A THERETICAL
AND EXPERT1'MNT
INVE7STIGATTON 0'P THE "DERFORMANCE
O7 7T,APPED
TTThDERS
by
Bohdan W. Orrnenheim
Undergraduate
M.S.,
Diploma,
Warsaw 7olvtechnic,
Poland,
1970
Stevens Institute of Technology, 1973
A Thesis
Submitted in
Partial Fulfillment of the Requirements
for the
Degree of Naval Architect
at the
MASSACHUSETTS
TNSTITUTE OF TECHNOLOGY
May, 1974
Signature of the Author:
Denjrtment of Ocean Engineering,
1 av, 1P74
M
Certified By:
Thesis Supervisor
Accepted By :
ARCHIVE9
Chaman, Dent.
JUL 16 .974
Comm ttee on Graduate Students
ABSTRACT
Two flapped rudders with .10
and .20
flao
areas of a typical
high speed vessel were examined exnerimentally in a water tunnel,
in
free stream,
and behind a oroneller.
metry are discussed
ration,
thickness,
A lifting
in detail, in
sween angle,
Effects of rudder geo-
particular,
effects of taner
and flan size.
surface nrogram was written and used in
compare and choose the best rudder planform.
order to
Pesults include
free-stream coefficients of lift, drag, moment and flan moment,
for a wide range of angles of attack and flan deflection angles.
Effects of variations
amined.
in
nropeller-rudder axial clearance was ex-
Presented is a discussion of effects caused by viscosity,
gans,and wall interaction.
Comnarison was made with the nerform-
ance of other flapped rudders.
A rudder with 20. flan was selec-
ted as one having the best performance characteristics.
A strong effect of rropeller wake on rudder characteristics
was observed.
TndeDendence
of rudder forces on axial clearance
between propeller and rudder was noticed.
CON TmENTS
1.
Introduction ...............
1
2.
Rudder Shape ...............
3
3.
Design of Test Equipment
17
...
4. Accuracy of Measurements and Data Reduc tion ..............
53.
Range of Parameters measured
32
6.
Results and conclusions ....
33
7.
Comparison of Theoretical an d Exrerimen .a1
Data . ....................... ...........
Acknowledgement ............ ...........
0.
61
Nomenclature ............... ...........
63
10.
References ................. ...........
11.
List of Figures and Tables
12.
Appendix
. ...........
(Lifting Surface Program)
66
1.
INTRODUCTION
Flaps have long been recognized and apolied in aerodynamics
to increase the lifting characteristics of control surfaces.
But prior to 1972, there appeared to be very little systematic
data available on flaped control surfaces, with aspect ratios
suitable for ships.
In
1968-1972,
a project was carried out at MIT,
to provide
the beginning of a systematic series of experiments yielding
flapped rudder data of direct use to the designer (1).
It
was an
experiment which determined the free stream characteristics of a
series of twelve rudders with systematic variations in the amount
Shapes of rudders were tvnical of
of flan area and flao balance.
a high sneed vessel.
to be 1.4.
Asoect ratio was chosen rather arbitratelv
Flap size varied from 20% to 60% of the total project-
ed rudder area, and balance -
(defined as the distance the hinge
line was moved aft of the center of circular radius flan leading
edge as a percentage
of the nominal flap chord)
-
from 0 to 10%.
Among other interesting results of that investigation,
it
was observed that the 201 flan, no balance all-movable rudder,
had the best characteristics.
It was also concluded that balan-
ced flaps produce disadvantageous
flow effects.
This information aroused a speculation that nerhans an allmovable rudder with a smaller than 20Y
flap, of zero balance,
would be more beneficial.
The objective of the nresent work was to obtain the steady
-l-
force coefficients acting on a 10
flap rudder in
free stream, as
well as behind a propeller for different propeller-rudder configurations, namely for single screw-single rudder, double screwsingle rudder,
and double screw-double
speed vessel.
-2-
rudder of a tyrnical high
2.
Rudder Shane
The initial
series
(1),
idea was to nreserve the rudder shane from the
decreasing only the flan area to 105 by moving the
flan hinge axis towards the trailing
edge;
and,
if
needed,
ing the model size to suit the geometrical reouirements
nronortions of the tunnel test
section,
scal-
of mutual
and nroneller and rudder
sizes.
The first
difficulty encountered
that the hinge line of the 10
flan,
during this attemnt,
(which was arbitrarely
strained to be nerpendicular to the root and tin sections),
was
conwould
intersect the rudder tin so far aft that the thickness of the tin
was too small to allow installation of any bearing of renuired
strength.
Several solutions were possible:
sween angle,
section,
to increase the rudder
thus moving the tin section aft relative to the root
so that the ratio of the flan chord to the tin chord
would be much larger at the tin than at the root , or to increase
the basic thickness of the tip section, or finally to increase
the taner ratio
to apply all
(ratio of the tin chord to the root chord);
these changes simultaneously
Here a matter of ontimization
or
In some suitable way.
Of these narameters became of prime
imnortance.
Decision at the thickness change was based on the exnerience
from the original experiments (1).
Those rudders tapered from a
root thickness ratio of 0.? to a tin thickness
-3-
ratio of 0.1 following
typical Practice.
However,
one of the reasons given in (1)
for
the poor maximum lift coefficient of the rudders comnared for
example, to the Whicher-Pehlner data (14)
a uniform thickness ratio of 0.15.
is that the latter has
'or this reason, it was de-
cided to adopt a uniform thickness ratio of O.15 over the whole
rudder span.
as in
Asnect ratio, it was decided, would remain the same
the original rudders,
namely 1.4.
The remaining values to
be determined were the sweep angle of the auarter chord line, and
taner ratio.
Asnect ratio of 1.4 is
too low for lifting line theory anoli-
cation and too high for low asnect ratio theory.
Tn view of the
lack of any analytical solution for ontimization of sween angle
and taner ratio values, a lifting surface Program was written (2).
The listings of this program is included in the Appendix.
This
is a rather general computer program for numerical evaluation of
lift slope, induced drag, rudder efficiency, moment coefficients,
and position of the center of Pressure for any flapped rudder
tra.erzoidalola
form, with a constraint that the hinge line does
not intersect leading or trailing edges,
the root and tin sections.
and is
nernendicular to
Characteristics of rudders without
flaps can also be obtained by snecifying in
the innut,
a dummy
flap area, subject to the above constraint.
Solutions for several nlanforms with systematic variations
in sweep angle and taper ratio were obtained using this nrogram.
Pinal choice of these narameters was based on three values
that appear in the output:
lift slone coefficient, induced
coefficient, and rudder efficiencv.
drag
An additional condition that
had to be satisfied was that the hinge line had to intersect the
tin section far enough forward,
so that at this point the tin
section of assumed thickness would be thick enough to nermit installation of hinge details of sufficient strength.
sweep angle
came out to be 170 aft, comnared to 110 aft used with
The optimum taner ratio came out to be, coinci-
the series (1).
dentally,
(Figure
The ontimum
namely 0.60..
identical with that of' the original series,
1).
the combination of a taner ratio of' 0.60 and a
Purthermore,
sween angle of 150 of the ouarter chord,
resulted in
the trailing
edge of the rudder being nernendicular to the root and tin chords.
