Theory of wings in slipstreams

THEORY OF WINGS IN SLIPSTREAMS

MAY, 1959

BY

H. S. RIBNER

Bibliotheek TU Delft

Factjejt der Luchtvaart· en Rumtevaartted1nlek K1uyverweg 1

2629 HS Delft

UTIA REPORT NO. 60

•

THEORY OF WINGS IN SLIPSTREAMS

BY

H.S. RIBNER

Bibliotheek TU Delft /LR

11111I111111

c

1855418

•

•

The rnaterial herein constitutes a slightly revised vers ion of a report prepared in 1957 by the author as consultant to The DeRavilland Aircraft of Canada Lirnited. The W ork was brought to publication with the support of the Defence Research Board of Canada, through the agency of Dr. G. N. Patterson, Director of the Institute of Aerophysics. I should like to thank Mr. R. D. Ris cocks and Dr. G. W. J ohnston of DeRavilland for their continued interest and co-operation during the course of the study •

A general potential theory has been developed for

determining the lift distribution and related properties of a wing in ane or more slipstreams of arbittary shape or position. The main limitations of most previous theories - e. g. restriction to either very high or (in one case) very loVJ aspect ratios are believed to oe removeq. (A recent theoretically sound, fair ly general theory due to Rethorst is limited to a single round slipstream and is

mathemat-ically much more comple~).

The basic idea of the method)made possible by use of a

"reduced" potential within the slipstream, is the representation of

the slipstream effects by means of a distribution of vortices (or

doublets) along the slipstream boundary. In ad'<iition, the weU-known

wing-plus -wake representation as a sheet of vortices (or doublets) is employed. Both vortex distributions are initially unknown. A pair of

simultaneous integral equations are set up for t:teir determination.

These replace the single integral equatiC'l'1 of ordinary lifting surface

theory. Additional slipstreams can be acccunted for by adding

further simultaneous eqqations (the details are indicated but not worked out) .

The integral equations are in

mo

dimensións: they cover

the entire wing-plus -wake anti the entire slipstream. F or practical application, a simplification along the lines of the "'Pisl.o.lèst~.I.· theory

is employed: the wing pattern is compressed ir!tc a single lifting

line at c/4 and the slipstream pattei~n into Cl. vertex ring, again at c/4;

bcth have s uitable trailing vortices . . A wi..ng flow-tangency condition

is specified along 3c /4 and a potential ccndition is specified at.the slipstream -boundary at x = 00 • The pair cf integral equations to ,be

sobTed have thus been greatly simplified by redu<::tion to one

dimensiol1- .

"""The further procedure - which has been used in ordinary

wing theory - is to divide the wing span intI) a nnrIJber of segments

and th~reby to express the integral over óe span as a summ ation. Similarly the integral aroll..l'J.d the vortex ring (vrlth the slipstream

specialized ,as circular) is expressed as a summation. The two

integral equations have thus become some 1Q to 40 simultaneous

algebraic equations . These equations have been explicitly formulated for inversion in a digital computer; thecoefficients (influence functions)

are simple algebrq.ic functions of the georn etry.

NOTATION ii

I. INTRODUCTION 1

Il. FORMULATION IN TERMS OF DOUBLET SHEETS ON

JET BOUNDARY AND WING-PLUS-WAKE 2

lIl. INTEGRAL EQUATIONS FOR DOUBLET DISTRIBUTIONS 5 3.1 The General Integral Equations for Arbitrary

Jet Seetion

~. 2 Pressure Distribution and Lift

3.3 Formulation in Terms of Lifting Elements in Place of Doublets

IV. REDUCTION TO ONE DIMENSION BY MEANS OF

5

6

7

PffiTOLESICONCEPT 7

V. WORKING EQUATIONS FOR SINGLE ROUND SLIPSTREAM. 9

5. 1 The Integro-differential Equation Pair 9

5.2 Reduction to S1.multaneous Algebraic Equations 10

5.3 First Forroulation for Digital Computer 11

5.4: Symmetry Considerations 12

!'J.5 Equationg for Digital Computer for Fully Centred

Slipstream (Case (b» 13

5.6 Statie Case 15

VI. OUTLINE OF ELEMENTS FOR TREATING MULTIPLE

SLIPSTREAMS (NON -OVERLAPPING) 15

VII. CONCLUDING REMARKS 16

REFERENCES APPENDIX A

Prooi that Wing Plus Slipstream Can be Represented

by Distribution of Doublets Over Wing-Wake and Slipstream Surfaces

APPENDIX B

18

20

Derivation of the Pair of Simultaneous Integral Equations 21

NOTATION

N ote: The words slipstream and jet are used interchangeably. b C

~lIp

)

g;,P

G~f

)

G~1jJ

!ll?q I

Hl

m}

n p q R

~

8

'}

C" >J S Si S

U

_o Uj-w 0(. J wing sPétn wing chord

influence coefficients defined in E q. (28)

indices locating vortex elements (see Fig. 1)

radius qf slipstream section w hen specified as circular

vector distances between.s'ource points and field points (Sketch 3)

surface of plan of wing plus -wake surface of slipstream boundary

spanwise length of bound vortex segment (Fig. 1) stream velocity

jet (slipstream) velocity

upwash velocity (perturbation velocity in direction)

rectangular coordinates: x, y as in Sketch 3, with z perpendicular thereto and upward. Positive x is downstrearn

wing angle of attack

jet angle of attack (positive when upstream portiorBof jet are higher than downstream portions)

oL

_p

·

/E

.

I'

r'

.

0'

O'

LlYn

Ll

~I

_m

$

~

f/

e

~

t

f

:i

cr-~

~j

Liep

po

~.

Subscripti l.I.

J

00

local wing angle of attack at point P. Fig. 1

nondimensional wing span

(1)/

R)

circ ulation on wing

circulatioll on jet boundary

nondimertsional circulation on wing

(r/~RtIo)

nondimensional circulation on jet boundary

.

(r;/

/f

I(

tJa)

nondimensional circulation of trailing vortex associated with point n (Fig. 1)

nondimensionaf circulation of trailing vortex

associated with point m {Fig.

1)

arc subtended by bound vortex segment (Fig. 1)

z/R

y/R

angle in cylindrical coordinates (x, R,

e )

locating point on circular jet boundary stream velocity /jet velocity {U o/Uj) x/R

dens i ty of air

point just inside jet boundary

jet-strength parameter

(7

J.~:)

disturbance potential outside jet

(cjJó

:=.

~)

"reduced" disturbance potential w ithin jet

(CPJ

==

~

lh)

potential jump across jet boundary (

(cPa -

QJ; )s')

(E qs. 8a, 8b) .

disturbmce potential outside jet

(~o

:::

c/J())

disturbance potential within jet running coordinat e

1. INTRODUCTION

For more than a quarter of a century the problem of the lift of wings in slipstreams largelr resisted efforts (e. g. Refs. 1 to 6) to

develo~

_

II successful theory 1 . The failure lay mainly in the general employment of a Prandtl-type lifting line approach suitable only for high aspect-ratiophenomena. However. it has become clear (Footnote 1 and R ef. 7) that an approximation (at least) to a lifting surface approach is needed, because the slipstream normally spans only a low-aspect-ratio wing segment. Several years ago Graham et al (Ref. 7) developed. among other things, a lifting-s urface theory for the ltmiting case of the very narrow slipstream via "slender-body" theory(2 . Recently Rethorst presented in his Ph.D. fuesis (Ref. 8) lhe first published theory without aspect ratio limitations .

