October, 19'71.
TECANISOtE HOGESCHOVl
f}J:*-r I
tuEGfUlGlOUW
W;:UNlJf
lIIuornEEK
AEHODYNAMICS
OF
HING-SLIPSTREAM INTERACTION:
A NUMERICAL STUDY
by
N. D. E1lis
f
~
3
.JUII
L
UTIAS Report No. 169
'J
AERODYNAMICS OF WING-SLIPSTREAM INTERACTION:
A NUMERICAL STUDY
by
N.
D.
E1Vs
Submitted September,1971 •
..
'ERRATA
Report No. 169
by
No D.
Ellis
Page 1. Para.
4,
Line
1.
It became clear(7) that an approximation (at least)
to a lifting
Page
2,
equation
(1)
pressure
:
Page 3, 3rd last line
Pa!;l!e
5
2
equation
{20}
1
w(x )
=
--f
47r
1
-
47r
Pa!;l!e
6
2
equation
{21l
Page
6,
equation
(22)
r'
(x'
.)
-
J
2
1 - I-l
P
00
+
2"
1
pU 2
0
.lp
2 0 0
(U
+
u,)2
1
(
_
1)2
-p
w
2
0
on
S' (1)
4~
is absorbed in the definition of strength) together
with another over
wing
and wake
or(x )
dS
-s
s
J
{rx
orC"s) }
+
r
(r)3
OX
y
oy
S
s
s
or(x' )
or(x' )
J {
r~
-s
+
r'
-s
cos(8') ox,
s
S
·
I
s
[
Or'
(x')
itn-
J
r~cos(8~) ax~s
S'
s
Yot
'
s
+
r'
y
dS'
}
s
(r' )3
or(x' )] dS' }
-
s
s
at
~
(r)3
ril' sin(8' )
+
ril' cos(8' )
}
Y
s
z
s
dS~
(r
l l l)3
(22)
(20)
Page
6,
equation
(23)
Inside
jet:
lift/unit area
(22.1)
Page
6,
equation
(24)
Inside
jet:
lift/unit
span
=
(22.2)
Page
6,
last paragraph, first line
Equatio~
(22
01) relates the
lift/~it
area to
o,/Ox
which is equivalent
·
Page 12, equation
(3)
Should be:
r'
cos(e' ) not r'
cos(e' )
x
s
z
s
Page
20,
paragraph
3,
line
3
ACKNOWLEDGEMENT
The author
wishes to
express his thanks to his supervisor, Professor
H.
S. Ribner,
for initiating this project and for his interest, patience, and
he1pful
direction leading to its completion.
The author is indebted to
the director and staff of
UTIAS
for providing
the
opportunity and
facilities necessary
for this project. Thanks a1so go to the
Institute for Computer Science for
providing the necessary computing faci1ity.
Thanks are
due
to
my
wife, Dorothy,
for her support and encouragement
throughout this undertaking.
Financial support for
the
work was provided
by
the
National Research
Council
of
Canada under grant
A-2003
and the
United
States Air Force Office of
Scientific Research under grant
70-1885. This support is gratefully ackn
o
w1edged.
SUMMARY
A fundament al theory of
wing-slipstream
interaction accounts for
slip-stream
:
of arbitrary cross-section by means of vortex
sheaths.
These sheaths
together
with
the
wing
circulation pattern are dictated by the boundary conditions;
the analysis
leads
to simultaneous
integral equations for
their
determination.
In a multiple
lifting
line
approximation these are
ultimately
reduced to
simul-taneous
linear algebraic
equations for machine inversion.
Programs for the IBM
360-65
digital
computer have
been developed
for
the case of round slipstreams
distributed with lateral symmetry
on
a rectangular wing.
