no means as clear as when only lifting surfaces areinvolved. Although this may seem to be a very complicatedundertaking, yet it is dangerous to underestimate thepotential of digital computers for well-stated, highlysystematized problems. One need only call attentionto the pioneering achievements by Smith and collaborators,of which Ref. 22 is an early example, on large-disturbanceincompressible flows around bodies of arbitrary shapewithout circulation.

-12-

III. Slender Configurations at Supersonic Speed.

When M> 1 the character of phenomena described byEq. (3) changes as a consequence of the inability ofsignals to propagate upstream. Although it is stillpossible to construct a kernel function for Eq. (12),and important work has been done in this direction, theauthor believes that a more fruitful approach tointerference problems lies via the inverted representationof the solution and the meth'o7 of aerodynamic influencecoefficients (AIC's, Refs. 23 and 24).

The AIC scheme is founded on an artificiality thatbrings about both a great deal of mathematical simplificationand some difficulty in physical understanding. This isthe use of source sheets, which produce symmetrical flowwith respect to the wing plane, in place of doubletsheets having the expected lifting antisymmetry. Anecessary accompaniment to sources is the addition ofhypothetical "diaphragm" areas, whose purpose is toisolate opposite sides of lifting surfaces whileensuring that no load acts on regions which cannot sustainit. Thickness effects are, incidentally, omitted alsofrom the present treatment of the supersonic case, butthey are easily superimposed as mentioned at the beginningof Section II.

By way of introduction to AIC's, the procedure forfinding the flow due to a vibrating plane configurationlike that in Fig. 8 is briefly reviewed. The disturbancepotential at any (x, y, z) in the half=space z > 0 isgiven by (Refs. 7, 23, etc.)

Here 13 / -1 ; w is proportional to source strengthper unit area and here equals the vertical velocityinduced by the source sheet Just above the surface.The operator R. P. calls for taking the real part onlywith respect to the change in sign of the quantity underthe radical. When this operation is performed,S reducesto that portion of the wing-diaphragm area interceptedby the upstream Mach cone from (x, y, z).

-13-

As in the figure, wing and diaphragm are overlaidtot-b closest possible approximation with elementaryareas ( b, by bl/O rectangles ) having diagonals parallelto the Mach lines. In terms of the dimensionless quantities

Y/ (25)6,

Eq. (24) reads

S(X, , Y,/ )(26)

Now let it be assumed (as will be correct in the limitbl-->O) that 2 is constant over each element and equalto the value R2/ at the center. By placing the originat the foremost 'of these centers, -v and are caused tobe integers counting centers rearward and to the rightfrom this origin. The potential may be written

-7(P (x.Y> l ~ ,_ , V- (27)

where the sum is extended over all elements ahead of oralong the hyperbolic intersection between z = 0 and theforward Mach cone. Making the substitutions

., y,-..-(28)

it is an easy matter to show that the AIC Jx 1 , y zl)actually depends only on the relative posit on bet eenthe "sending" area and "receiving" point. Indeed, itcan be expressed as

-114-

IC Co (9

where .=(x 1 h and (y -7)

n a similar fashion, the velocity components

v -q~and ii at a field point can be obtained from

z2Z 1~)>~ ~(30)

X, J~~=i (31)

V and W being essentially y and z derivatives of thequantity in Eq. (29). Formulas have been worked outfor numerically computing various AIC's along the linesof Ref. 214, Although space does not permit discussingall the complications brought on by Mach-come intersections,the basic forms for "Ucu area elements are reproducedbelow( V >c 0 z> o):

(32)

- /

~2 -I (33)

Cos'

-15-

- 5 2-,/ f,-(t

V --I

(34A)

+l 2 P7 -cosiderations-1-n')

22e/r=

(34B)

and 1 has the same form except that (u-1/2) Isreplaced 0yu + 1/2). Symmetry considerations can beused to establish that

In practice, the skgle integrals in Eqs. (32) - (34)must be evaluated by quadrature. This gives rise to noserious difficulties, and conveniently isolated computersubroutines have been successfully operated for each ofthe AICts (for example, by North American Aviation, Inc.*).

* Work of Andrew and collaborators, as yet unpublished.

-16-

The use of AIC's for loading calculation on a pwing is described, with rules based on extensive experience,in Refs. 24. In most cases, all design information canbe determined from the distribution of T over theplanform, so that only Eq. (27) and the velocity potentialAIC (Eq. (33) with 1 = 0) are used. ZT(Xl, yl, 0) iscalculated sequentially at centers of area elements,startin from (0,0) in Fig. 8. For the summation inEq. (27 ), . is known, in terms of the motion, atpoints on th "planform and can be determined from thecondition T= 0 at points on the diaphragm region, shadedin the figure. If the steps are performed in the properorder, it is never necessary to solve any systems ofsimultaneous equations or invert matrices.