Rethorst represented the wing by an array of horseshoe vortices. For each vortex he calculated the interfer:ence downwash field due to a circular slipstream. The results were prèsented as influence coefficients for use in a formulating a Weissinger - or Falkner-type system of simultarteous equations to be solved,ior the unknown horseshoe strengths . The influence coefficients are in the form of summations of infinite integrals of Bessel functions and

present a formidable computing'problem. even with a digital computer. A more serious drawback is the limitation to a single slipstream.

A different and quite general approach to the interference effect is developed in the present analysis. The limitation to a single slipstream is removed; moreover • the method can be extended to treat a non -uniform slipstream. The influence coefficients are relatively simple elementary functions .

(n

See also Peter Kriesis. Ph. D. dissertation. Polytechnic Institute and University of A thens. in German. 81 pages. about 1,945

{2)

, 'The present writer developed independently a very similar theory in the first phase of the investigation leading to the present

analysis. (The DeHavilland Aircraft of Canada Ltd .• urtpublished note, 1956).

Il. FORMULATION IN TERMS OF DOUBLET SHEETS ON JET BOUNDARY AND WING-PLUS-WAKE

S mall-dis turbanc e incompressible potential flow is specified as the basic

assumption. Thus the boundary conditions may be linearized,

(e. g.

Sketch 1

,

and may be applied at the pro-jection of the wing in the x, y

plane and at the unperturbed slipstream boundary (assumed cylindrical and of infinite extent), respectively. The wing flow is required to

satisfy the Kutta condition . (However , it is now known that supple-mentary boundary layer effects are important, but may be treated separately (Ref. 9) ).

Given a th in wing

,g

at an angle of attack 0( , partly

in a general stream of velocity Uo' and p:lrtly in a jet of velocity Uj' The jet is parallel to the general stream (this condition can be re-1axed - see footnote 4). The wing induces a perturbation potential

?,Po

in the free stream and ~ in the jet.

The pressure must be the same on both sides of the jet boundary

S' .

In addition, the flow inclination adjacent to the boundary

J"

must be the same on both sides. These two cOnditions may be written:

-ft/:

~I

tJ..!Jl

ol')

SI

pressure: -.:::;

- f

c

~X

Q J 'X

inc lina tion:

~'/17

_-

~/U

01

SI

() n

d

..J..

'

where x is parallel to Uo~plane of paper) and n is the outward normal to the boundary

S'

(in plane of paper). These equations show that both the axial and normal perturbation velocity components experience a jump on crossing the jet boundary.

The pressure equation integrates to:

-f~,

'Iv' ::

-r~

V.

with~. ~

taken as 0 at x = - 0.0 • The equations may now

be

writte~

more compactly as:

inclination: where .~

-;n

~A.

==

U/U:

/ - -

.

!

( 2 ) ( 3 ) I

The potential jump, Eq. (1) can be satisfied by a distri-bution of doublets on

S'

(or by an equivalent distribution of lifting elements: horseshoe vortices). The normal velocity jump, Eq. (2) can be satisfied by a dish"ibution of SGurces or sinks on

8

1 (Ref. 10).

It will sirnplify matters if the source distribu.tion can be

eliminated. This can be ef(ected by utilizing a special 'reduced

potential' within the jet. To thil3 end define new perturbation potentlals

( 4 )

I

( 5 )

Substitution into the jet-boundary conditions .

q

1 ): and q .2.) ) yields:

preS8Ure! ( 6 )

inclination: ( 7 )

The normal-velocity jump - and hence, the source distribution - have been eliminated. lnstead there is an augmented potentialjump of amoullt

v'o1~t~he@t

à,c>

U·

J

s

cpj

A

~

==

(~o

-

~

)SI

which by Eq. (6) is given as

L\

~

= ([

~Z)(ç4J.1

or

IJ.

rjJ

=

0-

-2 - / )

{fjJj.)SI

Sketch 2

( 8a )

This potential jump can be s atisfied by a distribution of doublets on

S'

of local strength

Aep

(a factor ~7T has been

absorb.~d in the definition of 'strength') . Furthermore. the result-ant flow field of the wing plus slipstream can be represented

entirely in terms of this doublet distribution together with another over the wing and wake. This is proved in Appendix A.

The reduced P9tential fJ!>;. now employed witl!-in the jet is just a fraction

)-l ::

Voli/~ of the actuat potential ~. T he derivatives of

~;.

- the reduced perturbat"on velocity lömponents are too smal! in the same proportion. Thus, the wing boundary

condition in terms of ~, .

w

~

~Z becClD3s in 'terms of _~rf rl-. I

.

w

=

!h -::. -

Va

0(

(XI

~)

~z ( 9 )

Compare the boundary condition for t.he wing-portions outside the jet:

The effect here is as though the velocity in the jet, U j' were reduced

to _{the free -stream value U o . However, all.of the jet interference}

effects have been retained through the inclusion of the doublet sheet on the jet boundary. Solution of this 'reduced' problem leads at once to the actuaJ. problam on rnultiplicaUon by ~-/within the

jE;t.

1---,1'--_---,1

---~ ~--~~~---~~

\ .

I

,

lIl. INTEGRAL EQUATIONS FOR OOUBLET DISTRIBUTION 3.1 The Gerieral Integral Equations for Arbitrary Jet Section

It is well known that the lifting surface theory of a thin

wing cari be developed from a doublet approach: the wing is represented as a sheet of doublets (with axis norm al t6 the sheet, negative side up) covering the wing projected plan area and· its entire wake. In the presence of a jet, then, there is added !:lnother doublet distribution covering the jet boundary. Both distributions are unknown, and we proceed to establish a pair of simultaneous integral equations for their evaluation .

Let the doublet strength on an element dS 1 at point P (xl' Y1, 0) on the wing (Sketch 3) be

r

dS 1 (see footnote (31) and let the

strength on an element dS 1 ~t pI ('J(/~

'1/)

~/

)

on the jet boundary be rd.5;~ According to Ref. 10 (p. 60, Eq. 12), the potential at a point a small distance 2 above the wing (Sketch 3) is

~(X;

!t

1!:)

=

'I'rr

{r

12,

(f)

ct

S,

-t-

't!"-;;

r'

?n/!=)

cts.

I

The upwash W at

~6<;

1;0)

is

~I-"~ ~f:

_

W()(J

LI)

0)

=

LII~n ~

6

,

[rf.z(-/)Js,

+

Irt

~ [(/~(;,)ds/l

, " z;'70

L

i,

IT~

' . 1 ' -:SI I

J

The potential at a point

leX;

~/i}which

is made to approach the inner surface of the jet boundary ".~' is found similarly (cf. Sketch 3); it is

rA ()(;

~;

Ij

=

f

J;!