The computed span
loadings
have
the shape
expected
from
experiment. A sequence of curves of added
lift due to the slipstreams
(integrals of
the
span loadings) show a progression
from 'slender body
theory'
for
very
narrow slipstreams
to 'strip theory' for very
broad slipstreams. Excellent
agreement is
seen with
the experimental results of
Brenckmann,
both
for span
loadings and integrated lifts. The expected decrease
in
added
lift
occurred
with
vertical displacement of the
slipstre~m;
however,
lateral
separation of the slipstreams
showed
little effect. Calculated flow
fields downstream
of the
wing were
qualitatively as expected, displaying
TABLE OF CONTENTS
Notation
INTRODUCTION
SLIPSTREAM AS A SHEATH OF DOUBLETS
GOVERNING
EQUATIONS FOR DOUBLET DISTRIBUTIqN AND WING
LOADING
REDUCTION TO ONE DIMENSION (MULTIPLE LIFTING LINE
APPROXIMATION)
REDUCTION TO COMPUTER ALGORITHM
CALCULATED EXAMPLES
CONCLUDING REMARKS
REFERENCES
APPENDICES
FIGURES
iv
PAGE
1
1
4
7
9
20
21
23
"
~urIAS
RDCII!
JO.
169
Institute for Aerospace Studies, University of T oronte
AIIIIcmJWIIC8
r.
1IIJÇ-8LIPSTIIEAIl mERACTIOlf:
A lfUMEIIICAL STUDY
IlUa, ••
24
p&ge'20
t1&ur'.
1. A.rocIyMa1cI
(inteterence) 2. V/STOL 3. W1D&
theory
4.
Propell.er
ettect.
I.
EU1I,
.N.
Il. IJrIAS Report
No.l69
A tuncta.Dtal theory
ot
ving-.lipstre .... interaction accounts for .Up.tre
...
ot
arbitrvy cros.-section by ... n.
of vort
ex .heath.. These
.heatha
together with
the
w1ng
circulation
psttern are
dictated by
th. boundary
condi
Uon.; the analy.iI
lead. to .imultaneollS integral
equationstor
th.ir determination.
In ..
multiple
litting line approx1mation
the
ae are
ultilll&tely reduced to dmultaneollS linear
algebra1c equation.
t
or
machine inversion.
PrOgrlLmS
for the IBM 360-65 digital
cOllputer
beve been developed
tor
tbe
case of
round
.lipstre
...
di.tributed vitb
lateral .~try
on a rectangular ving.
The
co""uted
span loading. have tbe
sbape
expected
from experiJDent.
A
sequence of curve.
of
&ddeei lift due to the
.lip-.tre
...
(integrals
of the
span loadings
)
show
a progression from
'slend
er
body
theory'
tor
ver
y
narro
v
.lip.tr ...
to
'
strip .tbeory'
for v.ry broad slip.treams.
Excellent agreeaent is
seen
vith .he experiaental
result.
of Brencklllann, botb for
_pan loadings end integrat.ed
Uf
.
.
T
he
expected decrea.e in added 11ft
occurred
vith vertical
displaceaent
o
f
.he
sUpstream;
bowever,
l
atera
l
.eparation of
the
_lipstreaaa .howed Uttle effec.
.
Calculated
flow fields down.tream -of th.
ving
~
,
l'
IIrW
RDCII!
110.
169
Institute for Aerospilce S1udies, -University ol T oronte
ADlGMJWal:S
r.
1IIJÇ-8UI'8TJIIAM IJrrIIIACTIOlI: .\
lI\NI:IlICAL
STtJDY
1lUa,
I.
24
pace.
-
20 ~1&ure.
1..
~ca
(intet
... )
2. V/STOL
3.
W1D& tbeory
4.
Propeller
attect.
1.
11U., 11.
Il. IJrIAS
açort
110.169
.\
~taltbeory
ot w1D&-.l1patream
interaction
accounts
for .Up.tr ...
of
arbitrv;y cro •• ·.ectionl1;y _na ot vortex
.beatb.. These .heath.
togetbor
with
th.
w1ng
c1rculation pattern are
dicteted by the bOunciar;y condi tion ..
;
tbe analy.1I
-le&da
to
.imultaneoua
integr&l eqUlLtions
far their determination. In
..
multiple
Utting lino &pprox1mat1C11 th ••
e &re
ulU
...
tely
reduced to
·
8illlUlteneoua llnear
algebra1c
aquat10na
tor -.cbino
inveraion.
ProgrlLmS
for
tbe IBM
360-
65
digital
c~ter
have
been developed
tor the
case of
round slip.tr ...
distributed
vith
lateral
. _ t r y
on
a
rectangul&r ving.
