VOL. 7, NO. 2, FEBRUARY 1969 AIAA JOURNAL

A Doublet-Lattice Method for Calculating Lift Distributionson Oscillating Surfaces in Subsonic Flows

EDWARD ALBANO* AND WILLIAM P. HODDEN|Northrop Corporation, Nor air Division, Hawthorne, Calif.

Approximate solutions from the linearized formulation are obtained by idealizing the sur-face as a set of lifting elements which are short line segments of acceleration-potential doub?lets. The normal velocity induced by an element of unit strength is given by an integral ofthe subsonic kernel function. The load on each element is determined, by, satisfying nor-mal velocity boundary conditions at a set of points oil the surface. It is seen a posteriorithat the lifting elements and collocation stations can be located such that • the Kutta condi-tion is satisfied. The method obviates the prescription of singularities in lift, distributionalong lines where normal velocity is discontinuous, and is readily adapted for problems ofcomplex geometries. Results compare closely with those from methods that prescribe liftingpressure modal series, and from pressure measurements. The technique constitutes anextension of a method developed by S. G. Hedman for steady flow. J

Nomenclature

AR = aspect ratiob = semichordc — chordK = kernel function ,3C = numerator of singular kernelk = reduced frequency,.&' = atb/UM — freest ream Alach numberNC — number of boxes on chordNS — number of boxes on semispanP = lifting pressurep = dimensionless lifting pressure coefficient, p =W = normal velocity at surfacew = dimensionless normal velocity (uormalwash), w =

W/Ut — times, a — curvilinear spanwise coordinates on the surface; s

also denotes span of planar surface= freestream velocity= Cartesian coordinates

y = dihedral anglew = frequency of oscillationp . = freestream density( " ) = complex amplitude

Introduction

THE linearized formulation of the oscillatory, subsonic,lifting surface theory relates the normal velocity at the

surface

U'0r,«,0 = URl[w(x,8) exi>(«oO]

to the pressure difference across the surface

P(x,s,t) = ±pU2Rl[p(x,s) exp(uo/)l

by a singular integral equation and the Kutta condition at

Presented as Paper 68-73 at the AIAA 6th Aerospace Sciences.Meeting, New York, January 22-24, 1968; submitted February13, 1968; revision received August 20, 1968. This researchwas sponsored by the Northrop Corporation Overhead TechnicalActivity Program.

* Senior Engineer, Structures and Dynamics Research Branch,Research and Technology Section. Associate Member AIAA.

t Consulting Engineer. Associate Fellow AIAA.| At the time of writing, the authors learned that a similar

extension had been developed independently by Stark,22 whichhas been reported in Hef. 23.

the trailing edge (TE):

w(x,s) = i TT fsf k(x£;s,<r',w,M)p(£,<r)d£d(r

. p[XTE(s),8l = 0

(1)

where (x,s)' are orthogonal coordinates on the surface Ssuch that the undisturbed stream is directed parallel to thex axis. • • • .

Rodemich1 has derived an expression for the kernel func-tion for a nonplanar surface in the form

K = (2)

where

= COS[T(S) — T(O-)]

X

?/ ' I

—-cos[7(0-)] ~ "~ si"[7W] |

X* = x —.£ yo = y - yi ZQ == z - T

and Landahl2 has simplified the forms of A^i and K^ to read

p.-riklUl

where

-/./»oo 0 — lk'iU

/., = I -—————————-' - J t i l . (1 + W2)6 / 2

M! - 0/fi - a:0)//32r

cori/( : /3 = (1

ft = (X02 + ^Vi2)1'

2 )3 /2

(3)

(4)

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E. ALBANO AND W. P. HODDEN AIAA JOURNAL

Fig. 1 Coordinate system.

The coordinate system is illustrated in Fig. 1. The symbol/ means that the integral in Eq. (1) is defined in the "finitepart" sense.3

The traditional method for obtaining approximate solu-tions for p when w is given is to assume a series approxima-tion

P

and to determine the coefficients a a by satisfying normal-velocity conditions on the surface. Known properties ofthe lift ing pressure distribution p such as the behavior nearsurface edges are built into the approximation by appropriatechoice of the functions in the series. This technique hasbeen used successfully for many, years, although most appli-cations have been for planar wings without control surfaces.