Since
the flan hinge is 'also normal tOLthe root chord,
simplification in
is
described in
This
the geometry of the flan was made nossible.
the following chanter.
Since the nlanform of the 1%
from the original rudders,
flan rudder had been changed
and one of the objective
nroject was to comnare this rudder with the 20l
model had to be built,
rudder without a flan,
by filling
flap
of the nresent
rudder,
another
namely one with a 201 flan, and the same
planform as the 10"' flan rudder.
the above,
significant
Tt was also intended to test one
and such a rudder could be ma.d.e of one of
the gap between the skeg and the flap with
a filler.
Span length was determined
tions of mutual proportions,
as well as of the tunnel test
from geometrical considera-
between the rudder and the nroneller,
section blockage limitations.
Tvi-
cal values of the clearance between the ship hull and the nropeller circumference is 0.3 of the propeller diamter D.
Rudder
tip is usually 0.25D above the lowest point of the proneller circumference.
With the nroneller of the diameter 7.4p"
for these tests,
selected
the rudder span came out to be 7.P75".
The sectional shape of the 66 series (5)
(Pigure 1).
was selected for the
original tests, because its maximum thickness is well aft of the
leading edge,
which was desirable
for large flan rudders.
Since
no large flap rudders are included in the current nroject, this
constraint no longer applies.
selected for the current
The 63 series shape was therefore
series,
because
it should have
stall angle and higher maximum lift than the 66 series.
a larger
The 632~
A015 sections have, however, been slightly altered in order to
develop a plain two-dimensional
This permits
leading edge,
prismatic flan,
and flan gap.
the flap section to consist only of a circular arc
and straight lines emanating from the tangency noints
of the leading edge to the sharp trailing
edge,
with a selected
edge thickness of 0.020".
The flap gap was chosen to be very small, namely 0.010", in
order to minimize the flow through it, inasmuch as such cross flow
decreases the pressure jumn across the hydrofoil, thus reducinp
7.031-73
+-0.
200% FLAP HINGE
10% FLAP HINGE
-.
__
____
MAC
4.219
Fig. 1
Planform of the 10% and 20% Flap Rudders
-7-
562
Comparison of Current Rudder with
Table 1:
rogram (1)
Rudders Tested in the Oriinil
Original
Rudders
Current
udders
Modified 63 2 A015
Section shape
6627020
66E10
0.60
Taper ratio
1.40
1.4110
Root thickness ratio
0.1507
0.21)
Tip thickness ratio
0. 1562
0.10
Geom.
aspect ratio
inches
Root chord,
inches
Tip chord,
20% Flap chord,
inches
10% 'Flap chord
degrees
MAC, inches
Area, So.
4.219
5.955
10
7. P75
Span, inches
Sweep angle,
8.925
inches
Tip shape
Location of stock axis
in 5M2AC aft of leading
edge
Flap size and flap
balance, 0 of MAC
1.125
0.562
15
11
5. 711'21
71.4
144.3
sauared off
sauared off
variable
37.0
0 flap
"
10%
20%
-
0 bal.
- 0
-
0
0 flan
"?
20"
"?
30%
30%
0 bal.
0
0
"'
"?
-
-
30
0
w40%it
-
0
140%
140%
50%
"?
-
"~
"t
78
?
1)
-
-8-
"?
I
.3:; T
the lift
force.
The minimum size of the gap is limited by a roughness of the
skeg trailing
edge,
and the flap leading edge,
due to a machining
Details
process, as well as due to biochemical effects of water.
and comparisons of the overall rudder and flap configuration and
the geometry of the oresent oroject rudders, and of the original
series, is included in
ject are presented in
Rudder shanes of the nresent nro-
Table 1.
Figure 1.
Modification of the Rudder to Suit Convenient Manufacture
The NCA 63 2 A015 section had to be slightly modified to oermit
an important simplification.
This section has a straight line
section shape from 75% of the chord aft of the leading edge to
the trailing edge.
In
order to further simolify fabrication
of the 20% flap it was desirable to have this straight line section extend from 70% of the chord to the trailing edge in order
to encompass both the flap as well as the mating surface on the
rudder.
Since the flan chord is constant from root to tip, it
was decided to make the flan strictly two-dimensional for the ultimate ease of fabrication.
In fulfilling this requirement, the
root and tip sections differ slightly both from each other and
from the NACA 63 2 A015 section.
A program outlined in the followirgparagraphs was written
to modify the 63 2 A015 section to oroduce the tio section and
the root section needed to meet the above reouirement.
The out-
put of this program was tabulated data in a form convenient to
the model from 6061-T6
the machinest-model maker who machined
in narticular,
aluminum alloy using a milling machine.
230 soan-
wise cuts were specified along straight lines connecting points
The final machine
of constant percentchord at root and tio.
marks were small and were removed by polishing the surfaces by
hand.
The resultant accuracv of the rudder surfaces
satisfactory.
Roughness does not exceed 0.002"
ance of the offsets
is
very
and the toler-
is smaller than 0.0057".
Modification of the tip foil section by numerical methods
(the referred sketches
was accomplished in the following steos:
are shown
(a)
an
'Figure 2).
The 63 2 A015 foil sections has a finite
edge thickness eaual to 0.032
trailing
(see Sketch 1).
(Prime
denotes values non-dimensionalized by chord length).
(b)
From this section a wedge was removed centered around
the plane of symmetry of the section leaving a sharo
trailing edge:
=
1
Y't=
(c)
(see Sketch 2)
' 0.032 - x'
0 < x' < 1
(1)
0
non-dimensional offsets of 612A019 section
Straight lines that form the trailing
edge of the last
250 of chord were extended beyond the trailing
edge
sufficiently that over the longer chord thus created
the last 30% of the chord would be linear (see Sketch 3).
(d)
A new wedge was added to remove the negative thickness
from the previous sten and the abscissa was rescaled
to go from 1 to 1000.
-10-
This
foil
has a shar
trailing
(e)
I and 5).
(see Sketches
edge
edge
Then a wedge was added to bring the trailing
thickness on model scale to 0.020"
(±0.010"):
(see
Sketch 6).
Y=
Y
As shown in Sketch 7,
flap region.
+
xt
(2)
root and tip sections differ in the
The root section was therefore next modified to
be identical to the tip section between 70% of tin chord and the
edge for this model with a 0.60 taner ratio.
trailing
"hese stens
are as follows:
(f)
edge thickness was removed by
The 6? 2 A015 foil trailing
subtracting a wedre then brought to 0.020"
(see Sketch P).
by adding a wedge:
=
(g)
thickness
Y-.032
- x' + 0'010 .x'3cr
c
Sct()
<x'<
edge wedge with that of the
Compare the root trailing
tip at the 700 of the tip chord from the leading edge
to evaluate a ratio, N=a/b
(see Sketch 9),
by which all
coordinates on the root section could be multiolied to
edge angles of root and tip identi-
make the trailing
cal on model scale.