Tbe co""uted span
loadings have
tbe ahape
expected
frOOl
experilant. A
.equence
of curve
.
of &dded Uft
due to
the
sUp-.tre
... (integrala
ot
the .pan
load1nga) sbow .. progression
from 'slender bndy
theor;y' far nry
aarrow
.Up.tr ... to 'strip tbeory'
for very
bro&d slip.treams.
Excellent aare_nt 11 leen
v1th
the experiaen
tal
r.sults
of Brencklllann, both
for
apan loadillg.
end
integrated 11tt.
The expected
decr.ase
in added
lif
t
occurred
Idth ver.tical diaplac_nt
ot
the .lipstream;
hov.ver, latera
l
separation of the
.lip.tra
. . . .
bowed
l1ttle
ettect.
Calcula.ed
flow
field.
down.tre ....
of
.tbe
I(ing
~
-A
A
A
s
A
'W
b
B
B
c
C
C,e
CL
CT
D
lill
D
E
F
F
g,
G
h
, H
J
.
'J
K
K
l
,e
NarATION
cut parallel to x axis
influence coefficients forming part of a linear matrix
slipstream aspect ratio (jet diameter/wing chord)
wing aspect ratio (wing span/wing chord)
span of
recta~gular
wing
cut perpendicular to x axis
infl
u
ence coefficients forming part of a linear matrix
chord of rectangular wing
influence coefficients forming part of a linear matrix
lift coefficient/unit span
overall lift coefficient
thrust coefficient
induced drag
~.ç
L
q .b.c
o
T
q •
b.c
o
increased induced drag due to the jet wit
h
total lift constant
influence coefficient forming part of a linear matrix
influence coefficient forming part of a linear matrix
force acting on a control volume!
=
(F , F , F )
x
y
z
influence coefficients derived from converting integrals to
summations
influence coefficients derived from converting integrals to
summations
influence coefficients derived from converting integrals to
summations
nurnber of jets
half the number of circumferential segments of the slipstream
above the wing
a constant
lift/unit span
~
L
,
•
L
6.L
M
n
N
p
q
6q
q
r
R'
R
S
T
T
T
0
T.
J
T
c
t
u
U
V
v
e
x
-dummy parameter for line integration
total integrated lift of
wing
half
the number of spanwise
segments
of the
wing
immersed in
the
slipstream
total added lift
half the number of circumferential segments of
the slipstream
normal to a surface; outward drawn from
slipstream; upward
drawn from wing
half
the number of spanwise segroents of
the
wing
local statie pressure
unperturbed dynamie pressure
differential dynamie pressure
6q
=
qj
-
qo
average dynamie pressure
q
.5
(q.
+
q
)
J
0
distance from a souree point (on
a surfaee to
a field point)
r
=
(r , r , r )
=
x
-
x
x
y
z
- f
-s
radius of jet
linear
measure
of size of eontrol volume
surfaee
of
wing
-plus-
wake
or jet
boundary
thrust of jet
part of eontrol surfaee in the Trefftz plane
thrust eoeffieient
T
T/(q .?T.R,2)
0
o
.
thrust eoefficient
T.
T/(q .• ?T. R,2)
J
J
thrust eoeffieient
T
e
T/(Q.?T.R,2)
tangential eoordinate of jet boundary
perturbation veloeities
u
=
(u,v,w)
unperturbed
veloeities
u
=
(U,V,W); V
=
W
o
control volume
tangential velocity of rotation in slipstream
point in space; in cartesian coordinates x
=
(x,y,z); in jet
coordinates x
=
(x,t,n)
ex
r
6,'
E
e
TT
p
Superscripts
"
IIIangle of attack of wing
slipstream angle of rotation; clockwise rotation (viewed from
rear) is positive
local slope of jet boundary
circulation (or doublet strength) on wing-wake or jet boundary
surfaces
values of circulation at selected spanwise points; used to
discretize the continuous sheets
width of a circumferential segment of the jet boundary
parameter used to indicate approach to a limit
z coordinate locating centerline of jet
y coordinate locating centerline of
,l
jet
angle of the clockwise jet boundary tangent to the y coordinate
values of trailing vorticity at selected spanwise points; used
to discretize the continuous sheets
velocity ratio
fl
=
U
jU.