Consideration of the efforts needed to develop a singlecomputer program to handle a fairly general class of non-planar problems has led the authors to seek a technique thatwould obviate the prescription of the behavior of p alongedges and corners and, in fact, would remove a priori restric-tions on the global behavior of the force distribution. The"box methods" for the supersonic problem provide examplesof the method sought.

Doublet- Lattice Method

We describe a method which is an extension of the one de-veloped for stead}r subsonic flow by Hedman4 in 1965. Theelements of the technique are to be found in the vortex-lattice method of Falkner.5 The present study may be re-garded as an extension of the method of Hedman in ananalogous manner to the procedure by which Runyan andWoolstbn6 extended the method of Falkner to the oscillatorycase. Since it appears at the present time that a rigorousanalytical basis for the method is not available, we presentan operational description as follows.

It is assumed that the surface can be approximated by seg-ments of planes. The surface is divided into small trape-zoidal panels ("boxes") in a manner such that the boxes are

DOWNWASHCOLLOCATION PO!NT~

arranged in columns parallel to the freestream (Fig. 2), andsurface edges and fold lines lie on box boundaries. The J-chord line of each box is taken to contain a distribution ofacceleration potential double ts§ of uniform but unknownstrength. In steady flow, each doublet line segment isequivalent to a horseshoe vortex whose "bound" segmentcoincides with the doublet line.

Let n be the number of boxes and / be the constant forceper unit .length of the J-chord line of a box. The amplitudeof the doublet strength of thejth line segment is

. (/y/4irp) A-d/i

where -djj, is the incremental length and // is the length of theline segment, and the amplitude of the normal velocity(normal wash) induced at a point (xi,Si) on the surface bythe jih doublet line is

Wifafr) = (J^- Un f^KbiBiVidti

The total normal wash induced at point (:r,,St)is the sum ofthe normal washes induced by the n doublet lines

47Tp*} f}/ «Mj

If Eq. (5) is applied at.n downwash points on the surface,the// are determined. The force on the doublet line is takenas the force on the box and the pressure difference across thesurface is approximated by

Pj = force/ (box area) = /,7y/(box area)

= ///AoJy COS A;

where &Xj is the box average chord and X, the sweep angle ofthe doublet line, so that

fj/4:7TpU2 = ^TTpjAXj COS A,

We note that the induced downwash calculated by Eq. (5)will be infinite if the downwash point lies on a doublet linesegment or downstream from its end points. Further-more, trie: Kutta condition has not been imposed.However, from numerical experimentation with this tech-nique, it has become .apparent that the Kutta condition willbe satisfied when each downwash point is the -f -chord pointat midspan of a box.

Equation (5) may finally be written

where

(6)

If-AH are the elements of the matrix whose inverse is thematrix of Di}, then

npi = 2 AijWj

j f - i -provides the approximate solution for the lifting pressurecoefficients.

Generalized force coefficients are computed approximatelyby

Fig. 2 Surface and panel geometry.§ See, e.g., Ref. 7, pp. 189 and 198, or Ref. 8, pp. 19, 130, and

211 for a discussion of PrandtFs acceleration potential.

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FEBRUARY 1969 LIFT DISTRIBUTIONS ON OSCILLATING SURFACES 281

where

Sk = area of box khk(i) = deflection in mode i of J-chord, midspan point of

box kpA.cy) = pressure coefficient in mode ;/' at box A:br = reference semicjiordsr — reference semi span

Working Forms

Approximate evaluation of the integral in Eq. (6) isachieved by approximating the integrand by a simple func-tion. We consider the down wash induced at a receivingpoint R = (XR^R.ZR) by a doublet line segment whose mid-point is Sm and whose inboard and outboard end points areSi and So, respectively. Let

JC =

Denote

Define the coordinate system (Fig. 3)

?? = y cosys + z sin 7,5f = — y smys + z

If the length of the doublet segment is small, it may beanticipated that a parabolic approximation for & along thesegment would be sufficiently accurate for evaluating theintegral;