Figure 9, adapted from Ref. 24, gives an exampleof the AIC estimation of spanwise lift distribution ata rectangular wingtip compared with series-expansionresults obtained by Watkins (Ref. 25). This figurealso demonstrates the improved accuracy which can beachieved by introducing additional terms to account forthe singularity of upwash that exists Just off theside edge where the diaphragm meets the wing (Ref. 24).The inclusion of such singularities at leading and sideedges has been examined in detail. As a general conclusion,it appears that comparable accuracy can be obtained inthe qp-distribution at less computational cost bydispensing with these special terms and reducing thearea-element size relative to the wing dimensions.Although mathematically less rigorous, this approach istentatively being taken with interfering surfaces as well.

Procedures for applying AIC's to nonplanar wingsand interfering wing-body systems have been outlinedby the author in an unpublished report (Ref. 26). Twoexamples are presented here to illustrate the principles.Consider first a single nonplanar surface of the sortshown in Fig. 10, oscillating in a known mode ofvibration. From Eq. (9), vn will be spedified over theplanform area S. To assure no communication betweenupper and lower surfaces, plane diaphragm areas areassociated with both the main wing and bent-down wingtip,each extending out to meet the Mach cone from the vertex.A vertical plane of symmetry is assumed at the vehiclecenterline, so that it is necessary to work over onlyhalf the planform, making automatic provisions to accountfor contributions from the opposite half. Clearly, thetwo tips do not interact.

-17-

Area elements like those in Fig. 8 are distributedover the upper and lower sides of all lifting and diaphragmregions, and oscillatory sources of constat amplitudeare placed on each element. Let Sj and S )denotesource-strength amplitude per unit area of the upperand lower sides, respec ively, of the main wing and itsdiaphragm. V4) and S(Iare corresponding strengthsfor the tip and tip diaphragm. The physical conditionsto be met are 1.) that/vhave the correct values onlifting area elements, and 2.) that both,and pressure(4in this case) be continuous through each diaphragmelement. For example, the normal velocity at points onupper wing area t induced by the sources onj can bewritten

-- "(35)

Care must be taken with the definitions ofi/AAhere.i7is the dimensionless chordwise distance aft from thecenter of the "sending" J-element to the "receiving"i-element, whereas and-O are dimensionless relativedistances measured tangential and normal to the plane ofwingtip J. Centers can usually be chosen so that-7isalways an integer, but/Z2will be irrational numbers( is here negative). )

Equation (35) can be recast, in obvious matricnotation, as

= - s(36)

Subscript "W" on the left-hand column matrix indicatesthat only area-element centers on the lifting portionof the wing are included, while "WD" on the right meansthat all wingtip and wingtip-diaphragm elements arerepresented. The rectangular matrices on the right willcontain many zeroes due to the law of forbidden signals,and there is an advantage to properly ordering thecomputations. The normal velocity at points on area Jfrom the presence of sources on the upper side of thewing and its diaphragm is

-18-

(37)

where 1 are now referred to the wing's plane.(,)Given the normal-velocity amplitudes n ) and

n (iproduced by the known motion of wing and tip,

the geometrical boundary conditions on these upperlifting areas read

(- s-(38)

- OS (Is~~~wov/)~

- " 16 (39)S711 -/W

There are two essentially identical relations for thelower surface, superscripts (i) and (J) being replacedwith (1) and (A), respectively.

The kinematic conditions of continuity of normalvelocity across the two diaphragms read

g~j+ L~'~3) (40)

+ (41)

Here substitutions like Eqs. (36) and (37), modified torefer to diaphragm rather than wing area elements, areused to eliminate the interference matrices and writeEqs. (40) - (41) entirely in terms of source strengths.

-19-

An additional set of equations covering thediaphragm centers is needed to construct a determinatesystem. This is provided by the requirement of pressurecontinuity, which is equivalent to '>-continuity in thepresent case. Thus,

6 .- (" -

b)}[ 0 +{ )S'[] 6 ' 'W (42)

D ,/,0WWD

'e) (e) (43)

The total of the linear equations can now beproved to equal the number of unknown area-elementsource strengths. Experience at North American Aviation,Inc., has shown that the foregoing computation can besuccessfully mechanized and that skillful orderingavoids the need for mtrix inversions. Once the unknownshave been determined, equations lIke the following willyield r on each side of each lifting surface:

£&~j~Ef 0i ]f?(7 +g M f() (44)Pressures and generalized forces can be found from1K - u 3 as for planar wings. An example of theinfluence of tip deflection on lift and aerodynamic-centerlocation is shown in Fig. 11. The AIC predictions areseen to follow measured trends quite satisfactorily.