~;

i!1R7T

~

r

?-./

Deis,

+

f,.,

[r' ih/

})cI

~'J

where points

L

X;

~,.

r)

lie on from the interior. )

I

t.

I

S

and /I?? is approached

~ -'11 (jt: ~; i/)

The boundary conditions on the wi.ng and jet boundary are recalled as W(X)

~)

0)

= -

(Jo

rx:

()0

~)

_{( 9 )}

LJC/; -

r'{k;

'Ij~)

==

(f-:-/12)

rfo

6(:

~; ~:J

_{( 8' )} ( 3 ) I

,

The doublet strength/unit area measures the local potential jump from bottom to top surface of the wing; hence it may be recog-nized as identical with the circulation

r .

The insertion of these boundary conditions now yields (4)

~U~(X/~)

=

i/rl?

fii,/i-[r/i

{-f)cI~ -r~)ifr/~

(f,)c/S/7

(10 )

il-t>o

~

~

~I ~''J

J

r(X:tt:~)

₌

L)m

J,"

f

r?i:(f)c/s

+1

((7/2

1-';)dSJ

(11)

/-;4~ .r..,~;~f~)

$

~

I

~71

JSI

~Ii,

(S

I

where

S'

is the entire slipstream bcundary and

S

includes both the wing and the trailing vortex (or doublet) sheets. Eqs. (10) and (11 )

constitute two simultaneous integral equaticns for the two unknown doublet distributions r a n d

r'.

(These equation_s are not unique; alternate forms can be obtained Ie. g. I by carrying the differentiation under the integral sign, and by integrations by parts. The differentation leads to divergent integrals which may be treated by the

=f

technique of Refs. 11 and 12).

3.2 Pressure Distribution and Lift

If Eqs. (10) and (11) can be solved (e. g. , numerically) for

r ,

the lift distribution on the wing can be obtained. Inside the jet

a 'reduced' potential was used, and the solution for

r

there must be amplified by a factor ~·t,.tJ,';U. The lift distribution i$ then givenby:

lift pIJ

"aL

unit area = 1/0 ~ 'K lift

= !

J),

~

,

()

~

u:'

unIt area v.~ ~\J v D

'1(

Inside jet: ('·12 ) Outside jet:

=

~-z fVo~

The lift/unit span results on integration to the trailing edge:

( 4 )

lift Ollts ide jet: _{unit span}

( 13 ) Inside jet: _{unIt span} lift

The analysis throughout - and these equations - refer to the case of the slipstream jet being aligned with the main stream.: lf the

-jet is inclined at an an,gle

ot;

(positive downwar9~ in the vertical plane the potèntial fields

cp;

and

rp.

must be supplemented by the two-dimensional field of an infinite cylinder of air moving down-ward with velocity

Vo

oC' . The upwash field associated with this motion must be addeJ to the right-hand side of E q. (10). For example, if the jet cylinder is round (radius R) and is centred

vertically on the wing the additional upwash is given by

~

w::-

-Uc>li

within the jet and

4W

=

+

Uol·

y~/RZ at anY lateral position y

3.3 Formulation in Ter;ms of Lifting Elements in

:p

lace of poublets

. , Eq. (12) relates the 'lift/unit area to ~r/J)(

.

For some

,purposes it may be desirable to refqrmu.late the governing equations in terms of ~rl'rJ1 in place of

r

,

.and Ir~)('in place of

r

l .~ One advantage is this: the wing lift, .an.Çlhence ~r/~x~ is zero behind the trailing edge, although

r

remains constant. Hence the area of

integration S may be limtted to the wing area when ~i'IJ)( is used,

rather than wing-plus -wake area as when

r

is used. The approp-riate pair of simultaneous integral equations can be obtained from E qs. (10) and (11) by means of 'an integration by parts.

IV. REDUCTION

Ta

ONE DIMENSION BY MEANS O~ISTOLESL_

CONCEPT

Two -dimensional integral equation pairs like (10) and ('11) can probably be solved by large capacity electronic computers after approximating the integrals as sumwations. T he procedure reduces

to the inversion of a very large number of simultaneous equations in as many unknowns. The undertaking is extremely fOI.'midable.

A great simplification results on employing the

Pistolesi '(RèL 16)approximation of the lifting wirrg by a lifting line: the entire wirrg circulation is compressed into this line, which is located at c /4, and the flow tanqency condition is applied solely along the 3c/ 4 line. Likewise, by analogy, we compress the bound cir-culation distributed ring-wise along the slipstream bqundary into a single lifting 'ring': a ring v9rtex located at the slipstream c /4 station. The boundary condition on potential (Eqs. (Sa), (Sb») is then applied solely on another ring about the slipstream (not necessarily at the 3c /4 station).

__

-_-_-_-_-_-_-_~

__

~

______ r __

I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

.

_.-_-~

X

_

.

k::~iZ'" ---~:

X

C/4

Sketch 4

Sketch 4 exhibits the qualitative build -up of doublet strength or circulation with distance in aplane y = con$tant. The

line -vortex, ring -vortex approximation is equivalent t~. replacing

the entire gradual variation by a step function localized'at c/4. Sketch 5 shows both vortices (as cut by plane y = const.) and the equivalent step-function doublet distributions :

c~

r=

roG

f1=o

J

1.6~

r

·

~

roo

.l

r:o

~pZlZZ=--

'

Cl4

_{Sketch 5}

The com'plète lifting-lîne and lifting-ring, together w ith their appropriate trailing vortices appear somewhat as in Sketch 6. (The subscript 0() has been dropped frorn

1:0)

r:,

,to simpÜfy

the notation) :

JfY

~---

~

+-~---~/~--~---77/

-~~========~/============================~~~~/

'- >

r

~~~---~---r

~~~==~~================~~~/

~---~---~---~--~--~

44

'

3c/Lj 'Sketch 6 rin<ó @x':..oO /

According to the PistolesL -Weissinger concept, the $ystem should satisfy the boundary c-ondition onupwash w (E q. 9) at x = 3c/4. Further, comparing Sketches 4 and 5, it would be most accurate to satisfy the condition on potential jump r'(Eq. 81

) at

x

=

00 (around a ring) . The required values of upwash at 3c /4 and potential at

oa

Cal). be evaluated by means of the Biot-Savart law (e. g. ref. 13). Upon application of the boundary conditions tl1ere results a pair of equationSin the form

- V.

r;(

(3CI'l

~)

-!Af(C)d;

+

f

(1

'ç '([:

a.'N?

(

14)

.

-~/2-

;:,

~

G

• I ( )

6'A

"

-

r

_(0);/

=

.rr~(r.)d~

-tfr?'

{[~~')d2

(

15)

/ f'-

-'Ia.

rln~

where

.(.fl

a

Q

1

are certain functions .