Arj2 + Bn + C ,/———rr-nr, dri

(7)

where

*?o = (yn — ysm) COST/? + (ZR — zSm) smys

f o = - (VR - ysm) shi7s + (ZR - zSm) 0087,5A = (Ki - 23Cm + OC0)/2e2

B = (3C0 - 3C,-)/2e C = 5CW

The result of the integration is

+ ifeB + tail

where r.i2 = For the planar case (f0 -> Q),

i?o — 770 +

+ 2eA

In order to converge to Hedman's vortex-lattice results forsteady flow and to improve the approximation of Eq. (7),the authors have found it necessary to subtract the steadypart (co = 0) from 3C before applying the preceding formulas,and then to add the effect of a horseshoe vortex. The errorin the parabolic approximation appears to be small if the divi-sion of the surface into boxes is such that the boxes haveaspect ratios of order unity. The steady values of Ki andK2 are

^(eo = 0) = 1 + Xo/R

K2(co = Q) = -2 - (x0/R)(2 + /3V//?2)In any event, it is necessary to evaluate numerically the

kernel function, Eq. (2), at a number of points, and hence toevaluate the integrals l\ and 72, Eqs. (3) and (4), respec-tively. Approximate evaluation of these integrals may beaccomplished in many ways. L. Schwarz9 has given an ex-

Fig. 3 Sending panel coordinate system.

pression for I\ in terms of infinite series. However, we preferto approximate the integrands by simple functions. It issufficient to consider nonnegative arguments because ofsymmetry properties of the integrand. Integrating Eq.(3) by parts gives

' L1

i_ \ — • .-- / __i

Watkins, Runyan, and Cunningham10 have given the formula

t/(l + £2)1/2 » 1 - 0.101 exp(-0.3290 -0.899 exp(-1.40670 - 0.09480933 exp(-2.900 s

which when substituted in the foregoing yields a simple ex-pression for 7i. The integral 72 is evaluated in similarfashion, requiring integration of Eq. (4) by parts twice.

In view of the lack of a rigorous basis for the foregoingassumptions, it is necessary to demonstrate the adequacy ofthe doublet-lattice method by comparison of results withsolutions obtained by other means, and with experimentaldata.

Results for Two-Dimensiqnal Flow

For planar flow, the integral equation (1) becomes theone-dimensional integral equation of Possio (see, e.g., Sec. 6-4of Ref. 7) and the doublet line segments become two-dimen-sional doublets on the chord. The results of the doublet-lattice method presented here have been obtained by dividingthe airfoil chord into equal intervals and locating each sendingand receiving point at the J-chord and -f-chord, respectively,of each interval.

For graphical presentation of results, the values of thelifting pressure coefficient p are plotted at the sending pointsand smooth curves are drawn going to zero at the trailingedge. In describing results, we use the term "boxes" tomean intervals in two-dimensional flow or small panels inthree-dimensional flow, and

NC = number of boxes per surface chordNS = number of boxes per semispan (planar surfaces)

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282 E. ALBANO AND W. P. RODDEN AIAA JOURNAL

. THEORY (EXACT) .0_ _ _o EXPT{REF. ID-

PRESENT METHOD

. 0 0.1 ' 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 O® 1.0

" x/c ' . • '

lm(p)

0 , 0.1 0.2 0.3 0.4 0<.5 -0.6 0.7 0.8 0.9 ' 1.0

Fig. 4 Lifting pressure distribution on airfoil with oscil-lating flap in two-dimensional flow, M = 0, k = . 1.0.

For incompressible flow', exact solutions are available.Solutions for an airfoil with oscillating flap are comparedin Fig. 4, where measurements reported by Bergh11 are in-cluded. < . . „

For compressible flow, Table 1 presents a comparison ofgeneralized forces for a flapped airfoil -from 'the present methodand "from the tables of Ref . 1 2. The coefficients 'are denned by

L « wpU2beikt(Aka -Cke)

Ia-KKtbbNI METHOD; NC = 6, NS=8I0EXPT, NACARMA51G31_____ I

M =

N =Bmb

Bnb

Cme)Cnc)

where

ij =M =

N =Ab =B =

C =

force of airfoil and flap, positive downwardmoment of wing and flap about midchord, positive

tail heavy 'moment of flap about hinge axis, positive tail heavytranslation amplitude, positive downwardamplitude of airfoil rotation about midchord, posi-

tive trailing edge downamplitude of flap rotation, positive trailing edge down

Results shown in Table 1 are for the case M = 0.8, A- = 0.9,T = (flap chord)/(airfoil chord) = 0.3. The tabulated re-sults indicate that the doublet-lattice approximation is validfor high subsonic Mach numbers and reduced frequenciesof order unitv.