As a second example, the hypersonic glider picturedin Fig. 12 is selected. No detailed calculations haveyet been carried through on such an elaborate configuration,which includes a body, but the procedure is not essentiallymore complicated than for the nonplanar wing. The figureshows, shaded, a suitable system of diaphragms to preventcommunication between the opposite sides of all thin surfaces.

-20-

As before, each side of each diaphragm and lifting regionwould be overlaid with rectangular area elements. Similarelements, also covered with constant-strength oscillatingsources, are affixed in a suitable pattern over thebody or fuselage. (The body base, being in a separatedflow region, must be treated empirically, but conditionsat this base cannot affect the loading forward.)

The illustrated fuselage is made up from essentiallyplanar areas. Each of these can be treated exactly likeone side of a lifting surface, with a flow-tangency boundaryequation being available to determine the source strengthon each area element. Were the body composed of curvedsurfaces, however, the normal direction for each separateplanar element would have to be found and carried throughthe computation. Airload determination would then besomewhat more complicated, because each element over theentire body will receive normal velocity contributionsfrom all others. There is no theoretical reason,however, why such interference problems involving bodiescannot be mechanized in a way that closely parallelssystems with lifting surfaces only. Finally, it shouldbe remarked that means for dealing with wakes arisingfrom subsonic trailing edges are well-understood and canbe deduced from the discussion and examples of Zartarianand Hsu (Ref. 24).

-21-

IV. Oscillatory Motion in Sonic Flight

As set forth in Ref. 11 and elsewhere, finitelifting surfaces flying near M = 1 are not susceptibleof stead linearized treatment, but when k'> 1-Ml thelinearifferential equation

(Pj 4Fp-+O (45)

governs the oscillatory field. This observation suggestedto the author the possibility that the concept of AIC'smight be adapted to sonic flight. Should such anapproximation prove practical, the results can readilybe extended to cover the "transonic range" between, say,M = 0.95 and 1.10 by means of Landahl's similarity law

The scheme which has undergone preliminary examinationio suggested in Fig. 13. Here square area elements, eachwith constant oscillating source strength, are distributedover the planform and diaphragm regions of a planarwing. The Mach lines at sonic speed are parallel tothe y-axis, so the diaphragms theoretically extend toinfinity. One must therefore assume the normal velocity%y,)M arbitrarily equal to zero for diaphragm elementsbeyond a certain distance away from each wingtip. InFig. 14 some analytical results for 7 off a rectangulartip in plunging vibration are plotted to suggest thatthis assumption is probably quite acceptable and thatthe necessary distance is a matter of a small number ofchordlengths. Figure 14 was computed incidentally,using the closed-form solution for a 'quarter-infinite"wing on x> 0, y ; 0 (Chap. VI of Ref. 11), from whichone can deduce

(flO 0) ('47

4_ _ ____ _ [E~ e6,_4

- ~~ ~ 2 rrX-- d'<x2..y7x

-22-

In Eq. (47) h is again the amplitude of verticaldisplacement, and q is a dummy variable of spanwiseintegration.

The formula for potential field which forms the soniccounterpart of Eq. (26) is readily found from Ref. 11 toread

(X, , Y,. ,

- lX _Y x [12(xr >i 2 y F)(8

77 Ex,- ,]_e0

The dimensionless coordinates here are referred to sideb, of the square elements and k, = Yb /U, Under therestriction-9 3 0, Eqs. (30) and (31J may be adoptedwithout change, and Eq. (27) is replaced byS(x, . ,) >' - __ -,Z l)-

bl /Z (49)

Except for easily-handled singular cases which occurwhen;P = 0, the AIC's are as follows (superscript (1)is used to denoteponic flow):

(i _ _1._ -:- . )

Z IMA

(50)

_; 1 ,,. - - I dr_

' I(4,

-,/- .(51)

-- (52).4m I f

-23-

The notation adopted in Eqs. (50) - (52) for the Fresnelintegrals is most conveniently expressed as

X

771 (53)

All single integrations will have to be carried outnumerically, but computer subroTyines are already availablefor the Fresnel integrals. 5 1 0 has already beenevaluated with enough accuracy and speed to assure thefeasibility of airload determinations for nonplanarconfigurations.

The only numerical application of sonic AIC's knownat present to author consists of an effort by the BoeingCompany* to compare the surface distribution of ' againstthe well-known exact solution for two-dimensional flow.A few results are presented in Fig. 15. The wing, performingplunging oscillations at k, = 0.05, is intended to haveinfinite span, but the contributions to q- from pointsmore than 80 area elements distant from the chordwisecross-section under consideration were neglected. Asthe computation proceeds downstream, the effectivereduced frequency k (based on winW chordlength ahead)increases while the 'aspect ratio of the rectangulararea by which the actual two-dimensional wing is approximateddecreases. Since there is no reason to anticipatecomputational difficulty at higher k in this range, theultimate deviation between the exact and AIC resultscan be attributed to the latter effect. Additional studiesshow poorer agreement at very low values of k, wherelinearized theory itself is questionable, but no practicaldifficulty is anticipated either with the total numberof area elements or with excessive spanwise dimensionsof the region which must be covered with them.

