Evaluation of lifting surface programs for computing the pressure distribution on planar foils in steady motion

0 P.

E-NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

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nderafdelingderScheepshouwkunde

Technische Flogeschool, Delft DCV-UMENTATIE

DATUM:

EVALUATION OF LIFTING-SURFACE PROGRAMS FOR COMPUTING THE PRESSURE DISTRIBUTION ON

PLANAR FOILS IN STEADY MOTION

by

Thomas J. Langan and Henry T. Wang

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SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT

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Scheepsbouwkunde

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EVALUATION OF LIFTING-SURFACE PROGRAMS FOR COMPUTING THE PRESSURE DISTRIBUTION ON

PLANAR FOILS IN STEADY MOTION

by

Thomas J. Langan and Henry T. Wang

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

TABLE OF CONTENTS

Page

ABSTRACT

ADMINISTRATIVE INFORMATION

..,

1.

INTRODUCTION or . -- Zo., rev 4 4 4 4:4 4C4 0' .6 444.. 9 9 0. 1

LIFTING-SURFACE PROGRAMS

*,..WM,iOl*1,.+,.o,WO0.C..0

.!*.o

4

THE POTENTIAL SOLUTION .. . . 14 .

COMPARISON WITH EXPERIMENTAL RESULTS. ' g ,00,14.110-0,n 37

CONCLUSION .p. 4 000.0.W A* * 43

ACKNOWLEDGMENTS

APPENDIX - SUPPLEMENTARY DATA ON SPANWIU AND CHORDWISE

DISTRIBUTIONS

o40.o.o.o. ...

o,o'o4oWo;o1000...."o.444o.00t,

45

REFERENCES 4,4

LIST OF FIGURES

- Vortex Sheet and Coordinate System 5

Chordwise Elementary Vortex' Distribution 9

Planform for the Tapered Wing 16,

Planform for the 15-Degree Swept Wing 16

Spanwisi Lift Distribution for Tapered Wing

at

11.4-Degree Angle of Attack ... 24

Spanwise Lift Distribution for Cambered Swept

Wing at 5,7-Degree Angle of Attack. oft 2S

Spanwise Distribution of Chordwise Location of Center of Pressure for the Tapered Wing at

11.4-Degree Angle of Attack .... 4 . 27

Spanwise Distribution of Chordwise location of Center of Pressure for the Cambered Swept

Wing at 5.7-Degree Angle of Attack . - 28

Spanwise Distribution of Induced Drag for the

Tapered Wing at 11.4-Degree Angle of Attack -... '29 Chordwise Variation of the Pressure Jump

Coefficient at y' = 0.58 for the Tapered,

Wing at 11.4-Degree Angle of Attack 30

Variation of CL with Respect to NSM far the

Tapered. Wing at 11.4-Degree Angle of Attack 32

Figure Figure 2

Figure 3

Figure 4 Figure, 5 Figure .6 Figure 7 Figure 8, Figure 9 Figure 10 Figure 11 1 -1

Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure 26 Figure 27

-Spanwise Distribution of Chordwise Location of Center of Pressure for the Swept Wing Planform, a = 1 Radian

Spanwise Distribution of Chordwise Location of Center of Pressure for the Swept Wing at

0-Degree Angle of Attack

Chordwise Variation of the Pressure Jump Coefficient at y' = 0.14 for the Tapered Wing at 11.4-Degree Angle of Attack

Chordwise Variation of the Pressure Jump Coefficient at y' = 0.80 for the Tapered Wing at 11.4-Degree Angle of Attack

Chordwise Variation of the Pressure Jump Coefficient at y' = 0.95 for the Tapered Wing at 11.4-Degree Angle of Attack

Chordwise Variation of the Pressure Jump Coefficient at y' = 0.20 for the Uncambered Swept Wing at

1 Radian Angle of Attack

Chordwise Variation of the Pressure Jump Coefficient at y' = 0.20 for the Swept Wing at 5.7-Degree

Angle of Attack Page 55 56 57 58 59 60 61 Figure 12 - Variation of CLa with Respect to NSM for the

Swept Wing

Figure 13 - Spanwise Lift Distribution for the Tapered Wing at 11.4-Degree Angle of Attack

Figure 14 - Spanwise Lift Distribution for Tapered Wing at 11.4-Degree Angle of Attack

Figure 15 - Spanwise Rate of Change of Lift with Respect to Angle of Attack

Figure 16 - Spanwise Rate of Change of Lift with Respect to Angle of Attack

Figure 17 - Spanwise Lift Distribution on Swept Wing at 0-Degree Angle of Attack

Figure 18 - Spanwise Lift Distribution on Swept Wing at 0-Degree Angle of Attack

Figure 19 - Spanwise Lift Distribution on Swept Wing at 5.7-Degree Angle of Attack

33 47 48 49 50 51 57 53 Figure 20 - Spanwise Lift Distribution on Swept Wing at

5.7-Degree Angle of Attack .... 54

-LIST OF TABLES, q

Page

iv

41 Table 1 - Major Differences in ,Functional Approaches

Table 2 Downwash Figures for Cambered Wing Cases,

12

La

Table 3 - Tapered Wing at 13-4-Degree Angle of Attack 19

Table 4 - tncambered Swept Wing

...

20.20

Table $, - Cambered Swept Wing at 0-Degree Angle of Attack'. 21

'Table - Cambered Swept Wing at 5.7-Degree Angle of Attack. 22

Table 7 Nomenclature for Figures in Appendix 46

-6

-ABSTRACT

The pressure differences, forces, and moments on the game two wings in steady subsonic flow were computed by 15

different programs. These results were compared with each

other and with available experimental data. Nine of the programs gave predictions that were generally in close

agreement with each other and with the experimental results,

four showed moderate differences, and the remaining two

showed significant differences.

ADMINISTRATIVE INFORMATION

This work was authorized under the Hydrofoil Development Program of the Systems Development Office of the Naval Ship Research and Development Center (NSRDC) and funded under Naval Ship Systems Command Subproject S4606,, Task 1703.

The Material was originally presented at a Symposium on Computers and Fluids, Brooklyn Polytechnique Institute, on 4 January 1973

INTRODUCTION 1

2

Since Falkner1 and Multhopp- first developed methods for calculating

the pressure distribution on wings in subsonic flow, many modifications and several alternate approaches have been developed. Landahl and Stark3 -present a comprehensive review of these various approaches, their problems, and the progress' made in resolving some of these problems for both the

steady and unsteady flow cases. In the steady case, these approaches are

based on two different, integral equations, the acceleration potential (the,

1Falkner, V. M., "The Calculation of Aerodynamic Loading on Surfaces of Any Shape," AerOnautical Research Council, Technical Report REM 1910 (1943), A complete listing of references is given on pages 62-64,

2Multhopp, H., "Methods for Calculating the Lift Distribution of Wings," Royal Aircraft Establishment Report Aero 2353 (1950),

3Landahl, M. T. and V. J. E. Stark, "Numerical Lifting-Surface Theory Problems and Progress," Journal of the American Institute of Aeronautics and Astronautics, Vol. 6, No. 11, pp, 2049-2060 (1968).

-more popular one) and the velocity potential. Neither has been solved analytically except in the case of a circular wing; moreover, the

diffi-culties involved in obtaining a numerical solution have required the use

of high-speed digital computers. Presently over 20 different computer

programs are available for obtaining solutions to the integral equations

and then calculating the pressures, forces, and moments on wings in steady

subsonic flow.

The purpose of the present study was to evaluate as many of these

programs as possible. During the fall of 1971, all government laboratories

and aircraft companies in the United States which were known to have a lifting-surface program were contacted and requested to compute wing coefficients and pressure distributions on two wings (aspect ratio 5) for which experimental data were already available. Because of time limitations

and probable logistic problems, no attempt was made to include programs

from outside the United States. The response to this request made available

computations of the distribution of the pressure difference, the spanwise lift and center of pressure distribution, and the total lift coefficients and pitching and bending moment center of pressure for the same two wings by 15 different programs; these calculations did not include either

thick-ness or viscous effects. Three of the lifting-surface programs were run

on the CDC 6700 computer at NSRDC; with the exception of the Lamar program, which was run by the McDonnell Douglas Corporation, the remaining programs were run by the organizations which developed them. Not only did this

arrangement eliminate the problems of adapting the computer programs to the

NSRDC computer, but--hopefully--it assured the proper execution of the

calculations. The present paper compares the computed results from the

4Lamar, J. E., "A Modified Multhopp Approach for Predicting Lifting Pressures and Camber Shape for Composite Planforms in Subsonic Flow," National Aeronautics and Space Administration, TN D-4427 (1968).

