A computer study of a wing in a slipstream

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A COMPUTER STUDY OF A WING IN A SLIPSTREAM:

by

-

.

_,

N. D. Ellis

~~H

'SC'

E HOGfSCP.OOl DEL

FT

VUEGTUIGB;;:U'>;!~UNr;;e

BIJL OlilEfi<

Bibliotheek TU Oelft

Faculteit der Luchtvaart-en Rulmtèmnttêch

.

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2629 HS Delft

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A COMPUTER STUDY OF A WING IN A SLIPSTREAM

by

N. D. Ellis

Manuscript received February 1966

Bibliotheek TU Delft /LR

~lIilliilmlll

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ACKNOWLEDGEMENTS

The method of solving the 'wing-slipstream interference problem used herein was proposed in an earlier work by Dr. H. S. Ribner, whose

interest and assistance in the present undertaking is gratefully acknowledged. The author wishes to express his gratitude to Dr. G. N.

Patterson, Director of the Institute for Aerospace Studies, and to Dr. C. C. Gotlieb, Director of the Institute of Computer Science, for providing the opportunity and facilities to carry out the work described herein.

Appreciation is also expressed to W. T. Chu, T. E. Siddon and other students at the Institute for Aerospace Studies for their helpful comments.

Financial support was provided under National Research Council of Canada Grant A2003.

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SUMMARY

A Fortran IV program for the IBM 7094-II digital computer has been form.ulated based on a theory of wing-slipstream'interference by Ribner which accounts for the slipstream effe cts by means of a vortex sheath. This sheath together with the wing vorticity lead a pair of simultaneous integral equations for the unknown circulations. A stepwise approximation to the circulations reduces the pair to a system of linear algebraic equations. The forma!-has been modified from that of the earlier work to facilitate inversion of the 'equations by computer.

This first program has been restricted for simplicity to the case of a slipstream centered on a rectangular wing. The printout yields circulation, span loading, integrated lift and other properties. The results show a

progression from approximately 'slender body theory' for very narrow slip-streams to 'strip theory' for very broad slipslip-streams and compare weU with experimental data.

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I Il III IV T ABLE OF CONTENTS NOTATION INTRODUCTION

SLIPSTREAM BOUNDARY CONDITIONS

TRANSFORMATION OF A PAIR OF SIMULTANEOUS INTEGRAL EQUATIONS INTO A SYSTEM OF LINEAR SIMULTANEOUS EQUATIONS

3. 1 Approximations Considered 3.2 Approximations Chosen 3. 3 Programming Difficulties

COMPARISON WITH THE EXPERIMENTAL RESULTS OBTAINED BY BRENCKMANN

V COMP ATIBILITY WITH OTHER AV AILABLE THEORETICAL EVALUATIONS

5. 1 Slender Body Theory 5. 2 Strip Theory VI CONCLUDING REMARKS REFERENCES APPENDIX A APPENDIX B APPENDIX C FIGURES 1 - 6 Fage v 1 1 4 4 4 5 7 7 8 8 9 10 14 18

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1) General a b c g

ti

L1.:.t

q 4q v x y z A B C G H

L

~L M N-NOTATION

dummy variabIe used as end point of integration

as above; also span of wing

chord of wing

influence coefficient

lift per unit span

Ja -{

dynamic pressure,

1/2fv2

qj - qo perturbation velocity downstream coordinate span coordinate vertical coordinate

aspect ratio: also influence coefficient

non -dimensional force coefficient

total lift of wing

La - Lb

measure of total slipstream pivot stations

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R S

u

E

?

e

Subscripts a b c j m n p q radius of slipstream area

unperturbed velo city

angle of attack; also end point of segment with constant c ir cula t ion

b/R; also end point of segment with constant circulation non-dimensional circulation

an incremental positive number y/R

Angular displacement around the slipstream

u/u.

o J slipstream strength (1 _1'2) / (1 +)'2)

x/R

z/R circulation augmented basic central

within the slipstream lift per unit span

slipstream integer position subscript wing integer position subscript

wing integer position subscript

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s w L 1 • Superscripts slipstream wing total lift

non-dimensionalized by angle of attack

continuous parameter; also integrated parameter

slipstream

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1 INTRODUCTION

The problem of determining the lift on a wing partially immersed in a slipstream pas in the past been apprpached from several viewpoints, but the expressions derived for the augmented lift tended to be specialized for cases not in the range of interest for each particular application (e. g., STOL aircraft at take-off or landing). The slipstreams considered were either weak (image approach), or extremely narrow (slender body theory) or broad (strip theory) compared with the wing chord.