(h)
the root foil with a sharp trailing edge
Re-evaluate
then multiply all ordinates by the ratio N:
Y=
4
-
3
x,
0.010".
0 < xt
cr
(cont.
-11-
overleaf)
<
(h)
2
4
I
II
100%
0%
100% 112%
100%
L.
I
I
0%
75%
~I~iI~5
0%
TIP CHORD
I00
7
ROOT CHORD".
t7~
+20%
FIG. 2
b
of c,
MIODIFICATIONS TO NACA 63 A015 SECTION
2
-12-
y
=
(i)
N
-
5
0
<
(5)
< 1
x'
4
The last step is to bring the trailing edge thickness
at the root to 0.020".
V
Y'/cr
5
=
+
0.010
(6)
0 < x' < 1
X
Cr
Table 2 shows the comparison between the unmodified NACA
63 2 A015 coordinates and the corresnonding coordinates for the
tip and root sections on the MIT flanned rudder model,
in
modified
accordance with the nreceding stens.
Propeller
The proneller used in the current steps is a tvnical modern
high-speed ship five bladed propeller model no. 4427.
meter is
7.48"
and the nitch P.03".
Its dia-
Design value of J is
0.8.
The onen water characteristics of this propeller, as measured
in
the MIT tunnel,
the same data in
are shown in
PigIure
3.
Tabulated values of
the region of the design J appear in
-12-
Table 3.
Table 2:
NACA
Comnarison of Coordinates
63 2 A015 Basic Thickness Form MIT Modified Section Shape
x
(per cent c)
Y
(ner cent c)
Y
T
V
Root
0
0.5
00
1.203
1.139
0
1.122
0.75
1.25
2.5
1.448
1.8P14 4
2.570
1.330
1.707
2.596
1.298
1.663
2.514
5.0
3.618P.833
7.5
4.3 2
3.715
10
4.997
14.511
5.171
15
5.942
6.1714
6.619
6.910
25
7.091
7.307
30
7.117
35
40
45
7.3814
7.496
7.1435
7.215
7.704
7.812
7.733
7.4-7
7.4'20
7.533
7
7.271
50
55
60
6.858
6.397
5.R20
6.915
6. 450
5.93
65
5.173
70
4.468
7.104
6.602
6.004
.339
4.616
3.813
20
75
.59
4.993
5.056
6.652
5.02h
3.731
3. 884
80
2.991
3.15
3.080
85
2.252
2.4214
2.346
90
1.512
1. 694
1.612
95
0.772
0.964
0.978
100
0.032
0.234
0.1-44
-1 4-
40
0
0.9
0.8
0.7
0.6
o0
I-a
0
0.5
0.4
0.3
0.20.\0.0.-0.1
-
0.0
Fig. 3
0.2
0.8
0.6
0.4
ADVANCE COEFFICIENT J
4
1.0
- Open-water characteristics of NSRDC Propeller 4427 as measured
in the MIT Tunnel.
1.2
Table 3:
15th August 1973
Onen Water Characteristics of Proneller h427
as Measured in ITTW'_ater Tunnel
MODEL 41127
5-BLAD)E
J-COR
KT
KQ
E
0.550
0.560
0.570
0.580
0.590
0.600
0.610
0.620
0.630
0.640
0.650
0.660
0.670
0.680
0.690
0.700
0.710
0.283
0.278
0.273
0.269
0.261
0.259
0.254
0.219
0.245
0.240
0.235
0.231
0.226
0.221
0.217
0.212
0.208
0.0471
0.0465
0.0458
0.0451
0.0444
0.0)430
0.0431
0.0425
0.0418
0.0412
0.0105
0.0399
0.0393
0.0387
0.0380
0.0374
0.0368
0.526
0.534
0.542
0.550
0.557
0.565
0.572
0.579
0.586
0.593
0.600
0.607
0.613
0.620
0.626
0.632
0.637
0.937795
0.08984
0.843023
0.799716
0.798886
0.72036P
0.680112
0.649679
0.617242
0.586592
0.557591
0.530162
o.504180
0.)479574
0.456224
0.434057
0.41204
0.720
0.203
0.0362
0.6)43
0.392965
0.730
0.740
0.750
7.760
0.770
0.780
0.790
0.800
0.810
0.820
0.830
0.8,40
0.850
0.860
0.870
0.880
0.890
0.900
0.910
0.920
0.930
0.940
0.950
0.960
0.970
0.980
0.990
1.000
0.199
0.194
0.190
0.185
0.101
0.177
0.172
0.168
0.163
0.159
0.154
0.150
0.145
0.141
0.136
0.132
0.127
0.122
0.118
0.113
0.100
0.103
0.090
0.094
n.o0o
0.0)4
0.079
0.074
0.0356
0.03'50
0.0344
0.0330
0.0322
0.0326
0.0320
0.0314
0.030p
0.0301
0.0295
0.0280
0.0283
0.0276
0.0270
0.0262
0.0256
0.0249
0.0242
0.0235
0.0228
0.0221
0.0214
0.020P
0.010A
0.0191
0.0183
0.0175
0.6)40
0.6Y4
O.650
0.664
0.660
0.673
0.677
0.61
0.60r1
0.688
0.691
0.6)Q
0.606
0.6(9
0.700
0.702
0.703
0.703
0.704
0.703
0.702
0.700
0.690
0.69)4
0..
0.6P)4
0.67P
0.660
0.37901
0.3557412
0.)9432
0.321915
0.2061415
0.291073
0.276659
0.262862
0.2)406)46
0.23675
0.22 418
0.213144
0.201925
0.191136
0.190750
0.170746
0.161102
0.151796
0.142012
0.131130
0.125730
0.117626
0.10079h
0.102201
0.004 P70
0.007790
~0.08002)
0.07 4203
-16-
KT/J**2
3.
Design of Test Equioment
The experiments were run
water tunnel,
in the 20"x20" test section of the
in the MIT Marine Hydrodynamics Laboratory.
This
facilitypermits tests with water velocities up to 30 ft/sec
(figure 4).
Rudder forces were measured on the six comoonent
force dvnamometer.
steel worm gear,
hole,
in
The base of it consists of a heavy stainless
set on a tapered plug that rotates in
a plexiglass test section window.
a matching
The rudder shaft
passes through a flexible seal, and is securely clamoed. to the
floating structure of the dynamometer (Figure 5).
ing is achieved by a tapered key.
connected to the dynamometer
strain gage load cells.
der, high strength
The floating structure is then
base,
by a set of Lebow Model 3345
The load cells are attached through slen-
steel,
flextures,
as possible to pin-ended support.
movable for calibration,
Angular clamp-
in order to orovide as close
These load cells are easily re-
and can be reolaced by elements of diff-
erent caracity, depending on the requirements of the test.
The
load cells are electrically connected to Lebow Model 6r digital
strain indicators.
The propeller shaft extends into the test section following
the section center axis.
In
order to model the ship bottom,
in
an appropriate scale to the propeller, and rudder sizes, and to
house the mechanism supporting the rudder model, a horizontal
splitter plate had been introduced.