15 different programs and their agreement with the experimental results of Thiel and Weissinger5

and

Jacobs.67,

Since the computations were made gratis, the number Of wings was

limited to two. An aspect ratio of 5 was chosen because it is in the range

of interest to hydrofoil designers, yet it is not 50 large that it causes convergence problems along the lines discussed by Jordan.9' One wing had taper but neither sweep, of the quarter-chord line nor camber, and the

other had sweep and camber but no taper. Since the experiments were done at low Mach numbers, (0,15 and 0.12), compressibility effects were less than 1 percent, and programs which did not make compressibility corrections

could still be evaluated. Moreover, since the two-dimensional experimental

data for the wing cross sections obtained from Abbott and von Doenhoffi° exhibit only a small difference from the two-dimensional thin wing potential results, viscous and thickness effects an the wing loadings are probably.

small

or

mutually cancel to a large extent,,

5Thiel, A. and J. Weissinger, "Pressure-Distribution Measurements on a Straight and on a 35° Swept-Back Tapered Wing," National Advisory Committee for Aeronautics TM 1126 (1947).

6Jacobs, W., "Pressure-Distribution Measurements on Unyawed Swept-Back Wing," National Advisory Committee for Aeronautics, TM 1164 (1947).

7jacobs, W., "Systematisch Sechskomponentenmessungen an Pfeilflugeln," I Mitteilung, Ingenieur-Archiv, Vol. 18, pp. 344-362 (1950).

8Jacobs, W., "Systematische Druckverteilungsmessungen an PfeilflUgeln Konstanter Tiefe bei Symmetrischer und Unsymmetrischer AnstrOmung,"

II Mitteilung, Ingenieur-Archiv, Vol. 19, pp. 83-102 (1951).

9 - -

-Jordan, P. F., "Remarks on Applied Subsonic Lifting Surface Theory,"' Jahrbuch der WGLR (Germany), pp. 192-210 (1967).

1 °Abbott, _{I. H. and A. E. Von Doenhoff, "Theory of} Wing Sections Including

a

Summary of Airfoil Data," Dover Publications, Inc., New York (1959).

LIFTING-SURFACE PROGRAMS

The programs are all based on the assumption that the flow is steady,

irrotational, and attached. When compressibility effects of the fluid are

included, they are represented by application of the Prandtl-Glauert similarity rule, i.e., by a contraction of the x-coordinate, where the x-coordinate is in the direction of the free-stream velocity (Figure 1). All but one of the lifting-surface programs were developed to solve the

same integral equation. This equation relates the distribution of the

pressure differences over the wing to the normal velocity or the so-called

downwash induced by the wing. It may be expressed as follows:

w (x, y) = p K (x - y - 7, M) d dri

4pU SW

where the integration is over the surface SW. Except in the Rubbert program,11 this surface is the projection in the xy-plane of the wing at

zero angle of attack (Figure la). Rubbert uses the wing mean camber

sur-face for S (Figure lb). The downwash w is given by the product of the

free-stream velocity U with the local streamwise slope of the mean camber surface; the kernel function K is a function of Mach number M and position

relative to the point (x, y, 0). The symbol Ap is the unknown pressure

difference. The remaining lifting surface program by Haviland12 was

developed to solve a similar integral equation which relates a jump in the velocity potential through the wing and the wake region to the downwash velocity.

Five of the programs for solving Equation (1) are based on the

vortex lattice method of Falkner.1 In this approach, the wing and wake are replaced by an equivalent vortex sheet whose vorticity is then concentrated

, into a finite number of horseshoe vortices arranged in a lattice. The

4

(1)

11Rubbert, P. E., "Theoretical Characteristics of Arbitrary Wings by a Non-Planar Vortex Lattice Method," Boeing Company Report D6-9244 (1964).

1

2Haviland, J. K., "Downwash-Velocity Potential Method for Lifting

Surfaces," Journal of the American Institute of Aeronautics and Astronautics, Vol. 9, No. 11, pp. 2268-2269 (1971).

tag

Figure la - Relation of Wing to Vortex Sheet in xy-Plane

Figure lb - The Rubbert Nonlinear Vortex Sheet Figure 1 - Vortex Sheet and Coordinate System,

-strengths of these horseshoe vortices are assumed to vary from one another in both the spanwise and chordwise directions, and this variation is repre-sented by a finite series of functions with unknown coefficients. Falkner computed the downwash due to each of these vortices at strategically placed control points and equated the sum of the downwash values to the known

downwash at these points. By solving the resulting equations, he was able

to determine the coefficients for the leading terms of the series and to

compute wing loadings based on the truncated series. The number of terms

in the series was e ual to the number of control points.

Tulinius13 extended the Falkner theory to the case where the bound vortices are swept, and he has programmed the resulting theory. His number of control points exceeds the number of terms in the series; hence, the

system of equations is over-determined. A least-squares technique is used

to solve them. By using a modern digital computer (a CDC 6600), Tulinius

is able to consider 420 vortices with 99 control points. In the

calcula-tions presented in the sequel, he used 420 horseshoe vortices whose strengths were related by six chordwise and six spanwise functions.

Dulmovits,14 Margason and Lamar,15 and Giesing16 developed similar

modifications of the Falkner method. Instead of assuming a series for the

spanwise and chordwise distribution of the vortices, they use a straight-forward vortex lattice method, that is, they let the strength of each

1

5Tulinius, J. R., "Theoretical Prediction of Thick Wing and Pylon-Fanpod Nacelle Aerodynamic Characteristics at Subcritical Speeds," North American Rockwell Report NA-71-447 (1971).

14Dulmovits, J., "A Lifting Surface Method for Calculating Load Distri-butors and the Aerodynamic Influence Coefficient Matrix for Wings in

Subsonic Flow," Grumman Aircraft Engineering Corporation Report ADR 01-02-64.1 (1964).

15Margason, R. J. and J. E. Lamar, "Vortex-Lattice Fortran Program for Estimating Subsonic Aerodynamic Characteristics of Complex Planforms," National Aeronautics and Space Administration TN D-6142 (1971).

16 .

Glesing, J. P., "Lifting Surface Theory for Wing-Fuselage Combinations,"

McDonnell Douglas Report DAC-67212, Vol. 1 (1968).

vortex be an unknown. Using the Biot-Savart law, they compute the

down-wash due to each horseshoe vortex at each control point. The sum of the

downwashes at each control point is set equal to the local slope of the mean camber times the free-stream velocity; the resulting system of

equa-tions is a set of linear simultaneous equaequa-tions for the unknown vortex

strengths. By solving this system of equations, they are able to determine

the vortex strengths and, in turn, the spanwise and chordwise loading on the wing.

Although Rubbert11 also lets the strength of each horseshoe vortex be an unknown, the nonlinear aspects of his vortex lattice method distin-guishes it from the Dulmovits, Margason-Lamar, and Giesing approaches. Rubbert approximates the mean camber surface at any angle of attack,

together with its wake, by a polyhedral surface (Figure lb). Each face of

the polyhedral is a quadrilateral with two of its sides parallel to the

xz-plane. Rubbert places a bound vortex line of unknown strength along

the quarter-chord of each quadrilateral and a semi-infinite vortex curve of the same strength trailing off from each end of the bound vortex line. The trailing vortices are parallel to the xz-plane and lie on the polyhedral

surface. Aft of the trailing edge, the polyhedral surface is bent at different steps until it is parallel to the xy-plane, at which point the

trailing vortices go off to infinity parallel to the xy-plane. The bending

is accomplished by an empirical method deduced from computer experiments

with the program. The strengths of the vortices are obtained from

satis-fying the tangential condition at the mid-three-quarter chord point of each quadrilateral; the remainder of the problem is solved as in the other vortex lattice methods.

Lopez and Shen17 developed the method of elementary vortex

distri-butions (EVD). They replace the wing planform by a polygonal approximation

in which the lines are either parallel to the x-axis or to the y-axis. The

17

Lopez, M. L. and C. C. Shen, "Recent Developments in Jet Flap Theory and Its Applications to STOL Aerodynamic Analysis," American Institute of Aeronautics and Astronautics Paper 71-578 (1971).