o
J
x location of plane for matching boundary conditions
density of air
parameters for integrating control area
control area
summation symbol
reduced perturbed velocity potential
perturbed velocity potential
pertaining to slipstream
velocities before conversion to reduced domain
effect at wing of jet disturbance
effect at jet boundary of wing disturbance
effect at jet boundary of jet disturbance
Sub scripts
a
b
c
d
f
i
j
j
j
L
m
n
p
q
s
ST
TE
00
00
augmented (i.e., jet present)
basic
(i. e ., wing alone)
continuous part of variabie
discontinuous part of variabie
field point
refers
t
o
a particular lifting line or ring
refers to a particular
jet
refers to conditions inside of
jet
general
sununation index
refers to left intersection of
wing
and
refers
to
source point on
jet boundary
refers
to
source point
refers to field point
refers to field point
source
point
Strip theory
trailing edge value
far field value
value at Trefftz plane
on
on
on
viii
wing
wing
jet boundary
jet
(
INTRODUCTION
The interaction of wings with slipstreams has been a field of interest
in
aerodynamics for many years. The problem of determining the auglJlentyd lift
due to this interaction has been approached from several viewpoints
,1-1)).
These
solutions
have, however, placed
relatively severe
restrictions on the ranges of
validity
of the
theories.
The
simplest
possible approach was to assume no induced effects, thus
each section
of the
wing
acts independently like a two-dimensional wing. The
lift
augmentation
obtained using this method is the limiting value for high
as~ec)t
ratio wings with
very broad
slipstreams.
Somewhat similarly, Smelt and Davies,3
developed a semi-empirical
theory
which
is
widely
used.
KOning(l)
first proposed a method to account for the interaction between
wing
and
slipstream.
The presence of the
wing was
represented by a Prandtl type
lifting
line
and the slipstream by an imaging technique correct only f
o
r high
aspect ratio
wing
segments. The resulting solution was valid asymptotically
for broad
slipstreams.
It
became
clear(7) that
an~approximation
(at least) to a lifting
surface
approach
was
needed
because
the slipstream normally
o
nly spans a
low-aspect-ratio wing segment.
As aresult
several
years ago
Graham
et al(7)
de-veloped
a
lifting
surface
theory for the limiting case
Qf
a very narrow slipstream
via "slender-body-theory"(2). More recently, Rethorst(Ö), in his Ph.D. thesis
presented a theory based on a refined imaging technique without aspect ratio
limitations.
A common feature of these
theor~e~,
an exception being in reference 7,
is the use of
an
image technique. Ribner,9J developed a general'exact' theory
that avoided the need for images by representing the
slipstream
boundary as a
sheath
of doublets (or vortices). This theory also eliminated the restrictions
on
number,
position,
and shape
of the slipstreams, but
was
in too complex a form,
as
it
turned
out, for
successful
implementation by computer.
Later, Ellis(lO) reduced one of Ribner's algorithms
to
a viable
compu-ter program.
The results
appeared to be very plausible, but a serious flaw was
found in
reviewing for publication; this is discussed later herein. The present
work was
an exploration of
alternative
algorithms
based
on the Ribner theory with
the
aim
of
developing
a valid computer program for computing both wing loadings
and
flow
field
for a
variety of cases of wing-slipstream interaction.
SLIPSTREAM AS A SHEATH 0F DOUBLETS
The basis
for
the analysis of this problem is the assumption of
smal~
disturbance
incompressible
potential flow.
5
SKETCH 1
Thus the boundary con.ditions may be linearized and may be applied at
the
projection
of the
wing and the unperturbed slipstream boundary in the y-z plane. The wing
flow
is reqpired
·to
satisfy
the Kutta condition.
The thin wing (whose
projection
in
the
y~z plarle
shown in sketch
1 by
the line
S),
at an angle of attack
a
is partly in a stream of
velocity
U
and
partly
in
a
jet
of
velocity
U..
The
wing induces a
pe~turbation potenti~l ~
in
the tree
stream and
~
.
in the
J
jet.