It was an aluminum rigid
flat plate displaced 2.5 inches from the test section ceiling.
-17-
Flf~
Ns
-.--------
---
Fig. 4
M.I.T. Water Tunnel.
-16-
FIG.5
MODEL IN THE TUNNEL TEST SECTION
-19-
Fig.20a
Rudder Model
in the Tunnel Test Section
Propeller-Rudder interaction
Fig.20b
(Photograph taken with a strobe-light)
-20-
This plate was rigidly secured to the unner window by an aluminum
streamlined foil,
area left
which occupied about 2/? of the cross-sectional
above the plate.
Inside this foil
connecting the model to the shaft,
was the mechanism
as well as the hinge moment
sensor.
The splitter plate extended across the whole width of
the test
section,
and was in
addition,
sealed to the test
section
side walls by means of rubber strips,
so that the flow entering
the test
above and below the plate.
section was split
comnletely
The leading and trailing
edges of the plate extended
upstream and downstream from the rudder stock axis.
were faired in
the plate.
order to minimize
As it
Both edges
the leading edge senaration on
later turned out,
and. caused significant trouble.
15 inches
the fpiring was not sufficient
Discussion of this is
presented
in the Section 7.
It
was honed that the solitter
thinner boundary layer,
olate would develon a much
than that 6n the test
section walls,
due
to being less extended upstream.
In
order t'o realistically
model a single rudder-double
screw
and single-rudder single-screw configurations, a rudder model
would have to be displaced off the vertical centernlane of the
test section.
This was accomplished by attaching the rudder model
rigidly to several different circular coverplates, which had
mounting holes in their different chords, in steps of 0.5 inches,
ranging from 0 to 2" from the plate center.
The coverplates were
in turn rigidly attached to a turntable, which was formed of a
wide and rigid collar at the lower end of the shaft
-21-
(Figure 5).
The bottom side of the covernlate was held flush with the
bottom side of the splitter plate.
plates was equal to l/P".
The gap between these two
A stainless 1.5" diameter shaft was
rigidly connected to the coverplate by means of a special bracket,
which was at the same time a housing for the flap moment sensor
and flan ball bearing.
rudder model,
All these mechanical narts, excent the
were completelv sheltered from the flow by the foil
connecting the solitter nlate and the unner wall of the tunnel
test section.
This arrangement also had an advantage of comnletef
ly removing the gap between the rudder model and the unner wall
of the test section.
As it
was observed in
(1),
such a gan has
very disadvantageous effects on rudder lifting characteristics.
Rotation of the turntable together with the coverplate would
change the angle of attack,
but at the same time would change the
clearance between the rudder and the nroneller in
lateral directions (see Figure 6).
both axial and
Tt was an easy task to keep
the axial clearance constantby simnlv moving the propeller shaft
by distance x,
since the nroeller shaft can be moved back and
forth by simply cranking a gear outside of the tunnel.
The lateral displacement (changing as 11 -
cosaI, Pigure 6),
it was hoped, could be neglected inasmuch as this distance is
small for the angles of attack of interest.
The flap was hinged to the skeg at two points: at the tip
and at the root.
The tip hinge was made of a shaft oermanently
pressed into the flap at the zero balance point, and was free to
rotate with a light sliding fit
-22-
in a stainlesshousing,
extending
from the skeg.
This housing had the outer contour faired to
match the rudder contour.
The unner hinge consisted of a shaft
permanently pressed into the flap root section, and housed in a
ball bearing, which in turn, was housed in the bracket connecting
The flan hinge moment sensor
the coverplate to the turntable.
was attached to the flap shaft.
it consisted of a tiller connec-
ted to the flan shaft, by a split collar, and clamning screw,
thus permitting adjustments of the flan angle.
The tiller was
instrumented with four strain gauges for measurements of flan
moment.
The gauges formed a full four arm bridge,
thus
giving a temnerature compensated output of significant amlification.
The outnut was read on a seventh digital Tebow strain
indicater.
The tiller
with the gauges was waternroofed,
ted completely submerged,
and onera-
thus eliminating the need for passing
the flap shaft through the dynamometer base nlate.
An electrical cable was lead through a drilled hollow in the
rudder stock shaft,
sealed and connected to the strain gauge in-
dicator.
Since two rudders were to be tested, each with a different
flap,
the housing bracket mentioned. above,
had a double housing
for two positions of the flap bearing, and flan tiller.
When one
flap was used, the hole in the covernlate for the other bearing
was sealed,in.order to eliminate any flow across the coverplate.
Both rudders had the same relative nosition with resnect to
the coverplate, and with the rudder stock axis.
able to locate the stock at the nosition where it
-23-_
It would be desiris
most likely
6
U
T JNNEL
SIDE
W
'ALL
XI
SPLITTE
PLATE
I
= sina
Fig.
6
y=I-cos a
R
(PROPORTIONS EXAGGERATED)
* = 0.133
a=30*
R |a=
Scheme of the mechanism changing rudder angles of attack
to be installed in
pondent
oractice.
on the rudder stock,
to zero torque
between 100 and 150.
This nosition is
According to (1),
roughlv corresat an angle of attack
this corresnonds to rough-
ly 30% of the MAC aft of the leadinr edge, at the MAC for the 201
flao rudder at a rudder angle of 12.50 , and a flan angle relative
to the rudder of 12.50 .Unfortunately,
because of structural
reasons it was not possible to locate the stock at this point.
In
the current series,
the stock was 37% of the MAC aft of the
leading edge at the TAC
-25-
4.
Accuracy of Measurements and Data Reduction
Much effort has been snent in order to assure the best accu-
racy of measurments and to eliminate most of the side effects.
Rudder stock diameter was increased from 1.0" - what the former
force dynamometer structure permiteed to 1.5" for better stiffness
of the model support and to compensate for the longer shaft needed
due to the introduction of the splitter plate.
redesign of the sealing and clamping systems.
the angles of attack more accurately,
This reauired a
In order to measure
the old system of a mechani-
cal counter connected to the dynamometer warm gear was abandoned
due to its lack of rigidity and a new ontical system was designed
and installed.
It consists of a circular scale mounted rigidly
on the room wall completely independent of the dynamometer rotation.
An optical telescone with a cross hair was mounted on ton
of the dynamometer and fixed to it.
As the dynamometer rotates
with the model, the current scale reading of angle of attack can
be seen in the telescone ocular.
approximately 0.010.
The accuracy of this system is
A special device had been made for measuring
and setting the flan deflection angle in place
test section.
in the tunnel
It consisted of a base attached to the rudder skeg
during the set up, with a circular rail in it.
The upper plate
rotated on the base rigidly, with the flap rotation following the
rail shane.
Rotation of the unner plate relative to the base was
read out optically through a magnifying glass, on a scaled vernier.
As the flap reached its nosition it was clamped by the flan tiller
-26-
clamp.
The accuracy of this device
0.50 or better.
is
Tunnel flow sneed was measured by means of pressure taps in
the contraction section connected to a Meriam !Iodel 33K A3R5 manometer with an indicating fluid with a specific gravity of 1.75.