(2)

8

approximate planform is then partitioned into a set of N rectangles, and the spanwise partition is obtained by lines parallel to the x-axis which span the entire chord of the approximate planform. A vortex distribution is placed over each pair of rectangles which are adjacent in the chordwise

direction. On each pair of rectangles, the distribution is uniform in the

spanwise direction and triangular in the chordwise direction with a peak

at the common side (Figure 2). An additional vortex distribution is added

to the triangular distribution over the rectangles that intersect the lead-ing edge of the planform; see the shaded area in Figure 2. The strength

of the nth elementary vortex distribution is unknown for n = 1, ..., N.

However, a system of N simultaneous linear equations is obtained by equating the known downwash at N arbitrary points to the downwash at each point due

to the N elementary vortex distributions. The unknown strengths are then

obtained by solving the system of equations.

Haviland12 calculates the subsonic forces on rectangular wings by using the relationship of the velocity potential jump to the downwash

instead of the pressure difference relationship, Equation (1). The integral

equation relating the velocity potential jump

0

to the downwash w is:

,s2

w (x, y) =

j717t4 (E, n)

[0( E)2 a2 (Y - n)2

9,2

-1/2

+ a2 z2] dE dn

where the integration is over the wing and wake and where a is a Mach

number correction. In his calculations, Haviland replaces the wing by a

system of rectangles similar to Lopez and Shen; he uses a finite wake of the order of 50 chord lengths and also partitions it into rectangles. He assumes a uniform potential jump over each rectangle on Sw, Figure la, and picks the number of points at which to satisfy the tangential flow

condition equal to the number of rectangles. Thus he replaces the integral

equation by a system of linear equations and solves to obtain the potential jump over each rectangle; in turn, the chordwise pressure distributions may be computed.

\

PANEL VORTICITY

DISTRIBUTION

-- --

-- TOTAL VORTICITY

DISTRIBUTION

ri

Lik/A

.A

I

A

i I

I 11 4

4

7 n - 1 7 n+ 1

Figure 2 - Chordwise Elementary Vortex Distribution

L.E. T.E.

A second general approach to lifting-surface theory is to replace the unknown pressure difference in Equation (1) by a truncated functional

expansion with unknown coefficients. Such a series takes the form

MN

Ap = E

Ea

A.

-

712 G(n) Hn(E)

m=1 n=0 nm

where the a 's are the undetermined coefficients. The Gm's are the assumed

TIM

spanwise functions or so-called spanwise modes, and the Hn's are the

chord-wise modes. If one then picks points (xk, yk) at which the tangential flow

condition is to be satisfied (the so-called pivotal points), the integral equation (Equation (1)) reduces to the linear system

M N

w (xk, yk) = E E

Inm (xk' yk) anm m=1 n=1

The function I (x, y) is defined as follows:

nm

-t

2/ Inm(x,y) =

ff

PC---771-1(y-n)2] Gm(n) Hn() [1 + Sw 10

(x-W13

_{/ dn} -Q2/132 4' (Y-T1) 18

Rowe, W. S., "Collocation Method for Calculating the Aerodynamic Pressure Distributions on a Lifting Surface Oscillating in Subsonic Compressible Flow," American Institute of Aeronautics and Astronautics

Symposium on Structural Dynamics and Aeroelasticity, pp. 31-45 (1965). (3)

where 8 is the Prandtl-Glauert correction factor. Equation (4) is solved

for the a by either a direct or a least-squares method depending on

nm

whether the number of pivotal points is equal to or greater than NxM. In addition to the 1/(y - n)2 singularity in Equation (5), there

is

also a logarithmic singularity in the integrand. The various programs

which use an assumed mode distribution for the pressure differ in their

treatment of these singularities. The methods also differ in the choice of

chordwise and spanwise modes as well as in their number. Jordan9 and Rowe18

have shown that the number of spanwise and chordwise modes is very critical (4

to the stability of the calculation. The methods are also sensitive to the choice of pivotal points; this problem is approached with various degrees

of sophistication. Table 1 indicates the major differences between the

six functional approaches for which data are presented in this report. The methods on which these programs are based are described in Jordan,9 Lamar,4

Widnall, 19 Rowe,18 and Cunningham;20 the theory for the Bandler program21 is described in Richardson.22

Jacobs and Tsakonas23 use the complete set of orthonormal functions

(I) = 1 - cos cp

0

4)1 = 1 + Zoos

(1)

m(E)

= cos mcP, m = 2,3,...

on the interval (0,

7)

to obtain a Fourier expansion of both sides of

Equation (1). For a given spanwise station y and local chord c(y),

cos

(1) = (-

+ Tilid)/c(y)

19 .

Widnall, S. E., "Unsteady Loads on Hydrofoils Including Free Surface Effects and Cavitation," Ph.D. Dissertation, Massachusetts Institute of Technology (1964).

20Cunningham, A. M., "An Efficient, Steady, Subsonic Collocation Method for Solving Lifting-Surface Problems," Journal of Aircraft, Vol. 8, No. 3, pp. 168-176 (1971).

21Bandler, P. A., "A Program to Calculate the Pressure Distribution on a Hydrofoil of Finite Span near the Free Surface," Engineering Research

Associates Report 53/4 (1966).

22Richardson, J. R., "Pressure Distribution on a Hydrofoil of Finite Span near the Free Surface," Engineering Research Associates Report 53/3

(1966).

23Jacobs, W. R. and S. Tsakonas, "A New Procedure for the Solution of Lifting-Surface Problems," Journal of Hydronautics, Vol. 3, No. 1, pp. 20-28

TABLE 1

MAJOR DIFFERENCES IN FUNCTIONAL APPROACHES

24Mangler, K. W. and B. F.

Spencer, "Some Remarks on Multhopp's Subsonic Lifting Surface Theory,"

Royal Aircraft Establishment TM 2181 (1952).

25Watkins, C. E. et al., "A Systematic Kernel Function Procedure for Determining Aerodynamic Forces

on Oscillating or Study Finite Wings at Subsonic Speeds,"

National Aeronautics and Space Administration

Technical Report R48 (1959). Program lin(E) cos 0 . U -. ;flid/C Gm(n) n i cos 8

Pivotal Points. ym Coordinate

Integration Jordang cot /2, n = 0 cot 0/2 - 2 sin 0, n . I H (0 ne n = , ... n-1 - 2 sin no. 2

(sin fim/sin e) sin (M . 1) 0

sin am/M . 1

Integration method of Mangler and Spence,24 which treats the logarithmic singularity analytically.

(M 4.1) cos (M + 1) 65 (cos 6 - cos em)

Lamar4

cot 1/2. n . 0

sin no, n . 1, 2, ...

Spanwise modes are defined at only M discrete spanwise stations.

sin mr/M . 1

Multhopp2 quadrature in the spanwise direction with a logarithmic singularity cor- rection; 20 to 30 point Gaussian quadrature in chord- wise direction.

WidnallIg cot 0/2, n . 0 sin no, n . 1, 2, ... m Ti Arbitrary choice

Gaussian quadrature for chordwise and for non- singular part of spanwise. Singular part of spanwise is done by means of finite parts as described by Watkins CC

al.25 Sandler21 [cos nO . cos (n -1).1,1 in Ti

cos (me/(2 M a-1)]

Gaussian quadrature in both directions with a correc- tion for the logarithmic singularity and with the integration and pivotal points interdigited.

sin 0 Rowe18 cot 0/2, n . 0 sin 110, n .

li

2, ... nm

Roots of the (M . 1) orthogonal polynomial having the weighting function

,ç2 2

Mehler-Gauss quadrature formulas in both direc- tiarawith the pivotal and integration stations interdigited.

Cunningham20 CO - OM * 0]'", n ' 0 e(1 - 01/2, n . 1, 2. ... nm sin mm/M + 1

Mehler-Gauss quadrature formulas in both direc- tions with the pivotal and integration stations interd1gited.

For the

treatment of the singular- ity, see Cunningham.20

*The x coordinate

weighting function

is arbitrarily chosen in the Widnall program; Rowe uses the roots of the

(N - 1) degree orthogonal polynomial having the

use 0fl . 21n/(2 N

..

I).