Because of the tendency of strong shocks to format intermediate chordwise stations in transonic flow,very severe limitations will have to be placed onthickness- amplitude ratios and k will have to berelatively large before the foregoing theory will bevalid for applications. It may, for example, proveuseful only for flutter analyses rather than loadsestimation. This is especially true on surfaces with

Work of Weatherill, as yet unpublished.

-24-

nearly unswept leading edges. It is therefore importantfor this speed regime that Landahl (Ref. 28) and othersare beginning to point the way toward suitable theoreticalrefinements.

V. Remarks on Special Topics; Conclusions

a.) Arbitrary Time-Dependent Motion

Aerodynamic loada due to sinusoidal vibration ofinterfering systems, predicted according to the methodsof this paper, have immediate utility in flutter analysesand when determining the transfer functions of a flightvehicle for such harmonic inputs as sinusoidal gustvelocity or displacement of the control surfaces.Instances frequently arise, however, when it would beadvantageous to get loads directly for more generaltime-dependent motions. At low frequencies, of course,one can make the quasi-steady approximation and use theforegoing theories with k = 0. In situations such asrapid maneuvers, impulsive inputs, and transient structuralvibrations this is often not permissible.

The author has given considerable thought to theengineering of unsteady aerodynamics for nonsinusoidalphenomena. He has concluded that, when elaborate theoryis to be used in connection with high-speed digitalcomputation, the most efficient approach lies throughFourier series decomposition of the inputs andresponses. First successfully mechanized by Bisplinghoffet al. in 1949 (Ref. 29) for aircraft problems, thisscheme has appeared in a number of versions includingnumerical evaluation of the Fourier integral by divisioninto finite frequency intervals and truncation. Whateverthe form, success seems to depend on retaining a ratherlarge number of terms in the summations, which issomething that can readily be systematized on a machinelike the IBM 7090. As an illustration, North AmericanAviation* has been able to determine flexible-airframeresponse to standardized design gust shapes, representingthe gust as a periodic input with quiewent intervals.Three-dimensional unsteady aerodynamics were employedin a seven-degree-of-freedom modal representation andthere was no excessive time required to sum 50 to 100terms of a Fourier-series approximation to the inputand outputs.

* Work of Stenton and collaborators, as yet unpublished.

-25-

b.) Intersecting Surfaces

In connection with adapting interference theoryto intersecting surfaces, an exhaustive study has beencarried out on exact linearized solutions in steady flow.Some of the configurations analyzed are sketched inFig. 16. Included have been general cruciform combinationsaccording to slender-body theory, cruciform supersonicdelta combinations with subsonic and supersonic leadingedges according to conical flow theory, and slender-bodyrepresentations of V and T-tails, etc. In all cases,no singular behavior is found near any corner where thesurfaces meet at an angle less than or equal to 1800.Indeed, one can speculate on a general conclusion thatseems reasonable physically: regardless of flight Machnumber, the crossflow in the corner behaves like thefamiliar, two-dimensional, incompressible corner-flowsolution r

To = s- (54)

being the angle and r, e polar coordinates at thecorner.

The v and w velocity components are proportionalto the power ( ' -1) of r, reducing to lineardependence in a rectangular corner. The three-dimensionaldisturbance pressure has an extreme value at the cornerand varies only slowly in the vicinity. Hence therewould seem to be no problem constructing suitablepressure series, generalizing Eq. (18), when analyzingsuch intersections, except at the outer side of aV-tail.

c.) The Question of Nonlinearity

The relatively thick lifting surfaces of some entryvehicles, the push to higher cruising Mach numbers, andthe appearance of heavily-loaded aircraft for flightnear the ground are among the many reasons why thesmall-disturbance restriction on aerodynamic theoryis often intolerable. Nevertheless, linearized methodslike those described in this paper are already socomplicated as to inspire pessimism regarding refinements.It is unwise to speculate too extensively about incompletedevelopments, but it should be pointed out that certainroutes exist by which nonlinearity can be introduced,both for planar wings and interacting surfaces.

-25-

b.) Intersecting Surfaces

In connection with adapting interference theoryto intersecting surfaces, an exhaustive study has beencarried out on exact linearized solutions in steady flow.Some of the configurations analyzed are sketched inFig. 16. Included have been general cruciform combinationsaccording to slender-body theory, cruciform supersonicdelta combinations with subsonic and supersonic leadingedges according to conical flow theory, and slender-bodyrepresentations of V and T-tails, etc. In all cases,no singular behavior is found near any corner where thesurfaces meet at an angle less than or equal to 1800.Indeed, one can speculate on a general conclusion thatseems reasonable physically: regardless of flight Machnumber, the crossflow in the corner behaves like thefamiliar, two-dimensional, incompressible corner-flowsolution 7r

S= K 5C- -g(5)

being the angle and r, e polar coordinates at thecorner.