In a technical report, Ribner (Ref. 1), proposed a new and

quite different approach to the problem without these limitations. The solution appeared as a set of simultaneous integral equations which were not analytically soluble, but were approximated by a set of simultaneous linear equations. It

was not expedient at that time to undertake a computer program for the

solution of these equations. That task has been carried over to constitute the main subject of the present thesis. In addition, certain modifications of the original equations have been introduced, together with minor corrections, to facilitate the inversion process. The limitations imposed in the analysis of this report (a single slipstream centred on a rectangular wing with no slip-stream angle of attack) are not due to the limitations of the theory but were imposed to simplify the computation of an example case.

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Diagram 1

It was expected that the variation of '(

(YJ)

would be smooth ex cept at the edge of the slipstream and that a similar situation would exist for

'('fe)

except at the surface of the wing. To solve for

Y(?)

and

1'(9)

by computer, discontinuities must be removed or proven nonexistent.

It is evident that the pressure is continuous across the slipstream boundary and thus it is shown in Appendix A that the following relation is

true:

'«(1-0):;

)/2

0(11-0)

Hence the wing circulation

'((1)

is discontihuous at the edge of the slipstream except for the case where

jI

= 1 for which the jet velocity is equal to the free-stream velocity.

defined in the

Because

Y(?)

is discontinuous a new continuous function is equation below:

't,

(ï):::

't

(?) -

~

Á

(1-

r/-)

(2a) Here h(x) is the Heaviside function defined in Appendix A. The value of

Y.s

is constant with respe'ct to

?

and chosen so that Y,(~) is continuous ,at both

7

= +1 and -1. The expression for '(5 is developed in Appendix A and is given below:

The value for

'(5

is therefore negative for the case where the slipstream velocity is greater than the freestream velocity.

(2b)

It is shown in Appendix A that the circulation around the slip-stream is discontinuous where it passes through the wing. Physically, if this were not so the upwash velocity field would be discontinuous due to a

finite trailing vortex; the :wing-sfope boundary condition would thus be violated.

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"

As in the case of the circulation of the wing bound vortex the discontinuity in the circulation was separated from the continuous portion through the use of the following equation where ~'(e) is a continuous function:

3 (a) The expression for

t

_{s ' is developed in Appendix and is rewritten below:}

3(b) Some of the properties of the symmetries of (($)are shown in Appendix A, and in particular the following equation is developed:

!'('((9,)d&,

=

0

-11"

4

In the theoretical development of Koning, (Ref. 7), where only weak slipstreams were considered, the augmented lift-curve slope obtained

was continuous at the slipstream boundary. As can be seen in the span Ioadings

of figures 1 and 2 this is applicable in the limit of vanishingly weak slipstreams

(/I

=

U

0/

U. ~ 1), but f,?r finite strength slipstreams the lift curve is dis con -tinuous. If is shown in Appendix A that the slope of the lift curve must be

discont inuous at

7

= 1 such that

),t

(;:),./10

~ /~('F/)7'(-O

and that correspondingly the slope of the reduced ç,inculation'

t.(,)

is continuous at

'1

= 1. The slope of

'It?)

has a delta function at

?

= 1 resulting fr om the

discontinuityof

'dl?)

at

1

= 1. Theexpectedshapesofl(,),

't,{?),

and~(y)

are shown below in diagram 2.

1

Î

1

?

Diagram 2

Two additional ·restrictions op the lift curve and hence on the circulation are that their slope must be zero at the centreline and they must vanish at the wing tips. It was expected t hat their slope:.>yVould be infinite at the. wing tips as in the case of the elliptic wing.