1(1 . x)/(1 - x).

All other programs

-.

-1

-The coefficients in the expansion of the right-hand side of Equation (1) are in terms of a double integral of .a kernel function times the unknown,

pressure difference. By expanding the pressure, difference in terms of the

Birnbaum, chordal modes, Jacobs and Tsakonas are able to perform the

chord-wise integration analytically and thus obtain the following Fourier expan-sion for Equation (1):

E cDm(x) Wm(y) = E

Mx)

I G(r) K. (y'

n) dn,

n m=G JTr=0 n=1 where clp (cb) W(4), 3")

4

y, n) =

fa iln(0d

I

m(0)

K(0, C, with

Hn representing the nth Birnbaum mode,

Equating coefficients in Equation (6) yields, an infinite system of line integrals which can be written

in

the form

'my) E

I G(h)

(Y, TO c1T1

n=1

Jacobs and Tsakonas terminate the number of these equations at N, the num-ber of chordwise modes,, and divide the wing into M spanwise strips oh each

of 'which they assume a constant spanwise loading. The set of integral

equations is thereby reduced to the algebraic system

"N

M

W (y.) = . G'

.m , . n

n=1.J=1

which is solved by a collocation method with pivotal points yl, yn.

1 Tr and

n)

(8) (6) Wm (y)

y;

d = Kmn (n.) K -mn

Lopez of Douglas Aircraft Company has developed a program for an IBM 360 which is based on a simple method by Kiichemann26 for calculating

spanwise and chordwise loadings. In this method, the distribution of the

pressure jump is assumed to have the form

Apg,

n)

= - [sin

7v(n)/7v(n)] ci(n)

[(1 -

)/]v(11)

where c is the section lift coefficient and v is an empirically prescribed

function of aspect ratio, sweep, taper, spanwise coordinate, and the

two-dimensional lift coefficient slope. Once v is obtained, the only unknown

in the above equation is

cz(n).

The resulting integral equation relating

c (n) to the known downwash yields an equation similar to the Prandtl lifting-line equation.

THE POTENTIAL SOLUTION

Although some of the programs can account for thickness, no thickness effects were included in the results of the calculations reported herein.* For these calculations, all the programs solved the problem of potential

flow around an infinitesimally thin wing. An exact solution to this problem

yields a pressure jump Ap(x, y)

which

satisfies the integral equation,

Equation (1). By substituting the pressure jump predicted by a program

into Equation (1), one obtains a prediction for the downwash w. The predicted

downwash could be compared with the known downwash at points other than the

26..

Kuchemann, D., "A Simple Method for Calculating the Span and Chordwise Loading on Straight and Swept Wings of Any Given Aspect Ratio at Subsonic 'Speeds," Aeronautical Research Council Technical Report R&M 2935 (1956).

*If thickness is taken into account by the usual linear approximation,

there is no contribution to Ap; on the other hand, if the nonlinear boundary

problem for a wing with thickness is solved, there is a contribution to Ap

because of the thickness. Since hydrofoil designers are especially concerned

with the pressure near the leading edge of the hydrofoil, they may need to

consider nonlinear thickness effects in computing pressure distributions on hydrofoils to obtain cavitation buckets.

control points and a measure of the error computed. This measure could be in terms of the maximum absolute error, the total absolute error, a least-squares error, or whatever measure one wishes to devise; in any case, the measured error would be due not only to errors in the predicted pressure jump but would also contain an error due to the numerical evaluation of

the integral. There would be no way to separate the integration error from

that due to the error in the approximate pressure jump.

Because of this integration error, a direct measurement of the error in the pressure jump is desirable; moreover, the primary purpose of these programs is to obtain the pressure jump on a given wing. No analytically derived pressure-jump distributions are available for comparison except in the case of a circular planform wing and it cannot be used to determine the

accuracy of a program at aspect ratio 5. Moreover, a number of the programs

cannot accurately predict forces on a circular planform wing since they were designed to handle only straight edges.

The present section of this report presents comparisons between the overall force coefficients, spanwise distribution of lift, pitching center of pressure, and induced drag, and chordwise pressure-jump distributions predicted by the 15 different programs; together these programs include a

minimum of eight different methods for solving Equation (1). If the results

of these calculations agreed, say, to five or, better, to seven significant figures, then one could conclude that Equation (1) has been accurately

solved from a mathematical standpoint; on the other hand, agreement to three significant figures is sufficient for most engineering applications.

The first wing studied is a symmetric uncambered wing with the

plan-form shown in Figure 3. It has a taper ratio of 0.5, that is, the ratio of

the tip chord length to the root chord length is 0.5. Its quarter-chord

line has zero sweep and its tips are parallel to the free-stream flow. Calculations were made for this wing at an angle of attack of 11.4 degrees

(0.19897 radians). For those programs which applied the Prandtl-Glauert

correction for compressibility, a Mach number of 0.15 was used; otherwise, the calculations were made for zero Mach number.

The second planform studied is shown in Figure 4. It has no taper (that is, a taper ratio of 1) but does have a sweep angle of 15 degrees.

mom, ammo mmm .mommt mmm mmor .mm mm 3.317°

Y

2.0 11.308° = 3 75 2

Figure 3 - Planform for the Tapered Wing

16

15.0°

= 5.0

---.111

Figure 4 - Planform for the 15-Degree Swept Wing

Ct = 1.0

= 2.0 15.0°

-a wing with this planform but no camber; the wing angle of attack was

1 radian in order to compute the rate of change of lift with respect to

angle of attack CLa Further calculations were made for a wing with this planform and an NACA 230 camber line. Two angles of attack were used for

these calculations, zero and 5.7 degrees. The downwash figures for these

two configurations are tabulated in Table 2. For those programs which

did not include camber, calculations were made for only the 1-radian angle

of attack. Mach 0.12 was used where possible; otherwise, the calculations

were for zero Mach number.

Tables 3-6 present the loading characteristics for the two wings as computed by the 15 programs and as obtained from the experimental results.

The programs are identified by name. In those cases where the program was

run with a different number of chordwise and spanwise modes, the results for the most representative number of modes are listed in the upper part

of the tables, and the remaining results are listed in the lower part. Mach

number for which the calculation was made, lift coefficient CL, pitching

moment center of pressure (PMCP) measured from the wing apex divided by the root chord, bending moment center of pressure (BMCP) measured from the wing

centerline divided by semispan, as well as the induced drag CD, are presented

for each computer run. Table 4 gives the rate of change of these quantities

with respect to angle of attack rather than the quantities themselves. For

cases where the program was run for two Mach values, the results for the correct experimental Mach number are listed in the upper part and those for

zero Mach number in the lower part. Blanks indicate that the particular

quantity was not directly available from the computer output.

In addition to presenting physical quantities, these tables provide three additional columns of information pertaining to the computations.

NCM x NSM refers to the number of chordwise rows of discrete elements versus

spanwise rows for the discrete element approaches. It refers to the number

of chordwise versus spanwise modes for the modal approaches. Computer time

is in terms of the execution time on the computer indicated in the last

column, except when the time is suffixed with a C to indicate that time for

Fortran compilation was included. It is not feasible to estimate execution

times for these programs since available compilation times for other programs varied as follows:

TABLE 2 - DOWNWASH FIGURES

FOR CAMBERED WING CASES

x' 0.0 0.0125 0.025 0.05 0.075 0.10 0.15 0.20 0.25 0.30 1.00 al -0.30508 -0.26594 -0.22929 -0.16347 -0.10762 -0.06174 +0.00009 +0.02203 +0.02208 +0.02208 +0.02208 a2 -0.20560 -0.16646 -0.12981 -0.06399 -0.00814 +0.03774 +0.09957 +0.12151 +0.12156 +0.12156 +0.12156 Notes: x'

is the distance measured from

the leading edge of the wing,

in the streamwise direction x,

expressed as a fraction of

the local chord c.

al corresponds to

the slope of the NACA

230 mean line. a2 corresponds to a constant downwash of 5.7 degrees = 0.09948

radians added to the NACA

230

mean line, i.e.,

a2(x1)