Results for Three-Dimensional Flows

Figure 5 compares calculated lift distributions with mea-surements reported by Kolbe and Bbltz13 for a swept, taperedwing at incidence in steady flow. For this problem, theseiniwing was divided into 48 boxes, each containing a horse-shoe vortex as hv Hedman's method. Results for the wingstations shown were obtained by interpolation.

Calculations and measurements for a wing with partial-span flap are.presented in Fig. 6. The curves labeled ."ex-periment" were obtained by reduction of graphical data ofHammond and Keffer,14 and, for this reason, are rather im-precise. . Calculations were made with 80 horseshoe vorticeson the semiwing, eight vortices being on the flap. Landahl15

has shown that both the form and strength of the ~ pressuresingularity at the hinge line can be determined analytically.The expression for the lift distribution for unit flap angle is

p = (-2/7T/3.) cosXe'log|(s - 'x<)/c\ H- 0(1) (8)where xr = stream wise coordinate of hinge, \c = hingelinesweep angle, /?„ = (1 — M2 cos2Xc)1/2, and c = local chord.For the wing considered here, Xc « 30° and J\f = 0.6; thesingular part of Eq. (8) becomes

p « -0.645 log|(z - xe)/c\ (9)The graph of this expression is labeled "local solution" inFig. 6a^ . Figure 6b shows the distribution of lift on theseiniwing calculated by the Hedrnan vortex-lattice technique.

Both kernel function calculations and measurements oflift distribution reported by Lessing, Troutman, and Menees16

are compared in Fig. 7 with results from the doublet-latticemethod. The rectangular wing (aspect ratio = 3) con-sidered is oscillating in a bending mode described approxi-mately by

h » 0.18043| (y/s 1.70255(i//«)2 -•1.13688|(2//s)8| + 0-25387 G//s)4

where h is the nondimensional deflection amplitude. In Fig.7, ah denotes the magnitude of the effective oscillatory angleof attack at the wing tip due to bending.

Table 1 Coefficients for oscillating flapped airfoil

Fig. 5 Lift distribution on swept wing in steady flow.

ka

bb

kcmam,bmr.naHbrik

20 "boxes" .

-0.0075 - 1.1 18i-1.571 - 0.029H

-0.4835 + 0.0789r0.3293 + O.lOOSt

-0.0391 - 0.8604r-0.4075 + 0.00217"

0.0584 - 0.0626t-0.0745 - 0.1456t-0.0912 - 0.0696r

30 "boxes"

-0.0197 - 1.119i-1.574 -0.0795i!-0.4824 + 0.0823^

0.3303 + 0.0924?:- 0 . 0572 - 0 . 8623?-0.4105 -f 0.0058?

0.0591 - 0.0638?:-0.0762 - 0.1474?-0.0919 - 0.0712?

Ref. 12

-0.0444 - 1.120H-1.5755 - 0.0267?-0.4803 + 0.0868;

0.3309 + 0.0760i-0.0946 - 0.8619?-0.4147 + 0.0248t

0.0601 - 0.0661?-0.0798 - 0.1504?:-0.0931 - 0.0739?:

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FEBRUARY 1969 LIFT DISTRIBUTIONS ON OSCILLATING SURFACES 283

Results of computations for another rectangular wing(aspect ratio = 2) with full-span, 40% chord, oscillating flapare shown in Fig. 8. These are compared with kernel func-tion calculations (with built-in hingeline singularity) givenby Curtis, Gikas, and Hassig17 and experimental data ofBeals and Targoff.18 The doublet-lattice calculations weremade with 99 boxes on the semiwing and, of these, 45 boxeswere on the flap.

The rolling moment of the horizontal stabilizer of a rec-tangular T-tail configuration is shown in Fig. 9,^ wheremeasurements by Clevenson and Leadbetter19 are repro-

7.0

6.0

5.0

4.0

3.0

2.0

1.0

- PRESENT METHODNC = 10, NS = 8

- LOCAL SOLUTION, EQN. (9)-i=_10° [ EXPT,-i = + 5° \ NACA RM L53C23

STEADY FLOWM = 0.6

SPANWISE STATION |- = 0.46

. \

0.2 0.4 0.6x/c

a) Comparison with Experiment

0.8 1.0

1.0

b) Calculated Li f t Dis t r ibut ion on Wing

Fig. 6 Lift distribution induced by deflected partial-span

^ Note that the side force results presented in Fig. 9 of AIAAPaper 68-73 were incorrect because of a mistake in the com-puter program. The rolling moment of the horizontal stabilizeris presented here in place of the side force since it is a morecritical measure of the accuracy of an interference theory. Thecomplete solution is given in Kef. 21.