The v and w velocity components are proportionalto the power ( - -1) of r, reducing to lineardependence in a rectangular corner. The three-dimensionaldisturbance pressure has an extreme value at the cornerand varies only slowly in the vicinity. Hence therewould seem to be no problem constructing suitablepressure series, generalizing Eq. (18), when analyzingsuch intersections, except at the outer side of aV-tail.

c.) The Question of Nonlinearity

The relatively thick lifting surfaces of some entryvehicles, the push to higher cruising Mach numbers, andthe appearance of heavily-loaded aircraft for flightnear the ground are among the many reasons why thesmall-disturbance restriction on aerodynamic theoryis often intolerable. Nevertheless, linearized methodslike those described in this paper are already socomplicated as to inspire pessimism regarding refinements.It is unwise to speculate too extensively about incompletedevelopments, but it should be pointed out that certainroutes exist by which nonlinearity can be introduced,both for planar wings and interacting surfaces.

-26-

Consider, for instance, the ideas of Covert (Ref. 30)and others regarding steady motion of supersonic wings.Reference )0 demonstrates that second-order effects ofthickness in the near field are properly accounted forif the disturbance potential q of the lifting flow ismade to satisfy the differential equation

(~/P-X -9 ± & 0(55)

where MT is the local surface Mach number due to thethickness distribuon and ? is often a very smallparameter. Covert shows how P can be then built upby superposition of source-like solutions depending onMT rather than free-stream M. Thus one corrects forvariations in sound speed and second-order convectioneffects, while still ignoring shocks which are third-order.

There is no evident reason why Covert's schemecannot be adapted to nonplanar surfaces and to liftingflows which are simple harmonic rather than steady.An attempt to mechanize his results for digitalcomputation could constitute a fruitful undertaking.

Already referred to above was the suggestionof Landahl (Ref. 28) for a refinement on linearized,transonic theory that parallels Covert's developmentvery closely. Landahl focusseL on simple harmonicmotion and again demonstrates how the unsteadyperturbation potential can be found from a differentialequation whose coefficients depend on the local sonicand particle speeds in the thickness flow. It mightbe observed that, for numerical computation, thereis no reason why steady data to be incorporated insuch theories cannot be obtained experimentally.

For finite wings oscillating at subsonic Machnumbers, the author is aware of no mathematicallyconsistent nonlinear analyses. This lack does not,however, preclude an attempt to make semi-empiricalcorrections for thickness and interference that mightextend the methods of this paper to more heavily-loadedsystems. The key idea is that proposed for two-dimensional airfoils by Allen (Ref. 31). Once theaggregation of vortices or acceleration-potentialdoublets representing the lifting flow is determined

-27-

by linearized procedures, one can compute the localstreamwise velocity ( VJ+ 9P ) at each such element.The load would then be estimated by replacing U with( V+ (P, ) in Bernoulli's equation or the Kutta-Joukowsky theorem. This has produced significantimprovements in a number of steady cases and meritsfurther examination in connection with the interferenceproblem.

d.) Conclusions

When concluding, it seems necessary to point outonly that a pattern is emerging in the treatment ofthree-dimensional loading of wings and interactingsystems by means of linearized aerodynamic theory.A host of questionably consistent approximations arebeing replaced by the systematic superposition ofappropriate singularities which, in the limit of aninfinite number of terms, would produce an exact solution.The unifying tool is the high-speed digital computer.

Illustrative calculations have been presented orreferenced for subsonic, sonic and supersonic flightMach numbers, involving both steady and oscillatorymotion. The number and variety of these is deemedsufficient to portend the widespread development ofthese methods and their forthcoming availability asanother string for the aircraft designer's bow.

-28-

VII. References

1. Ferrari, C., Interaction Problems, Section C,Vol. VII, of High Speed Aerodyn acs and JetPropulsion, Princeton university Press, Princeton,New Jersey, 1957.

2. Lawrence, H. R., and Flax, A. H., Wing-Body Interferenceat Subsonic and Supersonic Speeds-Survey and NewDeveloents, Journal of the Aeronautical Sciences,Vo1, No. 5, May 1954, pp. 289-328.

3. Ward, G. N., Linearized Theory of High Speed Flow,Cambridge University Press, 1955.

4. Bryson, A. E., Stability Derivatives for a SlenderMissile with Application to a Wing-Body-Vertical-TailConfiguration, journal of the Aeronautical Sciences,Vol. 20, No. 5, May 1953, pp. 297-308.