= a1(x') +

TABLE ,3 - TAPERED WING AT 11.,4=DEGREE ANGLE OF ATTACK Program Mach PMCP MCP' COf NCM x INSM Time sec 'Computer I Tulinius 0.15 0.817 0.240' 0.425 ' 0.043 42O CDC 6600 Dulmovfts 1 0.831 0.240 0.431 H 10 x 15 96 IBM 360/75 Margason-Lamar 0.831 0.240' 0.432 10 x 12 142 CDC 6700(0 i Giesing 0.15 0.829 4,237 . 1 0044 Ti x 17 120C IBM! 360/65' Rubbert 0.00 1, 0.798 4.238 0.420 ' 0.041 10 x 15 117 CDC 6600

Lopez & Shen I 0.00,

! 0.814 0.238 1 0.042 8 x.17 105C IBM 360/65 Haviland '0.15 0,848 0.242 ' 0.438 H Si x 8, 291 CDC 6400 Jordan DAD 0.820 0.211 0.043 2 x 15 H 9' UNIVAC 1108 Lamar 0.00 0.806 0,239 1 '0,423 0.041 4 x 13 98 1 IBM 360/65 Widnall 0.15 0.812 ' 0.218 0.431 10 x a 63 CDC 6700(f) Bandler 0.00 0,807 0.238 k 0.042 , 4 x 4 8 CDC 6700(f) Rowe 0.15 0.815 0.239, 1 0,428 4 x 6 1 13 CDC 6600 Cunningham 0,15 0.839 0.235, ' 1 0.424 4 x 5 14 IBM 370/155 Jacobs-Tsakonas 0..00 0A20 0.264, 0,439 H s(b) 40C CDC 6600'

Lopezk(KUchemahp) 0.15 0.852 0.242 0.047 16(c) I2C IBM 360/65'

_

" Experiment' 0.15 0.83 0.28

0.44

Additfonal Computer Runs

Giesing ' 0,00 0.823 0,240 0,044 1 11 x 17 121C IBM 360/75 [ , I Jordan 0.808 0.212 10.042 2 x '7 5 UNIVAC. 1108 1 1 0_819 0.211 0.043 I 2 x 15(d) 10 1 1 I 0.817 0.213 0.043 2 x 31 H 28 0.811! 0.211 0.042 3 x 15 31 Jordan, 0.00 0.798 0.213 0,041 4 x 15 68 UNIVAC 1108 Cunnimghim 0,15 0.849 0.233 0.425 1 3 x- 3 ID 18M 370/155

Lopez 0.00 0.845 0.242 [ 0.046 , 16(c) I2C IBM 360/65

Wtdnal1 0.15 0.826 0.217 0,426 10 x 4

71 CDC 6700

I

Notes:

(a) rulinius uses 420 horseshoe vortices with 6 chordwise and 6 spanwise functions relating

their strengths. The system is solved by a least squares method with 99 pivotal points.

Cbl Chordal modes.

c) Spanwise modes.

(d) The kink at the wing, centerline has been rounded.off for these calculations.

(el, Experimental results are examined in detail in the section entitled, "Comparisons with Experimental Results." There is a difference between the wing tips in the experiment and those in the calculations.

1/4,.(f) Times for CDC 6700 are in terms of COG 6400 time,.

TABLE 4-= UNCAMBERED SWEPT WING

(In the second last column, time is for all cases of the swept Wing. If a program was not run for the camber cases, then the time is just

for the uncambered wing. This should be kept in mind when comparing

times,)

20

.

___--Time

--Program Mach _Cur 1 PMCP I j BMCP, CDI NCM x NSM sec 1 Computer r-Tulinius 0.12 3.891 0.538 , 0.450 0.999 420 IL -..._-CDC 6600 I ' Dulmovits 0.12 3.987 0.541 , 0.455 10 x T5 104 IBM 360/75 Margason-Lamar Giesing 0.10. 0.12 4.008 , 3.973 0.532 0.540 1.053 10 x 12 1) x 17 71 123C CDC 6700(g) IBM 360/65 Rubbert(b) 0,00' 3.868 ' 0535 . 0',44,7 0.966 10x 15 351(c) CDC 6600 i Lopez A Shen 0.00 3.916 0.536 . II 1.000 S x 17 110C IBM 360/65 Haviland 0.12 4.021 01.583 . 0.,463 I

'

,8 x 10 463 CDC 6400 Jordan 0.00 3.910 0.531 0.999 2 x 15 13 UNIVAC 1108. ! Lamar 0,00 3.874 1 0.536 0.447 0.980 '.8

x ft

6.51 IBM 360/65 * Widnall 0.12. 4.1080 . 0.502 0:443 10 x 2 60 CDC 6700" Bandier 0.00 41J0301 0.528 . 1,052 4 x A 10 CDC 6700(9) Rowe 0.12 1.921 0,534 I:1 0.447 8 x 9 95 CDC 6600. Cunningham 0.12 4.091 0.523 0.444 5-x 5 19' IBM 370/155 .Jacobs-Tsakonas 000 4.253 01.553 i 0,464 5(d) 80C . CDC' 6600 I Lopez I 0,12' 4.065 0,550 i 1.109 1,6W

:-

_---Experiment ---Pressure 0.12 3,89 0.462 Force 0.12 3,85 L 0.466 Additional Computer Runs

_

Margason-Lamar 0.00 3,996 0.532 I 0,457 10"x 12 144 CDC 6700W Gfesing, 0.00 3.956 0.540 1..044 11 x 17 123C IBM 360/65 I Jordan MO 31.894 0.532 10.993 2 x 15(f) 20 UNIVAC 1108 Wfdnall 0.12 4.073 0.499 0.444 10 x 3 73 ,CDC 6700 Cunninghat 0.12 4.043 0.524 1, 0,445 8 x 5 115 IBM 370/155 Cunningham 002 4.145 1 0,523 01443 3 x 3 7 IBM 370/155 Lopez 0,00 4.047 0.55G 1.099

16(°

20. CDC 6600 Notes:

(a) See note (a) in Table 3, . '

(b) Rubbert's program is nonlinear, The above results were obtainedliv runntng the Uncamberedl

wing at a. - 0.1 randians.

()i

Each of the Rubbert runs for the swept wing took, 1117 seconds.

-(d) Chordal modes.

(e) Spanwise modes.

(f)

Rounds. off the kink at the centerline for this rum.

TABLE 5 - CAMBERED SWEPT WING AT 0-DEGREE ANGLE OF ATTACK

Program Mach CL PMCP BMCP _{CDI x} 103

_ NCM x NSM Tulinius 0.12 0.075 0.705 0.454 0.37 420 Dulmovits 0.12 0.078 0.673 0.459 10 x 15 Margason-Lamar 0.10 0.079 0.670 10 x 12 Giesing 0.12 0.076 0.717 0.39 11 x 17 Rubbert 0.00 0.077 0.667 0.451 0.38 10 x 15

Lopez & Shen 0.00 0.066 0.861 0.29 8 x 17

Haviland Jordan Lamar Widnall 0.12 0.085 0.739 0.447 10 x 2 Bandler 0.00 0.077 0.689 0.39 4 x 4 Rowe 0.12 0.075 0.699 0.451 8 x 9 Cunningham 0.12 0.076 0.737 0.449 5 x 5 Jacobs-Tsakonas 0.00 0.050 0.494 0.461 5 Lopez , Experiment Pressure 0.12 0.078 Force 0.12 0.058 0.64

ADDITIONAL COMPUTER RUNS

Margason-Lamar 0.00 0.078 0.670 10 x 12

Giesing 0.00 0.075 0.717 0.38 11 x 17

Widnall 0.12 0.085 0.735 0.443 10 x 3

TABLE 6 - CAMBERED SWEPT WING AT 5.7-DEGREE ANGLE OF ATTACK 22 Program Mach CL PMCP BMCP _{CDI x} 10 NCM x NSM Tulinius 0.12 0.462 0.565 0.450 0.141 420 Dulmovits 0.12 0.475 0.563 0.456 10 x 15 Margason-Lamar 0.10 0.477 0.554 10 x 12 Giesing 0.12 0.471 0.568 0.148 11 x 17 Rubbert 0.00 0.461 0.557 0.448 0.137 10 x 15

Lopez & Shen 0.00 0.456 0.584 0.136 8 x 17

Havil and Jordan Lamar Widnall 0.12 0.491 0.543 0.444 10 x 2 Bandler 0.00 0.478 0.554 0.148 4 x 4 Rowe 0.12 0.466 0.560 0.448 8 x 9 Cunningham 0.12 0.483 0.556 0.445 5 x 5 Jacobs-Tsakonas 0.00 0.473 0.548 0.464 5 Lopez Experiment 0 Pressure 0.12 0.483 0.64 0.467 Force 0.12 0.463 0.577

ADDITIONAL COMPUTER RUNS

Margason-Lamar 0.00 0.476 0.554 10 x 12 Giesing 0.00 0.469 0.568 0.147 11 x 17 Widnall 0.12 0.490 0.540 0.444 10 x 3 Cunningham 0.12 0.480 0.554 0.446 8 x 5 . I 1 '

All of the programs, except that of Rubbert which is nonlinear, combine

the downwash cases for the swept wing into one run. The total execution

time for all swept wing computations are recorded in Table 4. Since Rubbert

must make a separate run for each downwash case, the time shown for his

program represents three runs of 117 seconds each.