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III TRANSFORMATION OF A PAIR OF SIMULTANEOUS INTEGRAL

EQUATIONS TO A SYSTEM OF LINEAR SIMULTANEOUS EQUATIONS There are many approximations that can be made to reduce integral equations to linear equations. In each case either the circulations are approximated over a certain range by well known functions and then the result integrated or the given influence functions are replaced by an average over a certain interval and then the integrals are evaluated. Several of the approxi-mations are outlined below.

3. 1 Approximations Considered

(a) The Ribner (Ref. 1) scheme was a 'staircase' approximation to the span loading, resulting in a series of discrete trailing vortices.

Equations are ultimately forlllulated in terms of these trailing vortices as unknowns. This was found to be an awkward feature in the early computer programs.

(b) Another approach was therefore tried in which the span loading was approximated by straight-line segments. This eliminated the undesirable feature of the finite vortices in the trailing vortex sheet but the integrals involved in producing the linear equations were quite formidable. An even more troublesome problem was the limitation to finite slopes whereas the theoretical span -loading slope at the wing tip is infinite . Hence this

approximation was likewise rejected.

(c) Still another approach was tried, based on the following considerations. The span loading for an elliptic wing.is elliptic and it was expected that the shape of the span loading for the rectangular wing would be somewhat similar. It was observed from the previous calculations thaLthe shape of the circulation around the slipstream is nearly a cosine squared function. Hence it was thought that approximating these two circulations by their respective similar shapes over a series of short intervals would be quite accurate. However, the integrals necessary to form the linear equations could not be evaluated in terms of known functions and when with the use of numerical integration the circulations were evaluated there was enough difference from the approximation used to cause evaluated slopes to be of the wrong sign. Because the method did not appear to produce better results than the previous methods and the demands on cOlllputer time were excessive this method

was considered unacceptable.

3. 2 Approximation Chosen

The procedure finally adopted constituted a variant of Ribnerl.s (Ref. 1) 'staircase' span loading formulation. Thé wing was represented by a set of horseshoe vortices such that for each the bound and trailing vortices

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were of the same strength. The unknowns were then the strength of the bound vortices in each horseshoe. This avoided certain awkward summations

required in Reference 1. Mathematically the approximations are expressed

in the equations below:

'i, I ï)

=

n!-

'I.

j, (-

(0("

-7

8(7»)(/- -

'7

E

(?») )

5(a)

M+-I

~:

(e)

=-

m~ y~

B(cos

(e))

J"

HoI~

-

e

Big))

0:'" -

e

Bfe»))5(b)

In Appendix B these two approximations are used for the circulations

and the fOllOWi~~et of Simultan~~l's linear equations are derived:

110(1)

==

Y.,

Gnp

+

L

t~ G~

+

X

A-1l ;

P; "

~+-1

6(a)

n:

I 1'\'1=1

P

N+I

o

=- ; ; .

'("

Pin;

r;

r:r -,

+-

'IJ

BI

;

't

·1,11+

1

6(b)

o

=

(I

~).)

'(,

(1)

+-

~s

6(c)

The approximatlon chosen, while not the best possible, has several

features to its credit. For example, if enough points are taken the . approxi-mation approaches the exact value and thus a limit may be expected in the

calculations, providing rounding error does not grow too fast. Another feature

Î':;; the possibility of evaluating the integrals explicitly as seen in Appendix B. An important defeCt, on the other hand, is that the trailiilg vortex field is composed of many finite vorticies which contribute to a poorly conditioned matrix; this limits the number of intervals that may be used.

3. 3 Programming Difficulties

This is not a comprehensive list of all the programming difficulties encountered by rather a brief resume of some of the more signi-ficant ones. One of the first problems encountered was the method of choosing

the spacing to obtain the most accurate representation of ~,'YJ). It was .

decided th at the spacing distribution ~or ~'/f))was not as critical, as the range was fixed and had no really steep slopes. Several attempts were made to .try to define a spacing that would be adequate. However, this did not lead to what was considered to be a really good distribution so the next step was to use ,an iterative procedure whereby a specific distribution of the pivot stations is given. Using these points the circulation span loading is calculated. Using the span loading a new distribution of pivot stations is chosen so that the

circulation increases with approximately equal increments. This procedure is repeated until the increment sizes are approximately equal, to within a

specified degree of accuracy. The curvature varied too quickly in; some parts

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to interpolate between the pivot stations of the former approximation.