5. Spreiter, J. R., The Aerodynamic Forces on SlenderPlane-and Cruciform-Wing and Body Combinations,N.A. C. A. Report 962, 1950.

6. Adams, M. C., and Sears, W. R., Slender Body Theory-Review and Extension, Journal of the AeronauticalSciences, Vol. 20, No. 2, February 1953, pp. 85-98.

7. Miles, J. W., The Potential Theory of UnsteadySupersonic Flow, Cambridge University Press, 1959.

8. Ashley, H., Zartarian, G., and Neilson, D. 0.,Investiation of Certain Unsteady Aerodynamic Effectsin Longitudinal Dynamic Stability, U. S. A. F. TechnicalReport 59bb, 1951.

9. Statler, I. C., Derivation of Dynamic LongitudinalStability Derivatives for Subsonic Compressible Flowfrom Nonstationary Flow Theory and Application to an- OA Airplane, Cornell Aeronautical Laboratory Report

NO* TB-E95-F-9, 1949.10. Statler, I. C., and Easterbrook, M., Handbook for

Computing Nonstationary Flow Effects on SubsonicDynamic Longitudinal Response Characteristics of an

,plane, Cornell Aeronautical Laboratory ReportNO.-TBTI95-F-12, 1950.

U. Landahl, M. T., Unsteady Transonic Flow, PergamonPress, New York-oxford-London-Paris, 1901.

-29-

12. Zartarian, G., et al., Forces and Moments onOscillating Slender Wing-Body Combinations atSupersonic Speed, Parts I and II, U. S. Air ForceOffice or Scientific Research Technical Notes 57-386and 58-114, 1957.

13. Mangler, K. W., Improper Integrals in TheoreticalAerodynamics, British Royal Aircraft Establishment,Report No. Aero 2424.

14. Watkins, C. E., Runyan, H. L., and Woolston, D. S.,On the Kernel Function of the Integral EquationRelating the Lift and Downwash Distributions ofOscillating Finite Wings at Subsonic Speeds, N. A. C. A.Report 1234, 1955.

15. Watkins, C. E., Woolston, D. S., and Cunningham, H. J.,A Systematic Kernel Function Procedure for DeterminingAerodynamic Forces on Oscillating or Steady FiniteWings at Subsonic Speeds, N. A. S. A. Report R-4, 1959.

16. Hsu, P. T., Some Recent Developments in FlutterAnalysis of Low-Aspect-Ratio Wings, Proceedings of theNational Specialists Meeting on Dynamics and Aero-elasticity, Ft. Worth, Texas, Nov., 1958, publishedby Institute of the Aeronautical Sciences, pp. 7-26.

17. Stark, V. J. E., Aerodynamic Forces on RectangularWings Oscillating in Subsonic Flow, S aaW TechnicalNote 44, Slab Aircraft Company, SWeden, 1960.

18. Cunningham, H. J., and Woolston, D. S., Developmentsin the Flutter Analysis of General Plan Form WingsUsing Unsteady Air Forces froi the Kernel FunctionProcedure, Proceedings of the National SpecialistsMeeting on Dynamics and Aeroelasticity, Ft. Worth,Texas, Nov. 1958, pp. 27-36.

19. Saunders, G. H., Aerodynamic Characteristics of Wingsin Ground Proximity, Master of Science Thesis,Massachusetts Institute of Technology, June 1963.

20. Landahl, M. T., Ashley, H., and Widnall, S. M., SomeFree Surface Effects on Unsteady Hydrodynamic Loads

Hd Hdroelasticlty, Proceedings or the Fourthium on Naval Hydrodynamics, Washington, D. C.,

August 1962, Office of Naval Research ACR-73, pp. 490-518.

-30-

21. Crimi, P., and Statler, I. C., Forces and Momentson an Oscillating Hydrofoil, Proceedings of-th-Fourth Symposium on Naval Hydrodynamics, op. cit.,pp. 447-466.

22. Smith, A. M. 0., and Pierce, J., Exact Solution ofthe Neumann Problem. Calculation of Non-CirculatoryPlane and Axially SymmetrTc-F-ows about or withinAriary Boundaies, Douglas Airraft Gompany ReportNo.- , April 958.

23. Pines, S., Dugundji, J., and Neuringer, J., AerodynamicFlutter Derivatives for a Flexible Wing withSuper-'sonic and SUsonic Edges, Journal of the AeronauticalSciences, Vol. 22, No. 10, October 1955, pp. 693-700.

24. Zartarian, G., and Hsu, P. T., Theoretical Studies onthe Prediction of Unsteady Supersonic Airloads onElastic Wings, Parts I and II, U. S. Air Force, WrightAir Development Center. Technical Report 56-97, PartI, Dec. 1955, Part II, Feb. 1956.