P/a L

0 00 0 °

0.4 0.6X/C

a) Loading at Root, y/s = 0.0

— -0HA

THEORYREAL PART j FypTIMAG. PART [

NASATN-D-344

PRESENT METHOD, NC =8NS =8

M = 0/24k = 0.47

0.2 0.4 0.6 0.8x/c

b) Looding Neor Tip, y/s = 0.9

Fig. 7 Lift distribution on rectangular wing oscillating inbending mode.

^PRESENT METHOD (M = 0.1) NC = 11, NS = 9B KERNEL FUNCTION (M = 0.2) REF. 17A EXPT (0< M< 0.2) REF. 16______

Fig. 8 Flap hinge moment coefficient due to flap oscilla-tion for a 40% chord, full-span flap on a rectangular wing

with aspect ratio 2.

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E. ALBANO AND W. P. RODDEN AIAA JOURNAL

U.J

0.2

L

0.1

0

ir

Qo(CD? <} '

Uo^MuA PRESENT METHOD, NC=5, NS=8O EXPT, NACA TN 4402

°n 0

C

Fig. 9 Rolling moment coefficient of horizontal stabilizerfor simplified T-tail oscillating in yaw about fin midchord,

M = 0.

duced. To account in part for the tunnel wall, the imagesystem of the fin was included in the calculations; the imageof the horizontal stabilizer was neglected. Forty boxeswere placed on the fin and 40 on the horizontal stabilizersemispan. The discrepancy between calculations and ex-perimental data for increasing reduced frequency might bedue to the relatively small number of boxes used or to theincomplete modelling of the effect of the tunnel wall.**The T-tail results illustrate that nonplanar interferenceproblems are easily approached by the doublet-latticetechnique.

Concluding Remarks

Within the context of the linearized, subsonic, lifting sur-face theory, two types of approximations are involved in thedoublet-lattice method. The assumption that for purposesof calculating lift distributions the surface can be representedby a system of line segments of acceleration potential dou-blets is seen to be a valid approximation in view of the resultsobtained. As far as the authors are aware, an analyticalbasis for this approximation has not been established andwarrants further study so that its full implications may bebrought out. Such an investigation would also determinethe extent to which the box chord-lengths on a strip andstrip-widths across the span may be unequal. However,until a rigorous basis is found, further numerical experimentswill be necessary to define the limitations of the method. Thesecond kind of approximation is associated with the evalu-ation of the integrals in the kernel, Eqs. (3) and (4), and inthe normalwash-pressure influence coefficients, Eq. (6).These procedures may be improved and optimized accordingto standard techniques of numerical quadrature.

The advantages of the doublet-lattice method arise frombeing able to disregard the special behavior of the lift dis-tribution where the normalwash is discontinuous. So longas edges do not intersect boxes, a computer program basedon this technique does not need to discriminate among sideedges, fold lines, hinge lines, etc., and this fact is importantwhen problems of intersecting surfaces are considered.Furthermore, since the influence coefficients Z),-,- are inde-pendent of the properties of the normalwash distribution,the same matrix computed for a given wing will yield solu-

** The effects of viscosity and airfoil thickness, neglectedby the linearized theory, undoubtedly also contribute to thediscrepancy.

tions for a large class of normalwash distributions; e.g.,generalized forces for many different control surface con-figurations may be obtained from the same influence coeffi-cient matrix.

For applications in aeroelastic analyses, aerodynamic in-fluence coefficients that relate control point forces to de-flections have been defined by Hodden and Revell.20 Thedoublet-lattice method leads immediately to this definitionof influence coefficients since the control point force is givenby the product of lifting pressure and box area, and the normal-wash is the substantial derivative in the streamwise directionof the deflection. (The substantial derivative requires curvefitting "iii-the-small" along the surface strip; e.g., a parabolamay be passed through the control point and the points up-stream and downstream of it.) If a reduced number ofdegrees of freedom is desired for the aeroelastic analysis overthe number of boxes employed in the aerodynamic analysis,the number of control point forces and deflections may bereduced by a streamwise curve fit and the method of virtualwork as discussed in Ref. 20.