Another bothersome difficulty was the estimation of' (, (1) to ·be used in the calculation of

ts.

The value of

'

t,

(1) was approximated succes-siverly by the nearest available point, by a linear average of the two adjacent points, by aquadratic expression based on three adjacent points (which worked quite well but meant th at either the inside or outside of the slipstream was emphasized to the neglect of the other), and finally by a cubic approximation based on the two stations inside the slipstream and the two stations outside the slipstream closest to the edge.

It was mentioned earlier that the bound circulation on the w:ing

was discontinuous and that the discontinuity had to be evaluated explicitly. This

was necessary because the numerical solution: for t(~), without removing the discontinuity at Y) = 1 through the use of equations 3, yielded a distribution that

was continuous at the edge of the slipstream. The slope was quite steep and for th at reason a search was initiated which yielded the analytical expression for the discontinuity as shown in Appendix A. The inclusion of this relation eliminated the difficulty.

In order to obtain results that were comparable with those

predicted by the available theories it was necessary to evaluate lifts nearing

limiting cases. As can be seen from figure 4(e} this lift was in some cases extremely small and thus presented a numerical limitation to the answers that could be evaluated.

The total number of equations was limited by the fact that the matrix was not very well conditioned and as a re sult rounding error became significant, yielding some extraneous results. This was a significant problem only when a very large number of points on the wing were employed to show the errors introduced by a smaller number of pivot stations.

A schematic flow chart for the program is presented in Appendix C. ,A complete flow chart or a print -out of the program is not

included due to its length

\

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IV COMPARISON WITH THE EXPERIMENTAL RESULTS OBTAINED BY BRENCKMANN (ReL 2)

Perhaps the best-reduced experimental information available is presented in a report by Brenckmann (Ref. 2). This experiment consisted of an instrumented wing placed across a wind tunnel with a slipstream, produced by a model propeller, centred in the wind tunnel. The present theory is based on purely potential flow, and it cannot account for added lift due to boundary layer control or "destalling" provided by the slipstream. Brenckmann has

separated the added lift into added potential lift and added lift due to the destalling effect by their correlation on different parameters. Comparison was made herein with the potential lift of Brenckmann.

Brenckmann' s experiment had rotation of the slipstream which was not accounted for in the program. However, the rotation was not expected to contribute to the overall wing lift (because of the antisymmetry), but merely to distort the span loading. The variation of the added lift integrated over the wing with the strength of the slipstream is plotted in figure 3 for both

Brenckmann's results and the present calculated results. It was noted that although the theoretical values were low, the slope of the curve follows that of Brenckmann' s quite well. The much better agreement of slender body theory with experiment is surprising, since a diameter

f

chord ratio of unity is hardly 'slender' in the context of the theory. (The assumptions of slender body theory are discussed later). It is noted that the experimental data are not corrected for wind tunnel boundary interference: thus it is tempting to speculate on compensating errors of experiment and theory.

V COMP ATIBILITY WITH OTHER A VAILABLE THEORETICAL

E VAL UAT IONS

5. 1 Slender Body Theory

In making the comparison it is necessary to set forth the assumptions made in the derivation of slender body theory. First, the wing aspect ratio Aw is taken to be infinite and the slipstream aspect ratio As is taken to be vanishingly small. This makes a comparison quite difficult since the analysis of this thesis is ._a numerical one where values near zero and approaching infinity are quite difficult to deal yvith. In spite of this, at values that were as close . as it was possible to approach the conditions of the slender body theory the agreement was very close as can be seen in figure 5 where for the largest values of Awf As the curves are practically coincident, it will be noted, however, that the values for both are so small as to be negligible. Figure 6 shows the span loading for slender body and plotted with it is a re-presentative case of the present theory where the conditions of slender body theory are nearly realized.

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5. 2 Strip Theory

At the other extreme is strip theory where the aspect ratios are both assumed to be infinite. This allows two-dimensional theory to be applied at all points along the wing and to assume a lift curve slope of 21'r' at all points along the wing. This case was much easier to approach and with aspect ratios in the hundreds the strip theory was about

1/2%

different from the results that were obtained using the present theory. Figures 4 and 6 show the limit that the strip theory presents and the manner in which the present theory results tend to approach towards it for large As.