25. Watkins, C. E., Effect of Aspect Ratio on the AirForces and Moments of Har 0sciclating ThinRectangular Wings in Supersonic Potential Flow,. A. C. A. Report 102b, 1951.

26. Ashley, H., Supersonic Airloads on Interfering LiftingSurfaces by Aerodynamic Influence Coefficient Theory,The Boeing Company Report No. D2-22067, November 1962.

27. Rodden, W. P., and Revell, J. D., The Status ofUnsteady Aerodynamic Influence Coeff-citens, S. M. F.T-i- Paper No. FF-33, presented at Annual Meeting,Institute of the Aerospace Sciences, January 1962.

28. Landahl, M. T., Linearized Theory for UnsteadyTransonic Flow, paper presented at SymposiumTranssonicum, Aachen, Germany, September Tg2 (alsoissued as MIT Fluid Dynamics Research Laboratory ReportNo. 63-2, March 1963).

29. Bisplinghoff, R. L., et al., An Investigation of Stressesin Aircraft Structures under Dynamic Loadihg,Massachusetts Instituteof Technology, Aeroelastic andStructures Research Laboratory Report for U. S. NavyBureau of Aeronautics, January 1949.

30. Covert, E. E., The Aerodyinamics of Distorted Surfaces,Proceedings of the Symposium on Aerothermoelasticity,U. S. Air Force Aeronautical Systems DivisionTechnical Report 61-645, October 1961, pp. 369-406.

31. Allen, H. J., General Theory of Airfoil SectionsHaving Arbitrary Shape or Pressure Distribution, N. A. C. A.Report 035,--195 5.

-31-

FREESTREAM

TOP V IEW-~ POSSIBLE

/ / "----" "IMAGE

UK

/ POSIL

FREE WATERSURFACE

MIDPLANE OF S,yVERTICALCOORDINATEz (x,y,t )=Zo (y)

-Az (XTy,t) REAR VIEW

Fig. 1 - Top and rear elevations of' a thin, nonplanar liftingsurface performing small unsteady motions normal toa uniform subsonic or supersonic stream flowingparallel to the x-coordinate. The image of this wingin a ground plane or free water surface is shown.

-32-

8

0 MEASURED (22%-THICK AIRFOIL)7 0 0] MEASURED (1I%-THICK AIRFOIL)

- PREDICTED

6 0

5ASPECT RATIO 4

CLa 0

4

3

2 0 ASEC RA IO I

0 .2 .4 .6 .8 1.0 1.2h/Co

Fig. 2 - Steady lift-curve slopes of three rectangular, planewings as functions of distance above a ground planemeasured in wing chords. (Figures 2-7 refer to

constant density fluid.)

-33-

.12 ASPECT RATIO I

.08

.04 RTO

0CM -4 ASPECT RATIO 4

.4 -

- .08 4C

CM

-. 12

-. 20 -

-. 16

0 .2 .4 .6 .8 1.0 1.2 cch/Co

Fig. 3 - Curves of nose-up pitching moment coefficient, takenper unit angle of attack about an axis along thequarter-chord line, for three rectangular wings.Abscissa is distance above a ground plane measuredin chords.

-34-

5 -- __ EZIZ /Co .5

CLa ,.

3

2

01

-45o -30O -150 00 150 3008

Fig. 4 - Steady lift-curve slope vs. dihedral angle, forthe illustrated V-wing at three different centerlineheights h above a ground plane.

-.35-

2.6 I d

2.41

CL

i 2.2 e0.5

2.0 ASPECTRATIO 2,

RECTA NGULAR1.8 PLAN FORM

0 0.1 0.2 0.3 0.4 0.5k

-I = 0. 5

210Co

20(r OD

OL 1900 p

1800

1700 ,..0 0.1 0.2 0.3 0.4 0.5

k

Fig. 5 - Magnitude and phase angle of dimensionless liftdue to vertical-translation oscillation of arectangular hydrofoil, plotted vs. reducedfrequency k = 44/r. Three depths below a freewater surface at high Froude number are indicated.

-36-

0.8

0.6

0.4

0.2- __= .

1.0- DI

0 0.1 0.2 0.3 0.4 0.500 kI

ASPECT

RATIO 10,-200 RECTANGULAR

PLANFORM

-400

-60P Co 10

-800

0 0.1 0.2 0.3 0.4 0.5k

Fig. 6 - Magnitude and phase angle of dimensionlessquarter-chord pitching moment due to pitchingoscillation about the moment axis by a rectangularhydrofoil. Abscissa Is reduced frequency. Threedepths below a free water surface at high Froudenumber are indicated.