) ) v;

g

Subscripts OCJ are to be understood as applying to

/"':1

and

r' ;

they have been omitted to simplify the notation. The

r

integrals are taken along the wing c/4 line, and the r"integrals are taken around a ring about the slipstream at c/4. Eqs. (14) and (15) constitute, in schematic form, the desired pair of integral equations in one-dimension for the determination.of the unknown circulations (or potential jumps) r a n d

r'.

The kernal or influence functions

f'

-I'

~ Q' are yet to be specified.· ) I..,) rJ

\ _,

-(The same pair of equations could be obtained in principle. by &ubstituting the step-distributions of doublets of Sketch 5 into Eqs.

(10) and (11), specializing ~ and

!"

on the left-hand-sides to x =

3c/4 and x

=

ca ,

respectively, and integrating from x

=

c/4 to cx;l

The integrations are. however, quite involved and laborious and are avoided in the first-given method) .

v.

WORKING EQUATIONS FOR SING LE RO UND SLIPSTR EAM 5.1 The Integro-differential Equation Pair

The symbolic equation-pair, Eqs. (14) and (15) are not in a ~onvenient form. either for derivation of the kernel functions • or

. for numerical solution once they have been obtained. Ah alternative form, involving both rand d

r

/dy (and similarly

r"

and drl/cte)

under the integral sign will be used for th~ working equations . Integro-differential equations of this sort (e.g .• the Prandtllifting-line

equation) are common in wing theory.

The slipstream is now specialized as round. ç)f radius R. Points of the jet boundary are specified by co-ordinates (R,.

e ,

x~­

with the x-axis taken along the jet-axis. The jet axis lies, in the

general case. a distance z below the wing plane at the midspan. Points along the wing c/4 line have co-or<Unates(o, y. z)an~ points along the 3c/4 line have co-ordinates(x, y, zh where x = c/2.

For convenience we now introduce nondimensional variables :

~

= x/R

Y

=

r

_/4RUo

1

=

y/R

r'

=

r'

_/4RUo ( 16 )

5.2 Reduction to Simultaneous Algebraic Equations

Sketch 7 shows the approximatien of a continuous vortex sheet by a series of discrete vortices. This well-known idea is applied to the vortex sheets defined by

Y

and

Y',

associated respectively with the wing plane and the slipstream cylinder. (See

sketch 6). . ~ A 'V CS?,= L.. UOh

~-'

t\

6ou-nd

L

v A

a.

~. ~A a.. ~

•

, •. ü ~'-' y ... ~L I.---' ...

....

4 ~ ·cil1

Ccntinuous Sheet Discrete Vortices

Sketch 7

The specific scheme is worked out in conjunction with F ig. 1. Con-sider first Eq. (17); it is replaced by a series of equations; in each . 0( C~)1) is specified at a different value of

1 -

these are the points labelled "p" in Fig. 1. The integrals are replaced by summ ations; each of the tr:üling vortices (at points

n -

1/2) and each of the bound vortex segments (at points n ) contributes a term, and similarly for the vortices on the jet boundary. Eq. (18) is likewise replaced by a series of equatiens; in each

r'{

e )

on the left-hand side is specified at a different angular position q.

The fcllcwing scheme represents, in part, the trans-formation from integrals to summations :

Integral

Summati0I1-o(1,)

cl

r,

>-

y"

At;,

:=.

r;

S

~

dlf,

:;,

t10n

(:

;,

~

1p

=

pS

Terms in _~

117

=n.s

Terms

in~,

1,

_!>

~

=-{(l-

11z.)s

1, ·

tI-I/2

,

A similar scheme applies to the jet-boundary system involving

#y/tlt!J

and the series of points mand q. The basis of the transformation is this: the integrands are interpreted as expres-sing the Biot-Savart law applied to differential elements of the

continuous sheet (see sketch 3); the transformation makes the necessary changes to express the Biot-Savart law for the discrete vortex pattern. (.8 ee Fig; 1).

It wil! be noted that the pattern is contrived so that the trailing vortices (at points n - 1/2) straddle but never coincide with the points p. This arrangement avoid~ the singularity in the equations for coinciding "source" and "field" points. This singularity frustrates atteinpts at a straight-forward reduction of the integral equations into difference equations; it is the reason why the approximation is made in the physics of the problem rather than in the rnathemaUcs.

The transform ation of the integro-differential equations (Eqs .17) and (18) leads to the followirig set of simultan~ous algebraic equations : . Sketch 8 n

0;1

=

2

A

r;; )'

-~ ( 19 ) ( 20 )

5.3 First Fprmulati~n for Digital Computer

It wil! be convenient now to introduce ahbreviations for the "influence coefficients" in the sumrnations. THtis ~ptit

rr -I Xl

=

c:;" vI UI -L S V l i v V

6

~ I In

n

("#-1'/'

IJ

nni

( 23 ) O(p

=

L

r:,

b':d

+

L

~ Gn~

I?/I m r a//I"} ,- (24 )

1-

_~Hm

2.

4

rr~ J~P

+-

2.

Ll

~

Q"b oN" ~ r

~

=

t

L1

r: )

A

~+I

=

~

r,:, '"

-r:

+

i.:,

M~

tri '"

M }

==

y~

m-=M

( 25 ) ( 26 )

y~ ~

tr;'l,

==~

( 27 ) where (T

=

:i.j<=

111 -::: -

f/21T

~ ~Á-~~ ____ ~s~~

____ __

",. ==

-;;:--èns

-s'/j~f):+ (~- c.os~f')z.

C/n ...

=

t

á

CQS

n::z..C..

lL ( 28 )

,-

[la.

~ (/Os - S/;; mb)~

+

7Ç -

Cos m f~ 7'.1

• 0

G~JO

-

rr-

-I-

{n-;ls:J

'*

g" -

-[,sM (11I- I@l - es]

(,+

o

t

.\

"'p - (fi.s S.1n {m-1aJfP'

-t?, -

'os tm-1~,)t ~ 1r~(p$-rt/l

tm-'1.x.,.

T{' - cos (",-~)5J

9n"

=

(f1-41:~P)S

(/

+

.,~!:

+

(~-Ih..-t:r·st)

° •

m, q cover range - M, - M

+

1) - - - 0, 1, 2 - - - M - 1 '

n, p cover range - N, -N

+

1)---0, 1,2 --- N

Tq.at is, the slipstream is divided into 2M segments, wit,h 2M trailing vortices, and the wing into 2N + 1 segments,' with 2N +

2

trailing vortices.

v:.

If Eqs. (25) and (26) are regarded as eliminating the 1/.,

and

0"

as unknowns, the ~

r,:

and .A.

0,;

,

are the sole unknowns. The

totp.l number of unknowns is thus

2M

+

2N + 2. For their deter-mination we have the 2M + 2N + 1 Eq,s. ,(23) ánd(24) (witr E qs. (25)

and (26) substituted therein) plus the special equation

~~+I

=

r,;

for the wing right-hand tip vortex.