The typical column height at 20 ft/s
is
1408
mm.
Readings were
The velocities are corrected for
taken with accuracy of 1 mm.
both tunnel and gauge temperatures
effects.
The calibration of
pressure tans was obtained by comparison with a Pitot tube transColumn heigcht of fluid was corrected
verse of the test section.
to the zero velocity meniscus.
In the present case, with a new solitter plate and its supo-
ort body installed and also the presence of the prooeller drive
shaft, a new calibration was required.
A factor was determined
which related. the original calibration to the new test section
configuration.
In particular, the calibration factor, which is
2.183,
numerically
by the actual manometer reading
is rultiplied
and the resultant number is reduced to velocity by the nrevious
calibration procedure.
When this is done, the average water
velocity at the rudder station is
obtained.
The distribution of the ratio of velocity at the given noint
in
section at the rudder position to the average velocity
the test
at this section has been measured using a Pitot tube.
represents
a man of this ratio
dary layer
.
in
P'igure 7
the region away from the boun-
The results indicate that the velocity profile away
from the boundary layer is
quite uniform.
The average ratio of velocity in the current test section to
-27-
the velocity in the original tunnel section is 1.0q9.
ber stays constant
is
for auite a range of velocities (15 -
Typical value of velocity in
sec).
This num25 ft/
the test section during tests
about 20 ft/sec.
Boundary layer thickness on the snlitter plate has been
measured at two cross sections: at the nroneller and at the rudder
positions
(see Figure
9).
This thickness is smaller than at the
tunnel wall due to the shorter length of the solitter
not as small as it
the plate.
but
would be expected from the Reynold.s number on
The explanation of this relatively thick boundary
layer is given in Section 7.
tunnel wall boundary layer.
The plate itself is away from the
On Figure
P there are two olots of
boundary layer thickness at the mentioned nositlons.
represent
plate,
The numbers
again the ratio of velocity at a given point to the
velocity in the test section away from the boundary layer.
The
abcissa represents the distance away from the splitter plate in
inches.
The individual load cells as well as the assembled unit and
the flap sensor were calibrated by hanging weights.
A computer
program was written to provide the final data in the tabular form.
Force coefficient curves were plotted by hand.
The individual
load cell readings were first corrected for zero drift by linearly
interpolating
the zero reading before and after
the test.
The cell readings were then converted to forces in accordance
with the instrument calibration.
These forces were then conver-
ted from instrument axis to stream axes to yield lift, drag and
-28
moment forces.
attack.
Three corrections were aplied to rudder angle of
The first correction accounts for the torsional flexi-
bility of the dynamometer and rudder shaft, and it was assumed
to be a linear function of the measured torque, the constant
having been determined by calibration.
reaches
This correction typically
a maximum of 0.6 degrees.
The second correction accounts for the tunnel wall interference.
Since the test section is not square any more due to
the splitter plate, a special correction had to be made according
to (6) for the span and chord of the rudder and the height and
width of the test section.
It became,
Aa
=
0.9618 cT
ACT
=
0.01674 cL i
(deg.)
Typical values of Aa at the highest lift was 1.30 and this amount
was added algebraically to the measured value.
Typical corres-
nonding value of AcD was 0.032 and again this was added to the
measured drag coefficient.
The third correction was applied after initially
unsuccess-
ful trials to obtain an antisymmetric lift curve when there was
no flap deflection.
It is a correction obtained from the experi-
ments with undeflected flap in
measurements
uniform flow,
and consisting of an aoronriate
aoplied to all
shift
in
the abcissa
on the plots of force coefficients versus angle of attack.
-2 C)....
F--MAXMM
WU)TH OF FAMRED-j
Su. ORT
SPITRPLATE
-
O Pn
QJ4
9V96
40or$ .
LO
.. O
LOO
pi .9.0
o
.n.K.A
/*9
PRO
eel.00
I L..C.
CONTOUR
AKE SURVEY ABOUT PROPELLER AXIS
AT RUDDER STATION
LOOKING DOWNSTREAM
FIG.7
WAKE SURVEY
0.0
0.0
.
0.2
0.4
0.6
0.2
U
U00
0.4
0.6
0.8
1.0
0.8
1.0
1.2
1.4
1.6
1.8
z (irnches)
BOUNDARY LAYER AT
PROPELLER POSITION (UPSTREAM)
U
0.0
0.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1.0
1.0
1.2
1.4
1.6
1.8
z (inches)
BOUNDARY LAYER AT
RUDDER POSITION (DOWNSTREAM)
Fig. 8
Boundary Layers at two locations under splitter plate
-31-
5.
Range of Variables
To sum un the above, the following range of Darameters was
made available for testing in the nresent nroject set uo:
(1)
Two rudders were tested, with 10% flan and with 2flY
flap.
(2)
Plan deflection angles varied between 0 and 3 5
on one tack.
(3)
Rudder angles of attack varied between -100 and +?0o.
(1)
Reynolds number was kent constant and was approximately
equal to 0.9 * 106 based on the MAC length.
(5)
Axial clearance between the nroneller hub and the
leading edge of the rudder at the wAC was varied
continuously between x = 0.T) and x = I.0D where
T)
is
the proneller diameter.
(6)
Proneller revolutions were kept constant at
h50 'M
at the design J equal to 0.P.
(7)
The transverse oosition of the rudder could be adjusted
so that the rudder would be disolaced off the centerline,
by v=0, v=0.q"=0.067D,
y=1.5"=
v=.="=0.134D,
0.201D and v=2"=0.268D, where D is the oroneller diameter.
Unfortunately, the variations described in (7)
were never tested due to the budget and time limitations.
Results and Conclusions
6.
Results of the nerformance tests are shown in Figures 9 - 18
Figures 9 - 12 show the performance of 200
flow for all
angle,
flap rudder in
uniform
combinations of angle of attack and flano deflection
whereas Vigures 16,
19 show the comnarison of nerfor-
17,
mance between uniform flow condition and three different axial
positions behind the proneller location for 20fl% flap rudder.
The latter are,
Figures 13,
however, only for zero flan angle condition.
14 and 15 show the nerformance curves in free stream
for 10q flap rudder for all combinations of angle of attack and
flap deflections, as well as for behind the proneller condition
for two flan angles: n0
cient cL,
drag coefficient
moment coefficient
flo
and 35 0 .
cgo.
cD,
Data shown include lift coeffi-
rudder moment coefficient
The moment data in
ferred to a phantom stock axis located 491
ing edge at the MAC.
tially
cq,
and
Figure 11 are re-
of MAC aft of the lead-
At this noint rudder moment becomes essen-
independent of the flap deflection in the non-stalled range
of angles af attack,
and all
data collapses to a single curve.
The same data reduced to 19% of the MAC aft of the leading edge
of the MAC would result in
a family of curves with shapes simi-
lar to those of Figure 15.
A summary of principal characteristics of the rudders tested
in
this project,
and comparison to,
the 20%
original series (1) is given in Table 4.
flap rudder of the
Comparison of 20% Flap Rudders of the Present Project and of the
Original Series
(1).