0
J
At the boundary of the
slipstream~
which is a free
surface,
twocondi-tions must be satisfied. The pressure must be
equal
on adjacent points on the
inside
and
outside
of the jet
~undaryo
In addition the flow inclination on
adjacent points on the inside and outside of
S'
must be compatible as shown
in
·
sketch
2.
These two
conditions
may be written using
Be
rnoulli's
equation.
~I
A1
,
.
\1
Pressure:
~
f3
~
U
A~
cut
.
8-B
.'
=
p
+
1.
pr!-00
2
0
=
p
+
1.
ptt
• (Xl2
.
J
1
2"
1
2"
SKETCH 2
p
.
(u
+
0
u'
)2
·
0
(
+
')2
P U.
u.
J
J
Inclination:
v'/(u
+
u')
=
v~/(u.+
u.)
I
0 0 0
J
J
J
s'
1
2
2"
p(v' )
0
1
p(Wj)2
2"
=
When. these
equations
are linearized they become
Pressure:
-p
U
o
~'
=
0
Inclination: v:!u
o
=
Now
u'
=
o
,
-pU.u.
J J
v'./U.
J
J
êJif;.
=
J .
v'
=
Öx'
0
v~~~
J
on
g
.
B...;
'1
(cut
A-A
I(
,)2
2
1
W
0
"
.4
.
on S'
,
on
S'
o~
S'
on
S'
, where
n
is
the outward drawn normal to the surface
S'.
Af
ter
jU)
dx
the
equations
become
Pressure:
u
~
=
U.~.
o
0
J J
on
S'
êJif;
làn/u
=
êJif;.
/ènlu.
o
0
J
dJ
Inclination:
on S'
( 1)
(2)
U)
(4)
(5
(
6)
..
..
The potential jump, eq.
(5),
can
be
satisfied by a distribution of
doublets
on
S'
(or by an equivalent distribution of vortices). The normal
velo-city
jump,
eq.
(6),
can be satisfied by a distribution of
sources
(or sinks) on
S' .~9J
It is now
convenien~
to introduce a new parameter
~
=
U
jU.,
the ratio
of
the
unperturbed velocities.
~
The problem can
be
simplified if
either the
source
or doublet
distri-butions
can
be eliminated.
This can be
done
by
util~zi~g
a special 'reduced '
potential
within
the
jet. In
linearized
wing
theory~llJ
the
wing
and wake are
represented by a doublet sheet
so
the
'reduced ' potential should preferably
remove
the
source distribution
of
S'.
Define the new perturbation potentials
as
(7)
(8)
Substituting these new
potentials into equations
(~)
and
(6)
yields the modified
jet-boundary conditions
Pressure:
~
2
cp
0
CPj
on S'.
Inclination:
(xp
0
(xpj
dn
dn
on
S'.
(10)
The
normal
velocity jump
and
hence the source distribution have been
removed;
however, the potential
jump has the augmented
5
SKETCH
3
value
on S'
(11:)
The resultant flow
field of
the wing
plus slipstream can be represented
in
terms of
a distribution
of doub
lets on'
S
Iof local strength
f(
a factor of
4
~~
is
ab~orbed
in
the
definition
of
strength)
together with another over
wing
and
wake(9J.
The reduced potential
cp.,
and derivatives of
CP.
(the reduced
per-J
J
turbation velocity
components)
are just a fraction,
~
=
U jU.,
of the actual
o
J
>. »
potentia
l
and velocity components.
of
7f!
j
is
Thus the wing boundary condition
in
terms
êYIj;
.
.
~
= -
U
.a(x,y)
az
J
inside
the jet, which becomes in terms of
~j
Outsi
de
the
jet both forms of the wing boundary condition are
./
(xp
0
=
~
= -
U a(x,y)
az
0
on
S
(12)
on
S
(13)
on
S
(1
4 )
The effect of introducing the 'reduced' potential inside the jet
is to
make the
flow like one with an unperturbed velocity everywhere of U. All
of
the jet
interference, however, is retained by the inclusion of
the
~oublet
sheet
on the
jet boundary. Once the 'reduced' problem is solved the solution
to the
actual problem may be obtained by multiplication by
~-l
inside the jet.