~. 4 ' Symmetry Considerations

C ase (a): Slipstream cen'tred laterally on wing but not

\ _,

In this c~se the lateral symmetry yields:

'

Y-m

= IIrn

[

l V i

Y-n

_

=

Y"

\~symmetric bound vortex}

[

.

4

₄

Y!m=

v _

-4

11 'V'

d';."

(antisymmetric trailing vortices)

O-n - -1-1 Gn1"1 ' _

Case (bh Sl,ipstrearn centred laterally and

verti6ally qn wing

(? =

0 )

( 29 )

In this caSe the four equations of (a) rernain. In addition there ate the vertical symmetry relations

.,

r '

.

-= - '(,'.

Ll

.

'(.'

=- ~ ('

1'1-171 "., ) M-m·f! m ( 30 )

Sketch 9

'1.' In case (a) the un~nowns are cut in half (± one) by E ql .

(2"9); that is I the unknowns are restricted to asemispan and half the

slipstream. Jn case (b) the unknowns are fur.ther· restricted by Eq.

(30) to one quadrant of the s lipstrearn.

5.5 Equations for pigital Computer for Fully Centred Slipstream

(Case (b) )

Dr. Gotlieb of the Computation Centre I University of Toronto I

has suggested that a large saving in machine time may result upon

eli~inating elther the bound or the trailing vortices as unknowns in Eqs. ~23) to (27) . . F or exam:ple I relati(ms (Eq. 25) to E9-' (27) should

be formally substituted into (23) and (24) to elim~nate

'4",

and

Y"

as unknowns. A procedure whieh shoulq be eq,uivalent - and which

avoids the labor of sorting out the dO.ijble s ummations - is to start

from the basic vortex geometry of Sketch 7 and Fig. land reformulate

the equation~ directly in terms of the

.A

y~

.

-Tj1e symmetries when the slipstream is centred horizon-tally and vertically- on the wing permit a considerable reduction in the

:rlUmber of unknowns. To brîng in the syrnmetries it is helpful to

~L

\ , ,

\

, \..../ i ,1/ V I! ,/

_h

_{1\ ' /}

_Ir-.

I~ , / a 0 0 a 0 0 ₀ CJ 0

-N

-3 -2 ' -I 0 _{1 2} 3 ~ Sketch 10 Sketch 11

Examination show~ that eXG,ept for one term, the effect.s 'of a vortex o.n the underside of the s'Îipstream are (in this problem) identical with those of its mirror image on the upper side. Thus the terms for, the

,upper side need merely be doubled. The term Y ,..,:,

H

I of Eq.

"Tr;;,

(23) vanishes because of the antisymmetry in

S

.

'l'he finaJ equations worked out with the aid of these

sketches ar~: , n-_I

cK

=

'

,

t'

4

~

r(

I'l -

a

) -

~

G

~

J

f , n ~ ":J"/~ -:.s-n+~ P . -n+1 P

+

l

A '(.'

·2

[(al - QI

)

--t-'G' ]

hl' m -;J ~ P J-M+~ (J

-'"1-1

m'p

+

IJ

'(~+I

r(

q~~/>- q:~,)

-

(G~Ip]

' 1#' ' I

I]

~I

..,p.-I

o =

o--'[t,

'

4

r~

-

z

40"'+1 -

~

A

r"

'?111-/~17'f-Eq. (31) constitutes N

+

1 equations as p takes on ,the values 0,1,2,3 ---N. Eq. (32) constitutes another M

+

1

( 31 )

equation as q takes on the values 0, 1, 2, 3~ --- M .. The N rI-, M + 2 unknowns are

AG,)

Ll '( \

~ , ~

-r;;+-I;

At,'1

l::. "(// I l

·4:r,;+,.

For a plane

wing

~I =~1..=

.. , ..

~~::o<.

Slipstream rotation can be taken into

account by introducing an artificial

oc"

variation .

i

The factors in square brackets ('influence coefficients) are

numbèrs whiçh can be calculated once for all for a given wing-slip-stream geometry. Thus the entire set of equations is of the general form:

u:

=

2.

q

.

.

?( .

./" ; t.(j q

where the

4

~s

are the

Xrt'

and the factors are the

el,

'

t

l (a number of

which are zero). These equations can be solved for the-

Xi

by various

matrix inversion methods, e. g., as developed for high'-speed' digital

computers.

5.6 Static Case

When the flight speed is zero only the portion of the wing

"ünmersed in the slipstream carries lift; this is readily demonstrated.

''I'' .

Sinée the exterior portions of the wing can be ignored the lift

cal-culation is much simplified. For example, in the general case perhaps

25 segments might be needed for the semispa11. (N

=

24) and perhaps 5

for a slipstream quë.drant (M = 4). yielding some 30 equations in 30

unknowns. In' the static case, on the other hand, the 25 segrnents n'light

be reduced to 5, the total then being 10 equations in 10 unknowns.

We illustrate this case in Sketch 12.

o .,

h. 4

o 2

:a

Sketch 12

VI. OUTLINE OF ELEMENTS FOR TREATING MULTIPLE

SLIP-STREAMS (NON -OVER LAPPING) _, ,

The previous. .formalism can be generalized readily to apply to multiple slipstreams. A systern of unknown vortices is

placeq on the boundary of each jet, si~;rlilar to that for the original

single jet. A potential condition must be satisfied at each jet boundary

in addition to the downwash condition· to be mE1t at the wing. There are

thus J + 1 simultaneous integral equations to be solved. E ach

equation contains terms from the wing and all of the jets in terms of influence coefficients of the type already calculated for the single jet case. The transformation from single-jet to multiple jet is largely a matter of generalization of the notation . No new analysis is

required.

The basic elements underlying the generalization are as

follows. where allowance ~s made fpr different jet velocities

(asym-metric power condition) :

definition:

?: _

U

o

~ lj.