It can be seen from the last columns of Tables 4 and 5 that several

types of computers were used. This variety makes it difficult to assess

the relative computer time requirements of the various programs because

computing times vary for the different machines. For instance, the CDC 6600

is approximately three times faster than the CDC 6400 and approximately four

times faster than the IBM 370/155.* However, despite the above time

differ-ences between the computers, certain trends are indicated; for example, the Jordan, Bandler, Cunningham, and Lopez programs are significantly faster

than the others. Finite element programs, which encompass the first seven

programs listed, typically require 1 minute or more for each planform case. On the other hand, the time requirements of the modal programs are strongly dependent on the number of modes; for example, the Lamar program on the

same computer (IBM 360/65) required 98 seconds for NCM = 4, NSM = 8 and

651 seconds for NCM = 8, NSM = 11. Similarly, the time for the Rowe program

increases from 13 seconds for NCM = 4, NSM = 6 to 95 seconds for NCM = 8, NSM = 9 on a CDC 6600.

Figures 5 and 6 respectively present the spanwise lift distribution for the tapered wing at 11.4-degree angle of attack and the cambered swept

*The CDC 6400 comparison is based on experience at NSRDC; the IBM 370/155

comparison is based on a private communication from Dr. A. M. Cunningham, Jr. of General Dynamics.

Program Time (sec)

Jordan 14

Widnall 23

Bandler 27

Havi land 15

1.0 0.9 ' 0.8 0.7 0.5. 0.4

4- 63

00 -'V .. 11 1 _ I

0

113111111114 Ih"

11

is-1/0114

1111131.12111E111111111111allilea_

;

I 111716 u

IIIMAiNr_1111

4 ,..._., 6.01111111

Ell

c6

0

, ,

0

. , ! I 1 .

V

' CO SYMBOL PROGRAM

-

--i EXPERIMENT DULMOVITS CL 0.830 0.817 0.831 0.798 0.812 II.

0

1TULINIUS .,. RUBBERT _{i WI DNALL} , -

-1

I

-0

1 , .

--1---. 1 -CUNNINGHAM JACOBS-TSAKONAS 0839 ,' i 0.920 1

0

I i I I , ., ___-.,'

-0

1,

0,2

'0 3

04

05

0.6

0.7 0.8 0.9 ' I Y' -- Figure --

5--Spantfise Lift Distribution for

Tapered Wink_a_11.4-Degree Angle of

Attack

0

01 1.0 CL 0.6

0.6 0 5 0.4 0.3 0.2 0.1 0.0

0

6

41.--6

'

4

...oi

E

0

li

0

.

8

010 0

E ii

__..

saP

hp

0

0

00

.

0

cg

0

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SYMBOL PROGRAM CL

to

n

EXPERIMENT TULINIUS DULMOVITS RUBBERT WIDNALL CUNNINGHAM JACOBS-TSAKONAS 0.483 0.462 0.475 0.461 0.491 0.483 0.473

0

0

II.

0

0.0 01

02

03

04

05

0.6 0 7-0.8 0.9 1.0 Y'

Figure 6 - Spanwise Lift Distribution for

Cambered Swept Wing at 5.7-Degree Angle of Attack

-wing at 5.7-degree angle of attack. The results from all the programs are not included in these figures, but those shown are typical of the results

from the other programs. Figures 7 and 8 indicate the spanwise

distribu-tion of chordwise locadistribu-tion of the center of pressure for each of these wings, and Figure 9 presents the spanwise distribution of induced drag for

the tapered wing. A more detailed presentation of the spanwise data can be

found in the appendix.

The chordwise variation of the pressure-jump coefficient is shown in Figure 10 for a spanwise station near the midspan of the tapered wing

(y' = 2y/b = 0.58). This distribution is similar to the distribution at

y' = 0.14 and y' = 0.80. Since the results from all the programs were too

dense, results from only five of the programs are presented. These results

are fairly typical of the others. Additional pressure distributions can be

found in the appendix.

Compressibility effects are negligible. It can be seen from

Tables 3-6 that as given by the Margason-Lamar, Giesing, and KUchemann programs, the variation of the pitching moment center of pressure (PMCP) with respect to Mach number is zero to within three significant figures for the swept wing cases; it is less than 0.3 percent for the tapered wing. Corresponding variations in the lift coefficient CL are less than 1 percent

for both wings. Since the variation in the results among the various

programs are typically several times this figure, variations due to the Prandtl-Glauert Mach number corrections are not significant for the Mach

numbers considered in the present study.

As predicted by the Jacobs-Tsakonas program, the lift coefficient

for the tapered wing is 8 percent higher (Table 3), that for the uncambered

swept wing is 3 percent higher (Table 4), and that for the cambered wing at

zero angle of attack is more than 20 percent lower (Table 5) than the

predictions of any other program. Only in the case of the cambered wing at

5.7-degree angle of attack does it predict a lift coefficient in agreement

with the other programs. Even for this wing configuration however, the

spanwise lift distribution predicted by this program (Figure 6) is lower

than those given by the other programs for y' < 0.5 with the exception of Rubbert; for 0.75 < y' < 1.00, it is higher than all the others. Figure 5

shows that Jacobs-Tsakonas prediction of spanwise lift distribution for the

0.14 0.0 0.1 . 0,2

03

aA

oz-

Y'

Figure

7 .

Spanwise Distribution-of Chordwise Location of Center Of Pressure for the

Tapered Wing at 11.4-Degree Angle of Attack

1.1

1 1 , ,

111

I 1 1

111

-

I.

III

11111.11

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tiuI

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MIME

1SYMBOL ill PROGRAM PMCP ,

.:

..

--I,

EXPERIMENT REF 5 0.28

0

I

o

TUUNIUS 0.240 I,

outmovas

0.240 A MEMNG 0.237 P

#

HAVILAND 0.242 JORDAN

i 0/11

0

LAMAR . 0.239 WIDNALL

0/18

9

ROWE 0.239 A

----A

0

CUNNINGHAM

a235

JACOBSASAKONAS 1

a264

LOPEZ-SHEN 1

0238

KUCHEMANN

a242

-0,30 0.29 0,28 0.27 0.26 0125 0:24 0.23

xi

0/2

0.21 0.20 0,19 0.18 6.17 0,16 0.15 0.6 0,7 (0.8 0,9 1:0

0.34 0 33

032

10.31 030 0.29 0.28 r0.27

X'

0.26 025 0.24 0,23 022 011 0.20 0.19

018

0.1

04

06

.0 7

05

Nf _ Figure 8 -.. Spanwise

Distribution of Chordwise Location-of

Center of Pressure for the

Cambered Swept Wing at 5.7-Degree Angle of

Attack 1.0 1 L 1 1 , 1 1 .

ii

I

el

9

1 1 D

0

@

0

e.. _ A QD 40

n

-0

I

*

1 , 1 1 SYMBOL PROGRAM ouLmovns WIONALL EXPERIMENT

0

TULINIUS

0

CUNNINGHAM JACOBS1SAK0NAS1, REF PMCP

_ _

11 0577 , 0565 0.563 d 0.543 ! 0560 0.556 0.548 II , 1 1

---Clco

0

1:1 i ! i ROWE

i

.