(11)
Af~er
completion of the present work, the powerful method of Hess
and
Smith
,for a related problem, was brought to the author's attention.
Reference
11 dealp with the potential flow over arbitrary bodies; it employs
an unknown
sheet
'~f
singularities over the surface of the body,
to
be
obtained
via solution
of an integral equation. The Hess-Smith approach, however, lacks
any
mechanism for dealing with slipstreams.
It does seem, however,
that the
key notions
of the two methods could be combined to provide
very
great
genera-lity in the
flow situations that could be treated.
GOVERNING
EQUATIONS
FOR
DOUBLET DISTRIBUTION
AND
WING LOADING
It
is
well known that the lifting surface theory of a thin wing
can
be
developed from a doublet approach. The wing and wake are represented as a
sheet
of doublets whose axis is perpendicular to the sheet, with the negative
side
up. We have shown that the presence of the jet adds another doublet
dis-tribution
covering the jet boundary. Both of these doublet distributions
are
unknown;
however, the existence of two boundary conditions allows their
evalua-tion
by a pair of simultaneous integral equations.
r/
U
Z
të
,
X
/
/
/
I
I
SKETCH
4
,.
Let the doublet strength on an element dS
at
point
~
x
on the wing
s
-s
(sketch
4)
be
)'(x
).dS , also let the strength on
the
element dS' at point x'
-s
s
s
-s
on the jet boundary be )"(x'). dS'. The potential at a general point!f is
(12)
-s
s
q)(!f)
=
hJ
)'(~s) ~z (~)
dS s
+
h
J
)"(!~) ~n' (~,
)
dS~
S
s
S'
s
(15)
where
n'
=
(0
n'
.sin(8')
n'cos(8'».
Equation
(15)
is
not
of convenient form
-s
'
s s '
s
s
for
computer
solution;
a
modified
form
which
is convenient is:
1
J
r z
1
J
q)
(x
f
)
=
r-
)'
(x ).
--3
0dS
+
r.-
)'
,
(x' )
•
-
Lf7r
S
-s
(r)
s
Lf7r
S
'
-s
r'sin(8') + r'cos(8')
y
s
~
s
(r' )3
The three components of velocity at the
general
point !f are
dS'
s
( 16)
Af
ter appropriate
differentiation
of equation
(16)
and integration
by
parts
(see Appendix A) these velocities are:
d)'(X
)
v(x )
=
l-J
-s
- f
47r
ày
s
r
z
. --3 .
(r)
dS
s
r'sin(8')+
r'cos(8')
y
s
z
s
(r'
)3
- r'sin(8')
x
s
dS'
s
(18)
0)'
,
(x' )}
dS'
-s
s
"""'"'ax-s
:-' -
(
r
I )3
(19)
0)' , (x' )
o)'t (
x')}
dS'
ax~s
+
ry'~' at~S
s
3
s
s
(J)
(20)
where t'
=
(0, t' .cos(8 ' ), -t' .sin(8') ); n' is the outward drawn normal to the
surface S'
and!' the
clockwise
tangent.
Let
!j be a point
on
the
jet boundary and !f an adjacent point just
outside of the jet. Likewise, let
!w
be a point on the wing and !f an adjacent
point above the
wing. The
boundary conditions on the
wing
and jet boundary
are recalled as:
'
."
w(x ) - -
u
a(x ;
y )
-w
0
w
w
tiP
(X.)
:;.
)"(x'.)
=
(1-\-l2)
cp
(X'.)
-J
-J
0
-J
.
Substituting
equations (16) and (20) into these boundary coriditions yields:
-u
o
a(x
w w
,y
)
lim
1 -
~[r
Cl)'(x )
Cl)'(x
)
]
dS
-s
+ r
-s
s
=
dx
~
~f
--. X
-w
4
x
.
s
y dYs
~
-
t7r
J
[r~cos(8~)
dl' (x')
-s
.