f

~~~~]oI18.

q

pressure boundary condition

1,-at each of the

Si:

t..tf;

~ lope boundary condition

at each of the

S; :

_L~rJ

~,_ ~7

_{- ;1-i/J}

011

~.

ti"

win,g flow tangency condition

in different regions : It.~ __ / I

l

.

Ld"~

-

l.{c{~~outslde

the

~.

IW' ;:

-(l

,,//~

l.I)J

L

~

0 ""{,':lllIwithin

Si

(If the jets are inclined to the main stream by respective angles

the lË~ft-hand sides of E qs. (36) and (37) m ust be augmented by

additional upwash terms as described in footnote 4).

VII. CONCLUDING REMARKS

i ( 33 ) ( 34 ) ( 35 ) ( 36 ) ( 37 )

A general potential theory has been developed -for the

aerodyna~ics of a w ing in one (or more) slipstreams of..arbitrary

shape or position . . The central ide9-. involving the use of a Ilreduced"

potent.ial within the slipstream jet. is represyntation of the flow by means of two distributions of vortices (or dou'blets): one over the

wing and its wake. and the other over the jet boundary. A pair ·of

simultaneous ,integral equations are set up for their determination.

The integrals are reduced to one dimension (paralleling the Bistolesj.;.:,

\t}/4-3ç/4 .scheme~specialized to a circular jet. and approximated as

summations . . Thus the two integral equations have become so.me 10

to 40 simultaneous algebraic equations suitable for .solution by

digital computer. T he coefficients (influence functions) are.simple

The nonuniform axial velocity is not allowed for. but the theory could be extended to include it without difficulty. The slipstream could be represented by concentric shells with a stepw ise jump in

velocity from shell to shell. The form-alisrn for multiple slipstreams. adapted to this case; would require a vortex sheet at each of the shell interfaces. with an additional integral equation for its determination. Thus the cost would be a considerable increase in the number of

sim ultaneous equations . The nonuniformity will modify the distribution of lift. but it is not expected to contribute significantly to the total lift.

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. Recent experiments have shown that such "destalling" lift can be comparable in magnitude with the potential-flow lift; the two can be separated in the meas urements •

and they corre late on different parameters (Ref. 9). It is sus pected

that Üte disagreement of most previous theories with experiment is

1. 2. 3. 4. 5. 6. - 7 • Kbttirür r ' C. ö~ > Glauert, H. Smelt, R., Davies , H. Franke, A., Weinig, F'. v. Baranoff, V. Squire, H.B., Chester, W. Graham, E.W., Lagerstrom , P. A. Licher, R. M. Beane, B.J. 8. Rethorst, S.C. 9. Brenckmann, M. 10. Lamb, H. 11. Ribner, H.S. REFERENCES

Influence of the Propeller on Other Parts of the Airplane Structure, Aerodynamic Theory, Edited by W. F. Durand, Vol. IV, Springer, Berlin (1935)

The Lift and Drag of a Wing Spanning a

Fr~e Jet, British ARC R & M 1603 (1934) Estirnation cf Increase in Lift Due to

Slipstream, British ARC R & M 1788 (1937) Tragflügel and Schraubenstrahl, Luftfahrt-forschung, Vol. 15, No. 6 (1938), Also NACA TM 920 (1939)

Effect of the Propeller Slipstream on Down-wash, Luftfahrtforschung, British RTP Translation 1555, Jan. 20 (1942)

Calculation of the Effect of Slipstream on Lift and Jnduced Drag, British ARC R & M

2368 (1950)

A Preliminary Theoretical Investigation of the EJ:èfecls of Propeller Slipstream on Wing Lift, Douelas Aircraft Co., Rep. SM -14991 (1953)

Characteristics of an Airfoil Extending Thrcugh a Circular Jet, Ph. D. Thesis, Guggenheim Aeronautical Laboratory ,

CaJiiorniaInst. ofTech., Pasadena, Calif.

(1956)

F~x~Jerimental Inve3 tigation of the

Aero-dynamlcs of a Wing in a Slipstream, UTIA TN No. 11, 1957, also (abridged) Jour. Aero. Sci., May 1958, pp. 324-328.

Hydrodyn:1I11Ic'S. Sïxth Ed~ (1932), reprinted byhDove,r Publicatioris, ,N. Yi., 1945

Some Conical and Quasi-C onical Flows in Linearized Supersonic - Wing Theory,

NACA TN 2147 (1950). See remark of H. Mirels, p. 20, and Appendix D by.E...K.

lVI Qore

12. S~ars, W.R., Editor 13. Pr andtl, L., Tiet jens , 0 . 14. Milne-Thompson, L.M. 15. Glauert, H. 16. Pistolesi, E .

General Theory of High Speed Aerodynamics, Vol. VI, High Speed Aerodynamics and Jet Propulsion, Princeton University Press, (1954). See Sec. D by Heaslet, M. A. and Lomax, H., pp. 233-235

Fundamentals of Hydro - and Aeromechanics, McGraw-Hill, N . Y. (1934), reprinted by Dover Publications (Sec. 88, esp. p. 206,

third eq.)

Theoretical Aerodynamics, MacMillan &

Co. Ltd., New York (1947), p. 218

Aerofoil and Airscrew Theory, Cam bridge Univ. Press (1937), p. 50

Considerazioni sul problema del biplano, Aerotecnica, Vol. 13, p. 185, (1933)

APPENDIX A

Proof That Wing Plus Slipstream ean Be Represented By Distribution of D oublets Over Wing-Wake and Slipstream

Surfaces

A cut through the wing and slipstream jet is shown in the

following sketch

G

I

---II---~---

_D

_EI

_E

F

Sketch 14

The basic problem, from whose solution the flow may be

determined, is that of finding the potential in the various regions . By

virtue of the trick of the 'reduced potential' introduced in Section II, potential jumps may exist at the various boundaries, but there are no jurnps in normal velocity . Under thls circumstance it will now be shown that the potential in all regions can be èxpressed in terrns of doublet distributions on the boundaries.

First let the slipstream jet, assumed cylindrical, extend to infinity Upstream and downstream. In addition, consider the

infinite horizontal plane containing the wing-wake as a boundary (see

extensions AA,and E, E beyond. the wing tips). Above the plane AE

are reglons I and II outside and inside the jet, respectively; and.

below the plane A E are regions 1 and 2 outside and inside the jet,

respectively.

/on

I

else1V1, e re.

/~

1Z

dse;yAe~

( A-I)

~ ~

JfM:

_{4~eFf)+

(f) -

f:

~

(-FE

cis

=

[~ ~:WA:re

DF

.-11f~rW (~)

-

r4.

~

(f)]

cl

3

=

(~ I~

2

" 'l

f.

·~I1z.

r

~

è

0

els

e

wh~

re

where!D;-S"~ Sfl%J$IlJT.Sr;,)~~denote