'.-Ill

, 14'911

.

o

________ t _ _____J_ 1 1 i 1 _ V 8 0.2 0.3 0.8

09

0 .1541 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.0 .8429

0

0

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CI 9

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09

09

0

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SYMBOL PROGRAM

0

0

0(

0

TULINIUS RUBBERT LOPEZ-SHEN LAMAR(CIRCULATION) LAMAR(SUCTION) KUCHEMANN

v

0.0 01

02

03

04

0.5 06 0.7 0.8 0.9 1.0

Y'

Figure 9 - Spanwise Distribution

of Induced Drag for the Tapered

Wing at 11.4-Degree

Angle of Attack

ACp 3.5 3.0 2.5 2.0 1.5 1,0 0.5

00

09

-

1,0 l'SYMBOI: RUBBERT(211/b=0.50 JACOBS-TSAKONASI2Y/b PROGRAM -REF 5 -= 0.67) , ,

0

EXPERIMENT TULINIUS OULMOVITS WIONALL ROWE

1 CUNNINGHAM 1 1.

IP

1

V

IN

0

10 ill

Brill

al

1111

__....

.

..

0,

06

07

08

00

01

02

03

Q4 0.5 X' Figure 10 . Chordwise

Variation of the Pressure

Jump Coefficient at y'

,r.

.58

for the Tapered Wing at

11.4-Degree Angle

of

AttacK

0

T 1

1

I

-0

tapered wing planform does not agree at all with the other programs.

Moreover, Figures 7 and 8 show poor agreement between the spanwise distri-bution of the chordwise location of center of pressure as predicted by this program and by the others; it is high for x' = x/c < 0.15 and for 0.7 < x'

(see Figure 10). Tsakonas, Jacobs, and Rank27 indicate that for predictions

of chordwise pressure distributions, this method gives slow convergence particularly in the neighborhood of the leading and trailing edges. It

appears from these considerations that the Jacobs-Tsakonas solution to the problem of potential flow around an infinitesimally thin airfoil does not agree with the other solutions and probably does not solve the problem because of the slow convergence.

Consider now the results predicted by the other 14 programs. For

the planform cases (Tables 3 and 4), the variation in the predicted lift coefficients is less than 7 percent of the value of the smallest coefficient

in each case. The maximum variations in the pitching moment center of

pressure are large: approximately 8 percent of the root chord length for

the swept wing and approximately 3 percent for the tapered wing. If the

Widnall and Haviland results are not included, the differences for the swept

wing predictions reduce to approximately 3 percent. With the exception of

the Jordan value of 0.211, the PMCP's for the tapered wing vary only between 0.237 and 0.242 but those for the swept wing show a greater variation. For all programs, the variation in the spanwise location of the center of

pressure is 2 percent of the semispan length.

Figures 11 and 12 show the variation of the lift coefficient with

respect to NSM and NCM for the planform cases. The number appearing next

to the point in these figures is NCM. These figures also include additional

results for the Margason-Lamar program. Margason and Lamar15 found no

variation to three significant figures in the lift coefficient of uncambered wings for choices of four or more chordwise rows of horseshoe vortices; thus,

27Tsakonas, S. et al., "Unsteady Propeller Lifting-Surface Theory with Finite Number of Chordwise Modes," Journal of Ship Research, Vol. 12, No. 1, pp. 14-45 (1968).

0.93 0.91 0.89 0.87 0.85 0.83 0.81 0.79 5O<NSM I I I i SYMBOL PROGRAM

-0

El

0

A

L

b

0

6

0

6

0

cl O

TULINIUS DuLmovm MARGASON-LAMAR GIESING RUBBERT LOPEZ-SHEN HAVILAND JORDAN LAMAR WIDNALL BANDLER ROWE CUNNINGHAM

, CI 3 <58 4

04

v

,10

.--, 10

a

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Q4

610

610

P 4

D8

gtO

o

2

0

0 4

0 2

04

, _ rs14 10

_

20 22 26 28 24 30 32 0 2 4 6 8 10 12 14 16 18 NSM Figure 11 - Variation of CL with Respect to

NSM for the Tapered

Wing at 11.4-Degree

Angle of Attack

-180

0 2 4 6 8 10 12 14

NSM

Figure 12 - Variation of CLa with Respect to NSM for the Swept Wing

16 18 20 22

24 25<

I SYMBOL PROGRAM

0

0

0

A L. D, O

0

0

n

0

Q C TULINIUS

Damovns

MARGASON-LAMAR MESING RUBBERT LOPEZ-SHEN HAVMAND JORDAN LAMAR WIDNALL BANDLER ROWE CUNNINGHAM

05

a

1o

n

8

p

4 4

08

010

10

a

₄

t11

4

98

4

x 30-7

L8

o2

0

0 8

r10

,

41.---4.18 4.16 4.14 4.12 4.10 4.08 4.06 4.04 4.02 4.00 CLa 3.98 3.96 3.94 3.92 3.90 3.88 3.86 3.84 3.82 ' I

the Margason-Lamar results in Figures 11 and 12 also hold for any choice

of NCM greater than 4. These results show considerable variation in the

lift coefficient with respect to changes in the number of spanwise rows of horseshoe vortices; specifically, the lift coefficient decreases with increase in the number of spanwise vortices and appears to approach to within 1 percent of an asymptotic value for NSM > 30. For any given value of NSM, the vortex lattice method predicts a higher CL than the modal

methods for the planform cases. Finally, the spread in the values of the

lift coefficients in Figures 11 and 12 decreases as NSM increases.

It can be seen from Figure 5 and additional data presented in the appendix that the spanwise distribution of lift for each of the planform cases does not reveal any gross differences in the results of the 14 pro-grams. The Haviland predictions are noticeably higher for 0.7 < y'.

Cunningham predicts higher coefficients over the entire span. On the

other hand, comparisons of the spanwise variation of the chordwise location of the center of pressure (Figure 7) reveals that the predictions of the Widnall program are noticeably further forward than any of the others for the tapered wing and similarly for the cambered swept wing at 5.7 degrees

(Figure 8). An examination of the chordwise variation of the pressure-jump coefficients reveals an oscillation in the coefficients predicted by

the Widnall program. This oscillation is not realistic and is probably due

to the higher chordwise modes. In its present working form, the version

of the Widnall program used at NSRDC is numerically unstable for less than ten chordwise modes;* yet with ten chordwise modes, the predicted pressure distributions are not realistic.

The Lopez program predicts a lift distribution similar to that

predicted by Cunningham for the tapered wing. In the case of the uncambered

swept wing, the predictions from this program are much closer to the

aver-age predicted distribution; these results are shown in the appendix. As

*In a private communication Messrs. E. P. Rood, Jr. and P. K. Besch of

NSRDC have demonstrated the numerical instability of the Widnall program for less than ten chordwise modes.

predicted by the Lopez program, the spanwise distribution of the chordwise location of the center of pressure agrees with the others to within 1 per-cent of the chord for y' < 0.8. Klichemann26 correctly points out that the behavior of this distribution near the tip is flat and not a realistic

behavior for the solution to the potential problem. However, he feels that

this will not have a sizable effect on the overall pitching moment since

the lift coefficients are small near the tip. This is shown by the fact

that the Lopez values for PMCP for the uncambered wings (see Tables 3 and 4) are within 1 percent of the root chord of the other predicted PMCP.

However, in terms of finding a solution to the potential flow problem, the Lopez program fails from a mathematical point of view since its predictions for the chordwise location of the center of pressure are empirical and result in a flat distribution near the wing tip.

The Rubbert predictions are the results of a nonlinear vortex lattice method, and their accuracy should not be judged strictly by a comparison

with the other predictions. The accuracy of the Rubbert program is also

probably dependent on the number of spanwise rows of vortices as are the

other vortex lattice methods. Since he used 15 spanwise rows and 10

chord-wise rows, his calculations are probably good to two significant figures. Because of the apparent numerical stability of the vortex lattice methods for uncambered wings, comparisons between the Rubbert results and the other vortex lattice methods should give a good indication of the nonlinear

effects. At zero Mach number, the lift coefficients predicted by Rubbert for the uncambered wings are about 3 percent lower than those predicted by

the Margason-Lamar and Giesing programs. The apparent effect of nonlinearity

is to lower the value of the lift coefficient by 3 percent. The results

shown for Rubbert in Table 4 represent his results for a = 5.7 degrees

linearly extrapolated to a = 1 radian. Two differences between nonlinear

and linear theory contribute to this 3-percent difference. One is the

difference in the use of sin a in the nonlinear theory and a in the linear theory; at the angles of attack considered, 5.7 and 11 degrees, this

differ-ence would account for a 1-percent decrease in the lift coefficient. The

other difference is in the geometric surfaces on which the boundary

condi-tions are satisfied (Figures la and lb). This second difference apparently

The Lopez and Widnall programs do not solve the problem of potential

flow around an infinitesimally thin wing. The reasons for the differences

between the Rubbert program and the other vortex lattice programs have

already been discussed. It has also been shown that the results of the

Jacobs-Tsakonas program do not agree with the others. The variations in

the lift coefficients predicted by the remaining 11 lifting surface programs is 4 percent for the tapered wing and 6 percent for the swept wing at

non-zero angles of attack. If the Cunningham and Haviland programs are not

considered, these percentages are reduced to 3 percent for the tapered wing

and 4 percent for the swept wing. These percentages of variation represent

somewhat less than agreement to two significant figures. Clearly, there is

no indication from the results of this section that Equation (1) has been

accurately solved in a numerical sense by any of these programs.