Table 4 shows that the current 205 flap rudder does achieve
somewhat better maximum lift than the original rudder in (1).
The lift curve slooe of the current test rudder seems to be independent of the flap deflections for small deflection angles,
and is
slightly higher than that of the original rudder in
region,
while for larger flao
deflection,
this
the onnosite becomes
true.
Drag coefficient of the current rudder is lower at the maximum lift, as well as at the zero angle of attack at all flap deflections.
Stall occurs on the current rudder at similar angles
of attack for small flap deflections as on the original rudder,
but it occurs approximately 2 tions.
SO earlier for larger flan deflec-
Moment and flap moments are similar on both rudders.
The experiments confirmed the right choice of the sween angle
and taper ratio of the rudder, which was expected from the lifting
surface calculations.
the tests,
When the tunnel oressure was lowered during
thusih4-%v cavitation,
it
was observed that the
cavitation inception on the leading edge of the rudder was uniform over the entire span.
On the other hand, -the stall occur-
ance was earlier than expected on the current rudders, which, it
is suspected, could be exolained by non-optimum choice of the
basic thickness form of the current rudders.
Comparison of 20% and 105 FlaD Rudders of the Current Project
In view of a hopebased on the project (1),
-3 1-_
that the 10%
flap
0
0
0
0
0
0
0
00
0
0
0
0
-0
1
CL
60
aCL/9a CL
@ CL
max
max
a=0
max
@ CD
max
@ CD @
a=0
=
CL
max
L/D @
CL
CMF
21.0
20.5
20.0
20.5
21.2
19.6
-0.007
-0.008
-0.007
-0.007
-0.011
-0.017
3.37
0.190
19.6
-0.017
30
6.45
5.11
4.32
3.74
4.90
4.40
3.56
3.49
3.90
3.82
3.63
3.52
3.40
3.26
3.14
2
-0.110
-0.128
-0.150
-0.174
-0.192
-0.208
-0.222
-0.238
-0.105
-0.150
-0.180
-0.227
-0.260
-0.290
-0.310
-0,345
no
no
data
data
avail,
avail.
0.97
1.00
1.04
1.11
1.17
1.35
20.0
22.0
20.0
16.0
16.2
18.0
0
0.11
0.23
0.34
0.41
0.53
0.015
0.018
0.025
0.036
0.049
0.082
0.163
0.342
0.300
0.220
0.258
0.364
5.95
2.92
3.47
5.04
4.53
3.71
35
3.05
1.40
18.0
0.58
0,105
0.416
0
5
10% flap 10
15
20
rudder
25
30
35
1,
0
original 5
10
20% flap 15
20
25
rudder
30
lin (1) .35
2.75
2.75
2.75
2.75
2.86
2.95
2.98
3.06
2.07
2.74
2.82
3.01
3.04
3.06
3.08
3.24
0.80
0.92
0.95
1.01
1.07
1.10
1.16
1.20
0.78
0.88
0.98
1.09
1.19
1.27
1.32
1.40
17.0
19.0
18.0
19.0
17.0
17.0
18.0
18.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
0
0.07
0.15
0.22
0.24
0.25
0.26
0.33
0
0.05
0.12
0.19
0.27
0.35
0.43
0.47
0.124
0.015
0.180
0.017
0.220
0.022
0.270
0.025
0.218
0.031
0.250
0.039
0.326
0.045
0,34 4
0,052
0.20
0.021
0.23
0.023
0.27
0.028
0.31
0.039
0.35
0.050
0.39
0.075
0.42
0.085
0.110 10.47
*
CM
0.260
0.200
0.251
0.255
0.255
0.195
2.86
2.86
2.86
2.86
2.86
2.96
Principal Hydrodynamic
CM
@
ac
@*
ap
CMFmax
30
30
30
30
30
30
max
0
20% flap 5
10
15
rudder
20
30
Table 4:
*
max
max
31
31
31
31
31
31
31
31
30
30
30
30
30
30
30
30
max
0.002
0.005
0.007
0.010
0.012
0.014
0.016
0,017
Characteristics of Rudders
The moment coefficients are reffered to a phantom stock axis located at:
49% of the MAC for 20% flap rudder, aft of the LE at the MAC
24.8% Of the MAC for 10% flap rudder, aft of the LE at the MAC
18.0% of the MAC for the original 20% flap rudder, aft of the LE at the MAC
30
30
30
30
30
30
30
30
rudder might exhibit a more desirable performance than similar
rudders with larger flaps, the results of the present oroject
Table 4 shows that the maximum
seem to be somewhat dissapointing.
lift
is lower on the 10% flap rudder.
Stall occurs at lower
angles of attack on this rudder than on either the ?00 flan or
the original rudders.
Only the drag coefficient is more advantageous on the 105
flap rudder.
is
Plan moment coefficient data for the 10,
not available,
flan rudder
because the flan moment sensor waternroofing
lost its water tightness during the experiments.
Comna.rison of Rudder Performance in 1ree Stream and Behind Propeller
A dramatic increase in lift characteristics was exhibited
on the rudders when tested in the propeller wake. (Tables 5, 6).
Due to a helical shape of the wake and the fact that the rudders
were immersed mostly in the upper half of the proneller wake,
the rudder forces show assvmetry on two tacks, even when there
is no flap deflection.
The 201
flan rudder (Table 5) has a 20%
increase of the lift curve slop due to propeller wake.
lift increases 35% and 32
on two tacks resnectively.
angles occur at 50 and 9O later.
Drag coefficient
Maximum
Stall
at the maxi-
mum lift unfortunately increases as well, but only on one tack
this increase is
very dramatic,
namely 400 %.
On the other tack,
this increase is 40%.
A very interesting result concerns the drap coefficient when
the rudder has zero angle of attack,
-36-
and its
flan is
in the neutral
0
00
0
0
0
Table 5:
0
X
o prop
20% flapT 0.5 D
rudder 0.75 DJ
LL1.0 D.
0
0
0
0
L/ ;a
2.86
3.23
3.23
3.23
0
Q
Q
20?
I
0
0
Flap Rudder behind Propeller
C
CL
max
-a
+
-0.97 0.97
-1.5 1.43
-1.5
1.43
-1.5 1,43
-
CD
Lmax
+a
-20
-25
-25
-25
20
29
29
29
a=0n
0.015
0.004
0.004
0,004
CD
max
@ CL
MalX
aa
0.163 0.163
0.264 0.642
0.264 0.642
0,264 0,642
Table f:
Flap
behind Propell r
10%
10?/ Flan Rudder
Rudder behind Pro eller
Table 7:
Maximum Lift-Drag Ratio
I
tA
-4
1
position.
The drag coefficient then decreases from 0.020 when
in the free stream to 0.00h when behind the proneller.
This can
be explained by the interaction between the propeller hub vortex
and the rudder tip vortex.
In behind the propeller conditions,
rudders "feel" an angle of attack due to the proneller helical
wake, even when the geometrical angle
free stream direction is zero.
of attack relative to the
The rudder tip vortex thus in-
duced cancels partially with the nroneller hub vortex,
and there-
fore the induced drag on the rudder decreases.