References

1 Vivian, H. T. and Andrew, L. V., "Unsteady Aerodynamicsfor Advanced Configurations. Part I—Application of theSubsonic Kernel Function to Nonplanar Lifting Surfaces,"FDL-TDR-64-152, May 1965, Air Force Flight Dynamics Lab.

2 Landahl, M. T., "Kernel Function for Nonplanar OscillatingSurfaces in a Subsonic Flow," AT A A Journal, Vol. 5, No. 5,May 1967, pp. 1045-1046.

3 Mangier, K. S., "Improper Integrals in Theoretical Aero-dynamics," R&M 2424, 1951, British Aeronautical ResearchCouncil.

4 Hedman, S. G., "Vortex Lattice Method for Calculation ofQuasi Steady State Loadings on Thin Elastic Wings," Kept.105, Oct. 1965, Aeronautical Research Institute of Sweden.

5 Falkner, V. M., "The Calculation of Aerodynamic Loadingon Surfaces of Any Shape," R&M 1910, 1943, British Aeronau-tical Research Council.

6Runyan, H. L. and Woolston, D. S., "Method for Calculatingthe Aerodynamic Loading on an Oscillating Finite Wing in Sub-sonic and Sonic Flow," TR 1322, 1957, NACA.

7 Bisplinghoff, R. L., Ashley, H., and Half man, R. L., Aero-elasticity, Addison-Wesley, Reading, Mass., 1955.

8 Ashley, H. and Landahl, M., Aerodynamics of Wings andBodies, Addison-Wesley, Reading, Mass., 1965.

9 Schwarz, L., "Investigation of Some Functions Related tothe Cylinder Functions of Zero Order," Luftfahrtforshung, Vol.20, No. 12, 1944, pp. 341-372.

10 Watkins, C. E., Rimyan, H. L., and Cunningham, H. J.,"A 'Systematic Kernel Function Procedure for DeterminingAerodynamic Forces on Oscillating or Steady Finite Wings atSubsonic Speeds," R-48, 1959, NASA.

11 Bergh, H., "A New Method for Measuring the Pressure Dis-tribution on Harmonically Oscillating Wings of Arbitrary Plan-form," MP.224, 1964, National Aeronautical and AstronauticalResearch Institute, Amsterdam.

12 "Tables of Aerodynamic Coefficients for an OscillatingWing-Flap System in a Subsonic Compressible Flow," F.151,May 1954, National Aeronautical and Astronautical ResearchInstitute, Amsterdam.

13 Kolbe, C. D. and Boltz, F. W., "The Forces and PressureDistribution at Subsonic Speeds on a Plane Wing Having 45°of Sweepback, an Aspect Ratio of 3, and a Taper Ratio of 0.5,"RM A51G31, 1951, NACA.

14 Hammond, A. D. and Keffer, B. M., "The Effect at HighSubsonic Speeds of a Flap-Type Aileron on the Chordwise Pres-sure Distribution Near Mid-Semi-Span of a Tapered 35° Swept-back Wing of Aspect Ratio 4 Having NACA 65A006 Section,"RM L53C23, 1953, NACA.

15 Landahl, M. T., "On the Pressure Loading Functions forOscillating Wings With Control Surfaces," Proceedings of theAIAA/ASM E 8th Structures, Structural Dynamics and MaterialsConference, AIAA, 1967, pp. 142-147.

16 Lessing, H. C., Troutman, J. L., and Menees, G. P., "Ex-perimental Determination of the Pressure Distribution ou a

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FEBRUARY 1969 LIFT DISTRIBUTIONS ON OSCILLATING SURFACES 285

Rectangular Wing Oscillating in the First Bending Mode forMach Numbers from 0.24 to 1.30," TN D-344, 1960, NASA.

17 Curtis, A. II., Gikas, X. A., and Hassig, H. J., "OscillatoryFlap Aerodynamics—Comparison Between Theory and Experi-ment," paper presented at the Aerospace Flutter and DynamicsCouncil Meeting, Cocoa Beach, Fla., Nov. 1967.

18 Beals, V. and Targoff, W. P., "Control Surface OscillatoryCoefficients Measured on Low-Aspect Ratio Wings," TR 53-64,April 1953, Wright Air Development Center, Wright-PattersonAir Force Base, Ohio.