-37-

dd

4.0 oz05,F 10

CL. R-----

2.0 J 5, F -o,

0 0.5 1.0k

4.0

2.0 .- .5, F ,. 0

CLa [dlC~aI * d - F = I0-- -05Co "'1.F -oo

_ 0.;5 1.0

-2.0t.

Fig. 7 - Real and imaginary parts of dimensionless lift dueto pitching oscillation of a two-dimensional hydrofoilrunning parallel to a free surface. Four combinationsof depth-to-chord ratio and Froude number are shown.Infinite depth case corresponds to classical oscillatingairfoil theory for incompressible flow.

kv=O,p 0 '

OREGIO b tSHiADED)

- N kG

' 'i ,{ SIDE

EDGE

region Of a planeP . t

tratng rghtV reiAs{p toieflow%,eronhaindiagonals

suFe and ( # .Fig. ectn rectana-e (lines# S identify

elementary the 4ac of integers ae 8 t

paralel o ae Pairs enters Orespec and 3P~i s senals"send"aA

-39-

0 AIC, WITH EDGE SINGULARITYX AIC, WITHOUT EDGE SINGULARITY-SERIES EXPANSION

3.2 ---- EXACT TWO-DIMENSIONAL

2.4

LR,L /A

1.6 o

0.8 0/ ,,,' XX "LTW -IREGION

0 0.2 0,4 0.F 0.8 1.0 1.2DIMENSIONLESS DISTANCEINBOARD FROM WING TIP

Fig. 9 - Four theoretical calculations of the dimensionlesslift per unit span in the wingtip region of aplane, rectangular wing in vertical-translationoscillation normal to a supersonic stream at M7=/2.Reduced frequency based on semichord is4-/. 3.AIC refers to method of velocity potentialaerodynamic influence coefficients with fourarea elements along the chord.

-40-

M>I

MACH LINEFROM VERTEX

x

U.iAIRPLANECENTERLINE

z I MACH CONEFROM VERTEX

WING TIPDIAPHRAGM

Fig. 10 - Top and rear elevations of a triangularwing in supersonic flow with wingtipfolded downward. Shown shaded is asuitable diaphragm area for isolatingthe upper and lower surfaces.

( Figure 11 has not yet

been released for publication. )

Fig. 11

-42-

U(M >)

MAINI WING

TOP VIEWMACH CONEFROM VERTEX

A EDHYPERBOLA

// I D E A L I Z E D \

FUSELAGE

II I / /,/

\ I

IDIAPHRAGM\,J / /IN PLANE OF-- /TIP FIN

REAR VIEW RIGHT SIDE

Fig. 12 - Typical configuration of a hypersonicglider flying at low supersonic speed,with suitable diaphragm areas forisolating the opposite sides of eachlifting surface. Diaphragms areshown shaded.

-43-

I U (MCI)

DIAPHRAGMZ r*beo WING ,REGION

: ,-9 .. .... 4 ... . - 777-'

REAR BOUNDARY OFREGION INFLUENCINGCENTER POINT m,n

Fig. 13 - A plane wing flying at sonic speed.Illustrated is a pattern of square areaelements suitable for application ofoscillatory aerodynamic influencecoefficients. Under strictly linearizedtheory, all areas ahead of the spanwisedashed line can affect the central pointidentified by the integers m,n.

-44-

-0 0 0 0 0 1 I 1 1

'-44

S 0 00oo4.) %-4 r..,- 4.) - 4.)

1:4.) 0 14

44 rI

04.) Hr 0 0,I-H co 0 Ori bo

r- .. 0 1 4-) 04. 0

1 C) ) 46) 11 0 .V 0

> 4-4 A O 0.W4

o c) 0W4V-H 0 4a~*3000 W.4 P '

ca-It q- t2~4 .), 0 4.)0j~ >D 004-) X-C)IC)C

4) ;4 %H H 4)0

01 c" CO) 9 F.i 0

0 V $ '-H0 0H~

04 4 _ :3H- o; 0 HC0 )09w:

-45-

_ _ 0_ 0 a0 f 4)

04.) 0'-

I-. z i t - .0

(-2~1 0-> r. cr-r

>j~ W 4) 0)000

to) H. 4X: Ioz z U 4)4)-

C0 0 o4 a1Z0r-4 0) f24 H0.

04l -H (20-14.)O94-) 0 z5r 4) Q

W W 0 !ZH 0 0) - )

03 ~V M-4 O.)V $tM

w w 4 H4-)

CDO -JN6 6) 6 6) to 6 6I 0 -H Id In I-

ix~I I I)I'S. &ar

-46-

(a) (b)

IL 4

(C) (d) (e)

Pig. 16 - Top and rear viewu of various intersectingconfigurations whose loading can bedetermined by numerical methods of thispaper. Several simple cases like thesehave been analyzed in closed form forcomparison purposes.