The relative nosition of the rudder tin and the proneller
axis can change significantly the rudder drag.
of this geometrical relationship may be worth
The ontimization
some further in-
vestigation.
Another interesting result of the propeller-rudder
configu-
ration is that the steady forces on the rudder are comoletely indenendent of axial clearance downstream of the nroneller in the
range of locations tested 0.5D to 1.0D (D = proneller diameter).
In earlier work at MIT, Professor F.M. Lewis has shown that very
small changes in
axial clearance between the propeller and rudder
could result in a very large reduction in the blade frequency
vibration force on the rudder.
10%
flap rudder exhibits similar behaviour behind the pro-
peller (Table 6).
This rudder was tested at only one location
behind the propeller, namely at 0.75 of the propeller diamter,
since, as it was shown for the 20% flan rudder, the changes in
axial clearance
do not affect the rudder performance.
-2
0.-..
An effect
of one additional parameter was observed,
namely
of the flap deflection.
Increase of lift slooe of this rudder is 20% with undeflected flap and 17% with the 350 flap deflection angle.
Maximum lift is increased by 32% and 495% respectively, on
two tacks with undeflected flap and 38O on the ooerational tack
with flap deflected 350.
Stall occurs 50 later, and 110 later resnectively, with undeflected flap and 40 later on the operational tack with the
flap deflection.
Drag increase on the 10%
flan rudder is
also
significant, 300f and 4005 with undeflected flan on the two tacks
resoectively,
operational
and ?00" with flan deflection angle
of 350 on the
tack.
MaJor Conclusions
The following conclusions
anoly to the 7eynolds number of
the experiments which were about 0.96 * 10*
but there is
no evidence that they do not
(based on the MAC),
?poly to larger Rev-
nolds numbers.
1.
In
view of the above observations,
the 105 flap seems
to be less advantageous than the 20% flan rudder because of its
worse lifting characteristics.
2.
The 20%
flan rudder developed for the present tests is
better overall than the corresponding 20%
inal series (1).
flap rudder of the orig-
It has a higher lift, lower drag, similar mo-
ment and more uniform spanwise loading than the original rudder.
-39-
Since the 20% flan rudder in (1)
was selected as the best
rudder of the whole series described in
(1),
it
can be concluded
that the 20% flap rudder of the oresent project is
all
rudders considered in
both orojects and is
superior to
therefore recomm-
ended for nractical apolications.,
3.
Tt was observed, both in the nroject (1) and in the
current observations, that the ratio of flap deflection to rudder
deflection
angle to produce minimum drag increases with lift.
This indicates that in practical installations, it may be desirable to develor a linkage that incorporates
ween the flan angle and the rudder angle.
L.
a variable ratio bet(See Table 7).
Disadvantages of the all movable rudders with movable
flaps are their increased hinge moments, mechanical complexity
and nossible maintenance difficulties.
5.
Figures 9, 13 and 16 show a remarkable linearity of the
lift coefficient, completely unaffected by the flap action in
the nnstalled region.
*40
-
00
0 3s
z
-"
IL
a
00
-30_
ANL
Fig.9
20% Flap Rudder
-
-0
-to
FA
IL
ERE
in Uniform Flow
-o-03
0
0
0
0
1.2
L
LL
(U
ANGLE OF ATTACK -- DEGfEES
Fig.10
20% Flap Rudder in Uniform Flow
0
0
49
W
(A.
I I
I-
O.2
I
49% Cf MAC AFT CMAC LEADING EDGE
STOCK AXIS
-----
---
-
--
--
--
-
_
--
-
-
-
-
-
@0
a0
u
0
30
o.ao
I-'
as
--
- ------- - --Q2 - --
-30
-20
-- -- ----
-to
0
--
------
10
AIG E
Fig.11
20% Flap Rudder in Uniform Flow
-
--
--
-
W
W
T
.C|
-
z
--
.
0
--
---
-
-
0
30'
G
0 FLAP
C:
U
z
woso
a 10
.0!FLA
205
-30
-20
-0
0
ANGLE OF ATTACK
Fig.12
20%
Flap Rudder
10
DESREES
in Uniform Flow
20
30
0
A
'.5
-
owIle
F
1u
n
-0
-4
-
..-
-
....
-.
--
-
-
--
Z7*
U--
Fig.13
-loo0
ANa.A-
OFATIAK
D~4
10%Flap Rudder In.Uniform Stream and Behind a Propeller
S
VM
vatm11.
.04
.
1--
0123
gvce
FIg.14
10% Flap Rudder In Uniform Flow and Behind a Propeller
x
U
a
w-
.2 - - - - - -
-J--
-
-40
Ftg.15
~
-M
and Behind a. Propeller
10% Flap Rudder in Uniform Flow
9
0
V
PRO10
4x
I-o 5 ..
0.z
uIAp
-3I2
O203
x
-'
ANL O TAK
~ -DECE
..-..
10 ...
-30
- 0
-00103
ANGLE OF ATTACK
Fig.16
20% Flap Rudder Behind a Propeller
DLCM4LES
-
v
-
-
-
------
*X-0.75
aIX-I.
uK
4
-30
---
-
-zoOA
A
AN'.GLE OF ATTACK~
FIg.17 '20%
Flap Rudder Behind a Propeller
~
E
EEE
- - --
-
----
3
Q3
-I
STOCK AXIS
=
49% OF MAC AFT OF MAC LEADING EDGE
02
oL
*
-0.3-
-30
-20
-go
0
0
ANGLL OF AT lACK ~EGREES
Fig.18
20% Flap Rudder Behind a Propeller
20
30
7.
Comparison of Theoretical and Exnerimental
Table
Data
P shows some of the major hydrodynamic performance re-
sults of the 20% flap rudder obtained by the lifting
surface pro-
gram calculations and the exneriments described in the nrevious
chapters.
The center of pressure position is given as a percen-
tage of the MAC measured from the flap hinge line.
Moment co-
efficient curve slone is referenced also to the flan hinge and is
expressed per unit angle of attadcin radians.
Lift curve slopes are given per unit angle of attack in
radians too.
(Table
8)
The values in
the last
will be described later
in
row of the Table overleaf
this chanter.
The table shows that there is quite a strong discrepancv between the theoretical and experimental predictions.
The authors
of (2) believe that the theoretical results are correct to within 1% of the linear solutions and may, therefore, serve as a basis
for comparison with the experiment.
The values of the lift
curve slop
L
on the rudder with
no angle of attack were expected to differ significantly from the
theory, since all the lift was generated by the flap alone, operating in a fully separated, turbutent region, where the net velocity dueto the Von Karman effects
is
considerably smaller.
This
fact is not accounted for in the theory.