19 Clevenson, S. A. and Leadbetter, S. A., "Measurements ofAerodynamic Forces and Moments at Subsonic Speeds on aSimplified T-tail Oscillating in Yaw About the Fin Mid-Chord,"TN 4402, 1958, NACA.

20 Rodden, W. P. and Revell, J. I)., "The Status of UnsteadyAerodynamic Influence Coefficients," Paper FF-33, presentedto IAS 30th Annual Meeting, January 22-24, 1962; preprintedin Rept. TDR-930(2230-09)TN-2, Nov. 22, 1961, AerospaceCorp.

21 Rodden, W. P. and Albano, E., "The Subsonic AerodynamicLoads on a Simplified T-Tail Oscillating in Yaw—Summary ofCalculations and Comparison with Experiment," NOR-68-126,Aug. 1968, Northrop Corp.

22 Stark, V. J. E., private communication to W. P. Rodden,Nov. 10, 1967.

23 Landahl, M. T. and Stark, V. J. E., "Numerical Lifting-Surface Theory Problems and Progress," Paper 68-72, 1968,AIAA; also AIAA Journal, Vol. 6, No. 11, Nov. 1968; pp.2049-2060.

FEBRUARY 1969 AIAA JOURNAL VOL. 7, NO. 2

Aerodynamic Shattering of Liquid Drops

A. A. RANGER* AND J. A. NicnoLLsfThe University of Michigan, Ann Arbor, Mich.

New experimental and analytical results are reported for the problem of liquid drop shatter-ing. Breakup is observed to occur as a result of the interaction between a drop and the con-vective flowfield established by the passage of a shock wave over it. The purpose of this in-vestigation, which supplements and extends previous experimental and theoretical studies, isto establish the influence of various parameters on the rate of disintegration and on the timerequired for breakup to occur. Photographic, drop displacement, and break-up time informa-tion is presented for a range of conditions which involve shock waves moving at Mach numbersMa = 1.5-3.5 in air over water drops having diameters of 750-4000^. A model is formulatedfor the breakup phenomenon by considering that it results from a boundary-layer strippingmechanism. The experimental determination of the variation of drop shape and of dropvelocity with time is used together with the analytical results to compute the disintegrationrate.

Nomenclature

A — dimensioiiless interface velocitya = drop acceleration (dW/dt)CD = drag coefficientD = drop diameterM,m = droplet massMs = shock Mach numberP, p = static pressureq — dynamic pressureR = drop radiusRe = Reynolds numberS = drop frontal area (7rD2/4)t = time after collisionT = dimensionless time [t(U2/DQ)(p)ll2\U = fluid velocityu = boundary-layer velocityW = drop velocityx = drop displacementX = dimensionless displacement (a:/Do)a = boundary-layer shape factor(8 = gas-to-liquid density ratio (pg/pi)n = fluid viscosity (also implies micro and micron)

Presented as Paper 68-83 at the AIAA 6th Aerospace SciencesMeeting, New York, January 22-24, 1968; submitted February8, 1968; revision received June 17, 1968. The work reportedhere was conducted under NASA Contract NASr 54(07).

* Assistant Research Engineer, Department of Aerospace Engi-neering; now Assistant Professor, School of Aeronautics, Astro-nautics, and Engineering Sciences, Purdue University. MemberAIAA.

f Professor, Department of Aerospace Engineering. AssociateFellow AIAA.

v — kinematic viscosityp — fluid density

Subscripts

1 = initial conditions2 = shock conditions6 = breakupg = gasI = liquid0 = t = 0r = relative velocityoo = f reestream velocity

I. Introduction

THE fragmentation of liquid drops, resulting from theirsudden exposure to a high-velocity gas stream, has many

important applications in the fields of aerodynamics andpropulsion. For example, the phenomenon of supersonicrain erosion, which is caused by the impingement of rain drop-lets at high relative speeds on exterior missile and aircraftsurfaces, can be greatly alleviated through proper aerody-namic design. A reduction in the damage sustained fromimpacting drops is achieved by designing a body whose de-tached shock is sufficiently far removed to allow for dropshattering in the region separating the shock from the bodysurface. With regard to propulsion, the rate of mixing andcombustion of liquid fuel droplets can be greatly enhanced byvirtue of the fragmentation process. As a result of dropletbreakup, burning rates are obtainable which are higher than

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