VI CONCL UDING REMARKS

For simplicity the present analysis and program have been re stricted to the case of a single slipstream centred on an untwisted rectan-gular wing. The results are illuminating, nevertheless. First, they show that the present theory asymptotically approaches slender body theory and strip theory at appropriate limits, and spans the region between the two. lts

primary advantage over the other existing theories is that it covers ranges of parameters that are not touched by them and allows the introduction of still further variations, such as multiple slipstreams and the inclusion of rotation of the slipstreams. The purpose of this thesis was not, however, to produce masses of theoretical results, although considerable have been produced, but rather to show that the theory proposed by Ribner (ref. 1) was compatible

with two asymptotic theoretical formulations, (ref. 3, 6) that have been proposed and shown to be valid over certain ranges of parameters, and also to show agreement with the experimental results (ref. 2). This has been shown and a few intermediate results are shown that are not directly comparable with those of any of the other theories.

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---~~- ~-1. 2. 3. 4. 5. 6. 7. Ribner, H. S. Brenckmann, M. Graham, E. W. Lagerstrom, F. A. Licner, R. M. Beane, B. J. Van Spiegel, E. Wouters, J. G. Feirce, B. O. De Young, J. Harper , C. W. Koning, C. REFERENCES

Theory of Wings in Slipstreams, University of Toronto, Institute for Aerospace Studies Report No. 60, May, 1959. Toronto, Ontario.

Experimental Investigation of the Aerodynamics of a Wing in a Slipstream, University of Toronto, Institute for Aerospace Studies Tech. Note 11, April, 1957. Toronto, Ontario.

A Freliminary Investigation of the Effects of F ropeller Slipstreams on Wing Lift, Douglas Aircraft Co., Rep. SM-14991, 1953.

Modification of Multhopp's Lifting Surface Theory with a View to Automatic Computation, Nationaal Lucht-en Ruimtevaart-Laboratorium Tech. Note W2, June, 1962, Amsterdam Netherlands.

A Short Table of Integrals , Ginn & C()., Boston, Mass. 1929

Theoretical Symmetrical Span Loading at

Subsonic Speeds for Wings having Arbitrary Flan Form, N ational Advisory Committee for

Aeronautics Rep. 921, 1948

Influence of the Fropeller on other Parts of the Airplane Structure: Aerodynamic Theory, Edited by. W. F. Durand, Vol. IV, Springer, Berlin,

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APPENDIX A

:qiscontinuity of the Non-dimensional Bound Wing Circulation

1(,).

Since the pressure is continuous across the slipstream

boundary the lift per unit length .l(y) must also be continuous. Hence we may write

where é

Now

is an incremental positive number. 'l.

:1

(({+ (;)

=

f

t/o

rt"'+f:) :.

f

U. +1<

't(I+

f)

Hence

tL

t~-~)

-

f

UJ

rt

R-~) :.

f

liJ

4-

R '((1-6)

IJ

1Lf)%

4-

~

t

(0-6)

.à

r

u.~

4 ry-2

11,-

~)

'((I-I::)

::./~,((I+-~)

Definition of Heaviside Function and Associated Functions

h(x) = 0 x

_<-

0 x

=

0 1 ·X: _~₀ B(x)

=

x <. 0 0 x -:: 0 1 x

_"l

0

c1j:x.)

=

b

(x)

;

~B7<b<)

=

2 U"..)

j

Stx)

=

0 )

2:-1-

0

~

JJ'CX)~= /d(X.)<4~

1 )

-00 _{- 4} (Al) (A2a) (A2b) (A2c) (A2d}

Suppose th at z(b) =- 0 and there are no other zeros of z in the interval (~ B). then

J

Itx.)

di

d("t-)d4

=

A

d~

f

(&)

o

• ((b)

4î

_d>e

>0

c:;a __

0 d)(

~il

'- 0 (A2e)

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Evaluation of the Discontinuit in the Non-dimensional Win Circulation hence but and

• .

,

and

We have defined ~I (~) by the following equation.

'1,17)

=

tI?) -

'rs

~

),

(1-?l)

Y,(U-é)

~

'Q(lté:)

'tI {

/-t)

=-

I(/-é) -

'(5

t(I-~)

=

I'ê.