+ r'
Cl)" (x' )
,
·
-s
]
dS'
s
(21)
dX'
s
Y
dt'
s
(r'
)3
S'
.dS
s
1
J
z'''sin(8')+ r"'cos(8')
}
+
~
)"(x').
y
s
z
s
dS'
47T
S
, - s
(
r'')
3
s
(22)
S· is the entire slipstream boundary surface and S
is
a surface
inclu~
ding both wing and wake. Equations (21) and (22) thrs constitute two simultaneous
integral
equations for the two unknown distributions )' and
)".
If equations (21) and (22) can be solved
(i.e.,
numerically)
for
l,
the lift distribution on the wing can be obtained. Since
~nside
the
jet
a
'reduced'
potential was used, the solution for )' there must be amplified by a
factor
\-l~l
=
U./U. The lift distribution is given by:
J
0
as:
outside jet:
Inside jet:
lift/unit area
=
--!JU
~
lUt/unit area
=
Pu~
{ (
*
~~
1.2.
.
,
(23)
The lift/unit span is given af ter integration in the stream direction
Outside jet:
lift/unit
span
=
PUo)'TE
Inside jet: lift/unit span
=
p\-l-2UO)'TE
.
lu.1)
.
Equation
~relates
the lift/unit area to d)'/dX which
is
equivale
n
t
.
to a bound vortex strength (see Appendix A)).
From
this
relationship, since the
wing lift is zero behind the trailing edge,
i
Q)',i.@x
,
i~
..
likew~se
Il
Z~0
·
,
tnel'
e
c
and
t
he
effective region of integration is
reduced
to over the wing
only.
A
similar
reduction
is not possible on the surface of the slipstream.
.
...
..
REDUCTION TO ONE DIMENSION (MULTIPLE LIFTING Llm; APPROXIMATION)
An exhaustive attempt was made to obtain a numerical solution of the
full integral equationä (eqs. (21) and (22)), which was ultimately unsuccessful.
The method employed as well as some interesting details, applicable to any
solu-tion, and the detailed causes of failure are described in Appendix B.
In one
version the two dimensional formulation
w
o
u
ld have entailed a solution of 200
equations in 200 unknowns
w
i
t
h a packed matrix, and ot
h
er difficulties. Thus
there was astrong motivation for an approximate reduction to one dimension wit
h
a potentially very considerable reduc
t
ion in the number of unknowns (to 30 for
a comparable case). An earlier reduction to one dimension(9,10) proved to be
inaccurate (see Appendix C); hence a more physically realistic reduction to one
dimension was sought.
A great simplification of the integral equations results from
sub-dividing the wing chordwise int9 a)number of equal
~eg~ents;
each segment is
then treated as a separate wing\13. The Pistolesi,l)) approximation of a
lifting wing by a lifting line is employed for each wing
segment~
a fixed
percentage of the entire wing circulation is compressed into each of these
lines, which are located at c/4 of eac
h w
ing segment.
The flow tangency
con-dition is applied solely along the 3c/4 line of t
h
e rearmost segment (see
Appendix
C
for explanation of choice of loca
t
ion and determination of the fixed
percentages).
If we extend the concept, then by analogy, a fixed percentage
of the bound circulation distributed ring-wise alon
g
the slipstream boundary
is compressed into each of the 'lifting rings' located at the segment c/4
stations. The boundary condition on potential (equa
t
ion 11) is then applied
solely at a ring about the slipstream which is located at 3c/4 of the rearmost
segment.
Sketch
5 exhibits the qualitative build-up of doublet stren
gt
h (or
circulation)
w
ith x in a plane y
=
constant. The line vortex, ring vortex
approximation is equivalent to replacing
r'
---=~---~~---~x
1
---~~~~---~x
SKETCH
5
7
x,
the entire gradual variation by a series of steps equally spaced
in
the x
direction (each located at a segment c/4). Sketch
6
shows both sets of
vortices
(as cut by a plane y
=
constant, specifically for
5
segments)
and
the equivalent step-function doublet (circulation) distributions
o
l'
1=0
---~~~~---~x
SKETCH
6
The complete lifting-lines and lifting-rings, along
with their
appropriate trailing vortices appear somewhat as shown
in
sketch
7
0
X'f
-1-
-_
.#-
-
-1
1
-1
_ _
-,/ ' \ I
---SKEl'CH
7
,
.
Ol I