elements of the normal to dS drawn into the I, lI,

i

and 2 regions respectively. It follows that

:a... __

2-~I?z

-

~i?,

.L

_2-è>

I?z

.;f1% Ol?

on

_(A2)

tn,

-= -

~

DI)

J{Fj)

By virtue of ?he nature of the right-hand sides, the sum of Eqs. (A 1) will yield the potential

P

in all four regions . Since there are no jumps in normal velocity terms like ~~ /~I')r and

-

~~

..

Ie1~ etc., are equal whence all the source terms cancel in the

summat~.

The final result is

rJ

=

4;-

ffi~ -~)

{;;;c(/

)dS

+

Ir;

ffirfir -

~)

t;;zé;

)Js

AB roe 8[;

+

~ir

JJ(rfr

-~) ?~

(;)dS

+

Jtt,,-

/f~

-

~)~

r.))clS(A3)

13 C D a ta!) 'Z

According to this equation the potential anywhere in space is expres-sed in terms of doublets distributed over the slipstream boundary and over the plane of the wing and its wake. The doublet str~ngth is

given by

1/47T

times the local potential jump at the surface. Since there is no potential jump outboard of the wing tips, Al. El, the doublet strength in the plane AE differs from zero only on the wing and its wake.

APPENDIX B

Derivation of the P air of Simultaneous Integro-differential Equations

B-1 Slipstream-Induced Upwash at Wing B -1. 1 Contribution of the Ring Vortex

r'

The velocity induced at

(~/1IÇ)

by a vortex element of strength ('I and length

dS

is, by the Biot-Savart law (Ref.- 13),

ola...

=-

/,1

~

X

C

=

t'

Ju

+.,i

dv

+

h

dw

~

/f7rR

r:J

-

-Also 0

dl.

=

J~,

=

4:

Clrl&,Je-(

~

Ii

SIne,

d6,

C

:

f (

+

1

(1-~)

+1.

(~-~)

r

Thus '

-=

!:,l

É

+1.

(1-11;;&)

+-

J1

(e,

-cos

GJ

'

-

I

.t.

1t

c;

CO~ ~J~

-

Sll'~,~

~ ~ - .!

11'1

&,

~

-

~ ~I =:..

ie

!1~-

cos

8,)

GM~

+

~-Sll'/ ~).sII!Bj]d~

.

~.i:-

f

sl';-,B1dtJ,

-l!

i

c.t)~~lcll?

I

)z,

r;

),l'/z

=

[t:z.

+

(1-

~

I"

&1

of

\t?

-ecU

~ ~

al -

i.fj;-COSê,)C08 ()'_+g-S//J4)S:á1

4

J-j

fS/II&, -

B

ÉC$6,

(7ld&.

?$A -

,flT

R

$4!

-f

t,-

r/~

e;t

r{6 -

cose/J3

lz

I

The upwash at point

(4/

~

1

)

i

s

thl~

Jl.

con'),ponent~

'. oB -.1. 2,~' ContribuUon of Trailing Vo:rt~~

/ The circulation of the vortex fil'a:;:l1ents shed from the

arc

CIe.,

is

J.C.I

i

Ll

.

-

~Bt Cfq •

~

drd~

~

,

élrJ,

)-Sketch 16

(see arrow) and henc e par alle 1 to

1-(4

-eçtéJ;)

-IJ.

(1-~ ~/)

The upwash component of

d%

is thus

B -2 Wing-Induced Upwash at Wing

The bound vortex af strength

f'1(IJ}

is taken to be along

the

ft

axis; it sheds trailing vortices of

strengthq'r/~

per unit

length. The upwash induced ;:tt an arbitrary

'

win~

point

P

(4111)

has

been evaluated by many authorities (e. g. Ref. 14); ~n our

nondimen-sional coordinates it is

f ::

X/I(

B -3 Slipstream -Induced: Potential at Slipstream Boundary (Trefftz

Plane I

B-3.1 Replacement of Vortex She~t by Doublets

It wil! be convenient

to revert to the doublet forrnulation

of Section lIl. In this vie;w the ring

vortex

.:.

+

-

'trailing vortices approx":

imation''tJ9fhe slipstream _. boundary

vortex distribution is equivalent to

a cylindrical sheet of doublets of

local strength/ unit area r~. This

sheet extends from the wing c/4 line

to x = 00 witb the strength constant

along the generators I but varying

with

6

(Sketch 13).

Sketch 18

At J!: := 00 (Trefftz plane) an elernentary line of doublets

along one of the generators extends infinitely far upstream and

down-stream ,and hence locally simulates a tw o-dimensipnal doublet. The

potential at R, é> of a two-c4mensional doublet of strength m at R,

e,

on the slipstream boundary, .with axip in the direction of R, is (Ref. 15):

dfo

_-

-

.!!L.

~

Z". ,...~

r

where "

e-e

l tA

zR.

r

~ ~ 2. 1\ 1\ ""

1

-

r

~!!.::.E, ~ . II~

For an element of the slipstream doublet sheet m =

·

r'Rct::?,

upon

inserting this and the values of

~

and

~

above theeè results

A/IJ .

('I

J

((rp

= -

~lT ~

e

J

This formula fails at the singularity

8:::.

&,;

thus we'.integrate

e:x-cluding a smatl ara on either side of

B, :

.

....

6-~

j2fT

J

~

(2lT-26j

=

-Iir Jfl'dé7,

+

r'cl~

La

e-l-~

03\4)

B -3.2 Contribution of Arc 2

f

at

e

The exc1uded arc 2

é

of doublets in the vicinity of point P approximates a plane as

é ...

0; moreover ~ the variation of _. (?I _~

across the arc can be neglected. If we take P a small distance

1-above the arc and employ a limiting process ~ the contribution of the arc to the potential at P is

p

r----+----,ft.----, -

-.L..-_ _ _ ----I~---J _ _ _

~Ré~Ré~ 1 - - Sketch 19

where the limits are taken in such a way that

R6»

i

. The

integral is

R

6-1:=,

%

(2.~

=

f-:.

~' ~j>i~(R~)

~+O

~+o

27T

A

~ +((

'6, é- b u

~

E-

1

~-I

IPG

-

i6~+o

7T

T

:::: ÉI

-=

r/~)

ê.

2

(B5)

This result applies for point P on the outer surface of the cylindrical slipstream doublet sheet. For point P on the inner surface of the sheet ~ must be replaced by

-§

in the integral. This leads to

~

~~&)=

t-6-

~-O

(B6)

The sum of Eqs. (B4) and (B5) or (B6) taken to the limit

é

~ 0 gives the potential at any point P of! the slipstream boundary at the Trefftz plane:

~~

_...L

~

t:lfl

2. ~q

J

I

"t. v/' (I (outer surface) (B7)

,

_r~

_..L Î

rd6

2.

.ylT~

I I (inner surface) (B8) . T

B -4 Wing-Induced Potential éi\.t Slipstream Boundary at x = 00

Bere again we

. revert to the wing-wake

representation as a doublet

sheet. At x

=

eX:> a strip of wake of width

d1,

extends

in-definite'ly upstre am and

down-'strearn; thus it behaves

locally Iike a two.-dimensional doublet. The strength/unit

width is just the potential jump

through the wake.

r

~ Th~

potent.ial induced at P. (Sketch

20) is therefore

'

e/t

.

rJ'It

d.

2."..

.

r'"

wher~ ~

=

~

e _

~

r'"

=(11

-

~e)a.

+

~

_

~

&-)'2.

o

Sketch 20

(Here again;. aJl leugths are made non-dim ensional with R). Insertion of the values

of

'

~

a!èd

r~

ap:d integration over the span yields total

wing-induced potential at ~

f.

==

~B

-G

~rrW!~

.

W ;2.77" .

i,