O

(l+t)

(, t

1-

~)

:

.

Y,

.

(

I

+

t- )

'(,(;+1::.)

=/7.

f,(/-l-ê) -

(~

y~=

-(1/")

0(/+6)

,

Symmetry Properties of

t

((9)

~k

~

0=

f

(I?,)

7r:/

(e,'7,)d?, -

J.~ ['(~r;,)do,

-

(j_1

'(I(r;)

yt!A

- "

If we integrate this equation from (}=

-ï1

to

B=

7T

we have

71 Iï

o

= -

fr

'(e,)

d

B, -

cr-I

f (,'

(B,)d

e,

-TT -17

because H(~, ~I) = - H(1f-

8,

~,

)

and hence

f!I.z

TT

ft";,)

c;{

7,

f'lt

((J,?,)

de ::

Cl

~h

-

fT

ff

but

cr-~

I

t [)

hence

f

rfe)de

==

0

. -71 rewriting

1 (b)

we have

(J!2.

'('(f)):

cr-

f

'ti?,)

H(r;,

?,)

cA?1

-(J/l.

(Al) (A3) (Atb) (A4a) (A4b)

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Therefore if

'lil?)

~

0(-1)

I I

'1(e)=

r

(-e)

y'(rr-e)

=- -

rite)

Proof of the Discontinuity in

'1'((3)

Similarly it ean be shown

Now let

{(1J.~()) ~

.

-1

(1-/')

t,(,)

tr-~-o)::.

-1 (\-/')

'1,(1)

Y(-'f+

o)

~

-k

(1_;1)

1,(1)

rit

e)

=

~'(

e)

-+-

B (

C ()"'J

e)

~ ~'

where

t

(&)

is continuous

and hence

~S'

-=.

~

[I

jJ

~?

((

1

i-

0)

= -

±

'(5 (A4c) (A4d) (A5b) (A5c) (A5d) (A6a) (A6b)

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Derivation of the Discontinuity of the Lift per Unit Span Curve Slope at the Slipstream Boundary

Two boundary condition equations are available at the slipstream boundary. The first is the pressure condition which states that the pressure and hence the lift per unit span is continuous across the slipstream boundary.

Derivatives in the spanwise direction are precluded from this equation because the equation applies at only one value of the spanwise coordinate. The other equation states that the flow inclination must be continuous across the slip-stream boundary. Hence at the wing surface

=

for all values of x

or = but v = So we can write "2.. "\ ~ ()~" d

rJ·

~

~/

hr;

or

5;'

~

/

J;J

p but p = fUu or u =

Uf

Therefore.

~ ~

,,";/T.

5~i

(jo

dt

.

2Lj

r

and

~

=

I~~

hence

j:. " /'

~J

Lo=

L(R+D) ; LJ::

-I(R-O)

As a result of (A 7a) we may state

and

Î

r(f?+D)

'J'I

) '(1)+

D)

~')

,~

=

(A 7a) , (A7b)

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APPENDIX B

This appendix takes equations 1, applies the discontinuity boundary conditions of equations 2(a) and 3(a), and then applies the approxi-mation for

t ("))

and

'(,'(&)

respectively given by equations 5(a) and 5(b). The following symmetry conditions in the influence coefficients are also used.

-

--Ç~

-

g

'

Î

,-11--9)

-

=

-

Ç+

· l-,

-

9-,-

-8)

-g'(~,7,7T-e)= ~'(S,~,&)

-f-/(Tï-tJ,

~)

=

H

(8,

1)

-- -~

--0=

Applying the approximations of equations 5 and the relation between

Y5

and 'Is' of equation 3(b) the following equations may be written

N+1

d t,

(?) ::

L '/,

~

("t( ))

d"2:

d

1 .

17~

, n

.

?

d

?

, M-t-I (

d

do,

(e)

=

Z

O~

2 r

t(~

e)

d

C;9..k (2'(e))

+

de

m;/

B(ctnB)

J(~I(e)) ~~(f1)

J

(Blb) (Bic) (B2a) (B2b)

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B3 we have

~/.

l.J.t!..

f

7T

ei

(F,?)