Three-place accuracy is not obtained even when the same program is used with different values of NCM and NSM.

Consider now the results for the cambered swept wing (NACA 230 mean

camber line) at 0-degree angle of attack. Of the 11 programs which made

predictions for this case, the discussion of results will not include those

of Widnall and Jacobs and Tsakonas. With the exception of the Lopez-Shen

prediction, the predicted lift coefficient varies between 0.075 and 0.079, an approximate difference of 5 percent; PMCP varies between 0.670 and

0.737, a difference of 6.7 percent of the root chord length. The variation

between the predicted spanwise location of center of pressure is also quite noticeable (see Figure 23 in the appendix); there is very little agreement

between the shape and magnitude of the distributions. The predictions of

Tulinius and Rowe agree for 0.2 < y' but are definitely different for y' < 0.2. The other predictions do not agree with these two, nor do they

agree among themselves. The cause of these differences could be attributed

to difficulty in approximating mean camber lines which have a rapid change in slope (such as occurs near the leading edge of the 230 mean camber line); in particular, calculations with the Margason-Lamar program for NSM = 12 and choices of NCM = 6, 8, and 10 yielded lift coefficients of 0.081, 0.077,

and 0.079 and PMCP's of 0.632, 0.691, and 0.670, respectively. As a result

of the computations for the cambered swept wing at 0-degree angle of attack, our confidence in solving the problem of potential flow around cambered wings

at that angle of attack is considerably lower than in the planform case; this is especially true if there is a rapid change in the slope of the mean camber line.

COMPARISON WITH EXPERIMENTAL RESULTS

Although the programs may not accurately solve the potential flow problem in a numerical sense, they may still make a good prediction of the experimental pressure difference Ap(x, y). There are three primary sources of error between an experimentally obtained distribution and one obtained on a computer from potential theory. The first error is due to a difference between the computer prediction and the exact potential value; for a wing

at nonzero angle of attack, this error includes the nonlinear effects of

thickness, camber, and angle of attack. A second source of error is in the differences between a real and an inviscid fluid; the third error is the

experimental error. These errors can mutually cancel, and the predicted

and experimental loadings can agree sufficiently well for the programs to

be useful in design. The purpose of the present section is to compare the

computer predictions with experimental results.

Thiel and Weissinger5 made extensive measurements of the pressure distribution on a wing with a taper ratio of 0.5. The wing tips were

rounded off with the local half-profile thickness as radius. Without the

rounded tip section, the span was 1.5 meters, the mean chord was 0.3 meters,

and the aspect ratio was 5. The planform shown in Figure 3 corresponds to

the experimental wing without the rounded tip section. The tests were

con-ducted in a wind tunnel at 50 meters per second, a velocity which

corre-sponds to Mach 0.15 and a Reynolds number of approximately 106. The wing had an angle of attack of 12 degrees; at this angle of attack, the wing was

in an unstalled flow region. Accounting for the tunnel wall interference

and the elongation of the wires which support the wing, Thiel and Weissinger

give a corrected angle of attack of 11.4 degrees. The wing profile in a chordwise direction was an NACA 0012.

In this report Thiel and Weissinger5 give the normal lift'as 0.83; this value was obtained from integration of the measured pressure distri-butions and is the experimental value recorded in Table 5. Independent

yielded a value of 0.84 for the normal force; the spanwise loading could

be read to ±2 in the third figure. The bending moment center of pressure

given in Table 5 was computed from the spanwise loading in a similar manner. The experimental spanwise lift distribution is plotted in Figure 5. The

experimental pitching moment center of pressure as given in Table 5 is good only to two significant figures.

Although Jacobs and Tsakonas accurately predicted the bending moment

center of pressure, their predicted lift coefficient (0.92) is 11 percent

higher than the experimental value. The remaining programs predict lift

coefficients to within 5 percent of the experimental value. Figure 5 and

additional results in the appendix show a similar trend for the spanwise

distribution of the lift. For at least the inboard 70 percent of the span,

the Tulinius, Dulmovits, Margason-Lamar, Giesing, Lopez-Shen, Jordan, Lamar,

and Rowe programs predict the general shape of the experimental spanwise lift distribution; these distributions lie in a band approximately ±0.01 around the experimental distribution, whose magnitude lies between 0.8 and

0.9. For y' > 0.70, the predicted shapes do not agree with the experimental one; the predicted distributions fall off faster than do the experimental values, which are nearly constant for 0.95 < y' < 0.995.

Experimental values for the chordwise location of the center of pressure for different spanwise stations are shown in Figure 7 together

with error bars. The possible error is large since the experimental values

shown are obtained by integrating the experimentally obtained pressure

dis-tributions. These pressure distributions (which are shown in Figure 1 of

the Thiel-Weissinger report5) could be read to only two significant figures.

The ekperimental PMCP given in Table 3 was also obtained by an integration of these same pressure distributions; it is accurate to only one

signifi-cant figure. Examination of AC revealed that for x' < 0.6, AC (x, 0.14) = AC (x, 0.80) and that for x' > 0.6, there was sufficient difference to

cause the center of pressure at y' = 0.14 to be 1 percent of the local

chord nearer to the leading edge than the center of pressure for y' = 0.8. This shift in the location of the center of pressure is clearly shown in

Figure 7, yet none of the computer programs predicted it. Over the range

0.36 < y' < 0.80, the predicted centers of pressure are within 1 percent of the local chord of the experimental values with the exception of the Widnall

result and the Jacobs-Tsakonas result. In this region the experimental

center of pressure is closer to the leading edge than any of the predicted

values (except that of Widnall) since the experimental AC is higher for x' < 0.5 and lower for 0.7 < x' than the predicted LC.* Tip effects are

likely to be responsible for the difference between the experimental and potential results for y' > 0.80.

Part of the difference between the experimental and computer results might be attributed to the addition of the rounded tip on the experimental

wing. Whicker and Fehlner28 found that the lift coefficient of a wing with a square tip is decreased less than 1.6 percent when a rounded tip is added. This is for a wing with an aspect ratio of 3, a taper ratio of 0.45, and an

NACA 0015 foil section. Moreover, their results indicate a decrease in the

difference with increasing aspect ratio. Certainly, larger differences will occur in the spanwise lift distribution near the wing tip than in the overall

lift coefficient. The available data are not sufficient to give a

quantita-tive estimate of the error in the present case.

Jacobs6'7'8 has measured the pressure distribution on a 15-degree

swept wing of aspect ratio 5.18 with rounded tips but no taper. The wing

tips on this wing were rounded off with a radius equal to one-half the local wing thickness; the foil shape was an NACA 23012. Without the rounded tips, the span was 0.750 meters (0.770 meters with the rounded tips), its planform corresponded to the one in Figure 4, and it would have had an aspect ratio

of 5, for the chord length was 0.15 meters. The measurements were made at

an airspeed of 40 meters per second which correspond to Mach 0.12 and a 5

Reynolds number of 4.2 x 10 based on the chord length. Force measurements

*If the experimental AC near the leading edge is consistently higher

than the predicted LC for other wings, predicted cavitation buckets should

consistently indicate higher inception speed than obtained experimentally.

28Whicker, L. F. and L. F. Fehlner, "Free-Stream Characteristics of a Family of Low-Aspect Ratio, All-Moveable Control Surfaces for Application

to Ship Design," David Taylor Model Basin Report 933 (1958). .