~~~)~+

(/

_C()rJfj-)'J.. -,d/& 'VI ~

(B9)

B -5 The Upwash-C ondition Integral Equation

. The upw.ash W induced at an arbitrary wing point P (~)1 )

may be written

w-').7ï J. ..

W-(~J1)

-:=

fc/tu;,

+-1

tbn

1-

Wë: (BlO)

where tJ.:e first integral is i;he contribution of the slipstream ring

vortex (Eq. B 1), the second is the contribution of the slipstream trailing vortices (Bq. (B2» and the t:1ird is the contribution of the wirig-wake vortex sheet (Eq.

CB

-3). The upwash boundary condition to be satisfied aiong the x = 3c/4 line is

-

lAr'

.;;::

-

c<

(4)

1)

{Io

') Up on substituting for

Ju.;..

~and

WC

in Eq. (B 11) and substiting in the upwash condition Eq. (.:9); tbe.re results

.

~

~

lTl'J

ti"

0<. -

/,r

-r"

4

C#

&ï

d

&, .

- " ()

-!

L1~

-r

{jf-~~r·+- (~_U"të)a.J:1/:z.

+

-

I~

de,

1&,1

~

&,-

'I)

(I

.

~

.

t

)

!

LJ?-~8;)l.+ ~-~~Jj

+

if'+

(,-~B,)"-t(t,-~6;)7

'

.

~ r~r(l,l)

c/.'-J

+

I~;-?

'"

.

!

€

)

_

tBll)

where

~

is taken to be 3cJ4R. The

~ubstitutions

(1=

rl'1~

U '

r

=-r/~Ra, reduce this to Eq. (17).

B -6 THE POTENTIAL-CONDITION INT EGRAL EQUATION

_,

, i.' •

The resultant poterttial at a point P on the

inn

'

~

;

~

sbrface (subscript j) of the slipstream boundary at ~ = 'oe is given by

The potential boundary cöndition, Eq.

t

,,

8b

:,

,

~

reads, with

0fJk-

1

replac~çl by

r;

"

,

rtç;)

==0~Z-1~'J,

r~J ~

0-2-

-I)L~

+

{}wl.>'ldc

r"~J

=

Ar'--I)[-

~

-

fir

FW&

-t-4?r]

byvirtue of Eq. p~-e). This simplifies to

\ '

,.

The ful! equation reads (cf. Eq. {B ...

-gq,

, ~

1

1.. IJ 2- I. ",

/~'

"

'J ' COS' L!J. _

:~

' ~

'-"

,l1e)J

't ..

,,;ev

r

[t7) ::: -

~

/,,1

ër&

-f

C7

~,., I )

/

_Ll'Z.

/-.

,

2Ji.

o

,,'

'

r

-W~ l~-.r/,,~t-+rt,-tfJs8

,

(B-12)

The ,subsÜtutions

(I

-rt'll(~)

p

==

r/rl(

/Ia

reduce this to Eq. (18) .

/ /

;....-./

/

/ I

/

~~

,

,

~'

I

,

,J../~

\

\

"

--

...-. /

'(~,?~

I I

.

/ / ~ ~ . / . . / ~?

f~~

. / ../

Figure 1 - Notation for vortex summations representing

2-wing in a slipstream, f(;r programming in a digital compute1'.

t.. ~ ~ Cl? 0".. ~

--UTIA REPORT NO. 60

Instltute of Aerophysics, University of Toronto

Theory of Wings .in Slipstreams

H.S. Ribner, May, 1959, 1. Wings, Power Effects

3. Propellers, Wing Interference

I Ribner, H.S.

21 pp., 1 fig.

•

2. Slipstream, Wing Interference

II UTIA Report No. 60

A general potential theory has been developed for the aerodynamics of a wing in one (or

more) slipstreams of arbitrary shape or position. The central idea, involving the use of

a "reduced" potential within the slipstream jet, is representation of the flow by means of

two distributions of vortices (or doublets): one over the wing and its wake, and the other

over the jet boundary. A pair of simultaneous integral equations are set up for their

determination . The integrals are reduced to one dimension (paralleling the Pistolesi

c/4-3c/4 scheme), specialized to a circular jet, and approximated as summations . Thus

the two integral equations have become some 10 to 40 simultaneous algebraic equations

s uitable for solution by digital computer. T he coefficients (influence functionsl are

simple aigebraic functions of the geometry.

Available copies .. f this report ore limited. Return th is card to UTlA, if you r .... uire a copy. UTIA REPORT NO. 60

InstÏlute of Aerophysics, Unlversity of Toronto

Theory of Wings in Slipstreams

H.S. Ribner, May, 1959,

1. Winlfs, Power Effects 3. Propellers, Wing Interference

Ribner, H.S.

21 pp., 1 fig.

2. Slipstream, Wing Interference

n UTIA Report No. 60

•

A general potential theory has .been deve10ped for the aerodynamics of a wing in one (or

more) slipstreams of arbitrary shape or position . The central idea, involving the use of

a "reduced" p:>tential within the slipstream jet, is representation of the flow by means of

two distributions of vortices (or doublets): one over the wing and its wake, and the other

over the jet boundary. A pair of simultaneous integral equations are set up for their

determination . The integrals are reduced to one dimension (paralleling the Pisto1esi

c/4-3c/4 schemel, specialized to a circu1ar jet, and approximated as summations . Thus the two integral equations have become some 10 to 40 simultaneous a1gebraic equations s uitab1e for solution by digita1 computer. T he coefficients (influence functions) are simp1e algebraic functions of the geometry.

Available copies of this report are limited. Return this card to UTIA, if you require a copy.

UTIA REPORT NO. 60

Institute of Aerophysics, University of Toronto

Theory of Wings in Slipstreams

H.S. Ribner, May, 1959,

1. Wings, Power Effects

3. Propellers, Wing Interference

I Ribner, H. S .

21 pp., 1 fig.

•

2. Slipstream, Wing Interference

II UTIA Report No. 60

A general potential theory has been developed for the aerodynamics of a wing in one (or

more) slipstreams of arbitrary shape or position . The central idea, invo1ving the use of

a "reduced" potential within the slipstream jet, is representation of the flow by means of two distributions of vortices (or doub1ets): one over the wing and its wake, and the other over the jet boundary. A pair of simultaneous integral equations are set up for their

determination . The integrals are reduced to one dimension (paralleling the Pistolesi

c/4-3c/4 scheme), specialized to a circu1ar jet, and approximated as summations . Thus

the two integral equations have become some 10 to 40 simultaneous algebra ie equations

suitable for solution by digital computer. T he coefficients (influence functionsl are

simple aigebraic functions of the geometry.

Available copi ... of th is report are limited. Return this card to UTIA, if you require a cop.y. UT IA REPORT NO. 60

Instltute of Aerophysics, University of Toronto Theory of Wings in Slipstreams

H.S. Ribner, May, 1959,

1. Wings, Power Effects

3. Prope1lers, Wing Interference

Ribner, H.S.

21 pp., 1 fig.

"

2. Slipstream, Wing Interference

n UTIA Report No. 60

A general potential theory has been developed for the aerodynamics of a wing in one (or

morel slipstreams of arbitrary shape or position. The central idea, involving the use of

a "reduced" potential within the slipstream jet, is representation of the flow by means of two distributions of vortices (or doublets): one over the wing and its wake, and the other over the jet boundary. A pair of simultaneous integral equations are set up for their

determination. The integrals are reduced to one dimension (paralleling the Pistolesi

c /4-3c /4 scheme), specialized to a circular jet, and approximated as summations . Thus

the two integral equations have become some 10 to 40 simultaneous aigebraic equations

s uitable for solution by digital computer. T he coefficients (inf1uence functions) are simple algebraic functions of the geometry.