=

~

f

{G(f,?,'}t)+ G(i, ,,-?,)]d?1

~~

.

r~o(

..

{GlJ,~,

?)+

C(

J,

?i?,)}d?,

+:!.

r,'1

fr;.'(J,?,

9,)-

Crf,

?;

-o,J]de,

+:

2h!~! ~1[

CII,

'J,(1)·C~j,?,

&,)1

de,

+

n

~I

b'.

f

rt{

I,

?,flol

-'I

(I,?,

<>10 )

-'i(!,?,.;,)

+

fa,

1,

-a(.,)!

11+/

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t!,

N+/

0=

'I,!

[Hf

13,

'I.)

+

H le,-'7.l]

d,.

<§

't~

[We,'),)

t

H(e,-'),J}

d'),

M+I (

-cr-/2~>t~

\3(ctPeî

Jv(2:I(e))+~:s [fJ-l(e'?I)d~/+~cY'B(t'~(7)]

(B4b)

M"'I -I

To proceed further we need to have the integrals of the above functions. These are evaluated from tables of integrals (Ref. 5) below.

b

jen,

'7,

?.Jd'7,

~ !Jd?,/(f~(?,_?),)'h=

f?r /(

hf

7.-,d,Z,.b

=:

G,

(f,?,a,b)

[#/'7

5 ]

(B5a) OR (B5b) (B5c)

(25)

Now the pivot stations will. be chosen for positive

7

and for

e

between 00 and 900 • Let the pth pivot station along the wing have a span

coordinate of

'7

p. Similarly the qth pivot stati:=>n op the slipstream win

m

ve an angle of f) q degrees. Note that

0(,-=1'

and

rX,:

ti .

Let

G

r} P

=

".

.

Cl (

1,

r'J p ,

-ft '

f')

+

~

(

f,

Y)p

'fJ -

? (

f)

IJ;>, -;, );

n:

1

~

G,(

fdp)

al",

/ol

+

C,r

J,

7" - "" ,

"IJ

+

~

(

f,

7?,

In ) -

~

(f,

1(>,

o(~)

- 9

(f, ')

f ,

-;'1)

+

? (

f,

iJ,., -

cl,) ;

n

t

I

G~p ~2G:

n,

'1-.

-f:,

f:

)+~ ~

lf,

,,,t-') -

2.p'([,

'1nf,')

~

=

2. f

G;

(f,

YJP'

o(~ ,~~)

+

&,'f

I,

?PJ

-~;, 1~

)

+-

~

I (

f,

1 r,

f~)

-

~

I (

f

~

?

p,

f1'

~

)

- '1'(

f,

'7n -/;.)

+~'(~

'7"

-a(~)}

;

/Yl"*

I

fl

_p

=

G,

(I,

7,.,

-I,

I) -

G,'(f,1p,

-1j,~)

,

/

f/ntj

=

H,

(&1'

-~

,1

1 ) ( •

,

n

=

I

=

H,

(13"

d~

,In)

of

H, (

e, ,

-cl" ,

In ) ;

n

i'

B,

=-

Hl (

f39;-

1 - /,

/)+

~

0- -I

=

i

[1+cr-1J

'

"

With the addition of equation 2(b) we have

;"h-/

M+/ lTo(p::'

Z

~

G"'f

+-

L(~ C~p

+-"(s

Af>

h:/

m=

I (B6a) I'Y) :. / (B6b) (B6c) (B6d) (B6e) . (B7a)

(26)

(B7b)

(B7c)

Choose

f

=

c/2R to make the horseshoe vortex

(27)

APPENDIX C ISTARTI

l

Input the parameters As' A

w' M, N,

..

,.,

_,;V,

_{and any necessary precision}_and

control parameters.

t

Set up a spacing if this is the first case of a run and otherwise

adjust an existing one.

t

Calculate the matrix composed of

the coefficients of the system of

r+- equations 6. Also calculate the boundary composed of the left hand

sides of equation 6.

,

Solve the resulting matrix for both the basic and the augmented

lift.

1

Are the differences between adjacent values of the augmented lift (i. e.

Y

n and '( n-l) equal to the desired accuracy?

t

yls no

t

Calculate the lift per unit span from the circulation distribution obtained for both the basic and the augmented lift.