On the other hand, the experimental lift slone coefficient,
3CL/aa
,
which is 9% lower than the corresoondant theoretical
value, suggests that a strong side effect must have been taking
S
0
0
0
20% flap
3CL/ca
aCL/as
CL @
a=00
6 =ii
CL @
CD @
a=1008=10o a=10 6=
gCM/aa
XCP/MAC @
CD @
rudder
6=00
a=0 0
theory
3.134
1.771
0.547
0.856
0.0476
-1.893
-60.4 %
0.0085
2.86
1.17
0.48
0.73
0.049
-1.609
-61.6 %
0.0150
0.057
-1.725
-58.7
%
0.0152
experiment
6=00
6=00
6=00 a=j
experiment
iew plate
2.947
-
0.514
-
U,
Table 8:
Comparison between the Experimental and Theoretical results
place during the tests.
It was suspected that the boundary layer on the snlitter
plate (see Figure 8) might be the cause of this decrease in the
rudder load at the neighborhood of the root section.
later confirmed,
As it was
the unexpectedly large boundary layer was caused
by the splitter plate leading edge separation due to a too small
radius of curvature of the leading edge.
In
order to obtain a
auantitative information of how much this phenomena had been
changing the rudder characteristics,
the solitter plate shape
has been changed and one additional test of the 20% flan rudder
was performed.
The forward
3" of the solitter plate was curved unward to
provide a smooth, faired entry to the flow, and the.sunnort of
the splitter plate located between the plate and the unper tunnel
wall was extended aft, up to the solitter plate trailiner edge.
Since this new geometry of the test section forces more flow to
go under the plate, a new wake survey was required.
that the velocity increases now by 15.58
the previous splitter plate shape,
It
determined
more as compared with
or by 15.57% as compared with
the original tunnel test section.
New boundary layer thickness measurements
thickness decreased from 1.4" to 0.35".
showed that the
The latter value is in
a close agreement with the theoretical prediction, which for the
considered Feynolds number on the plate is
results of the test section of the 20
equal to O.34".The
flan rudder in
flow condition is oresented in the last row of
-53-
Tmable
this new
P.
The agreement between the theoretical and exDerimental results is now much better.
with
The lift curve slope on the rudder
undeflected flap differs by 6% from the theory,
according to (8)
which
is within the reasonable limits. It was expected
that the theoretical and new experimental values of the drag coefficient differ now more.
It can be explained by the fact,
that more of the rudder sur-
face is now exposed to the high velocity field, because the wall
boundary layer is thinner, thus causing more of the rudder area
to be subjected to the viscous stresses.
Also the induced drag,
which is proportional to the sauare of the lift coefficients, is
now larger, since the lift is larger.
Having a workable and well checked lifting surface Program,
it was considered worthwhile to confirm the above analysis theoretically,
in other words, to obtain theoretical results of the
rudder characteristics for the condition of non-uniform distribution of spanwise inflow velocity.
The original version of the lifting surface program has an
assumption built into it, that the incoming velocities at all
spanwise positions are the same, non-dimensionalized to unity.
In order to account for the non-uniformity, the velocities
from Tigure 8 were specified at the soanwise stations.
Also, to
increase the sensitivity of the solution to the flow field near
the root section, where the wall boundary layer was sunnosed to
affect the loading, a slightly different vortex and control point
grid was introduced in place of that given in Figure 3 of (2)
In particular,,four uper control noints stations were displaced toward the root section into the region of the boundary
layer.
This change required some changes in the vortex line
distribution in order to obtain a converging solution.
The result of this new lifting surface program calculation
confirmed fully the experimental result,
namely,
that the re-
duction in the boundary layer thickness from 1.4" to 0.35" for
the 7.875" rudder span resulted in 3% increase of the lift curve
slope.
(Figure 19).
Unfortunately, the formulation of the spanwise mode functions
in (2) is such, that all the modes produce a final value of the
circulation at the root section.
This was correct for a uniform
velocity field soecified at the control ooints on the wing.
In
order to obtain a solution for a non-uniform spanwise distribution of the velocity, which is the case if a tunnel wall boundary
layer is Present, another definition of the soanwise modes is
needed, namely, that the mode harmonics have a period twice that
of the existing modes over the same span and are symmetric with
respect to the root section.
Since this modification had not
been introduced to the program, the resultant lift slope coefficient curves in Figure19 do not go the zero at the root section,
where the actual velocity is zero due to the boundary layer, but
rather to the same finite value.
Nevertheless, the author believes that the above numerical
estimation of the overall lift slooe coefficient decrease
-55-
due to
00000000
0
0
0
0
U'
a'
S ,
0.1
root sect Ion
FI G.19
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
tip section
SPAIll SF Di STRIMITio0
OF LIFT SLOPE COEFFICIENT (THEORY)
0
9
the presence of the boundary layer on the tunnel wall is reasonable, since the same result (3%) was confirmed by the experiment.
Since the real rudders operate in the shin hull boundary
layer and wake,
in
the author believes that the results presented
Figures 9 to 18 do represent the realistic rudder performance
characteristics in spite of the fact, that the experiments were
performed on the rudders, which had been unintentionally subjected
to a large boundary layer on the splitter
plate.
It remains to be explained why there is still about a
difference between the theoretical and the new experimental value
of the lift curve slopes.
The lifting surface program (2),
is
based on the linear inviscid theory, which assumes, that for an
uncambered airfoil all the lift is generated by the angle of attack
alone, and that the thickness effects do not contribute to the
lift.
But it is known from a two-dimensional analysis that, in
fact, a viscous fluid does affect the lift by introducing some
secondary effects.
When the wing operates in a viscous fluid,
and is subjected to some angle of attack, the so-called displacement thickness of the boundary layer on the wing is affected by
the potential pressure field around the body, and the streamlines
a.re displaced into the outer regions.
As the displacement thick-
ness is bigger on the suction side, especially near the trailing
edge, due to a lower pressure on this side, an uncumbered wing
becomes effectively cumbered with a negative cumber.
generated by this negative cumber subtracts
The lift
from the lift genera-
ted by the angle of attack. Schlichting in (9) states, that this
phenomena may decrease the overall lift curve slope by large
amounts depending on the Reynolds number, aspect ratio and the
wing thickness.
Another factor contributing to the discrepancy between the
theory and experiment is the three-dimensional effect, namely,
behaviour of the flow near the tip section.
It was shown in (7) that a change of the tin cross-sectional
shape from a square tin to a rounded tip may change the lift by
as much as 3.5% and this effect is not accounted for in the lifting surface theory.
Tests of airfoil models in a closed test section of wind or
water tunnels, where the models are placed at the tunnel walls,
are inevitably encumbered with an error caused by the wall boundary layer effects.
It seems appropriate, therefore, to conclude
this discussion with a short review of some of the major works
done in this field.
In
1944, Preston (10) calculated analitically the loading
on a two-dimensional model spanning a closed wind tunnel.
He
assumed that the loading would be decreased at each end of the
model, in proportion to the square root of the local velocity
through the tunnel wall boundary layer.
in
that work was,
that this change in
Another assumption made
loading will produce
induced effects over the whole model.
From computations of the vortex strength in
terms of the
experimentally obtained tunnel wall boundary layer thickness,
-58-
an
estimate of the induced angle of attack loading over the wing
was made by a consideration of the tunnel width and model chord.
A typical result
of that theory was that for a two-dimensional
wing of the span chord ratio of 2 and the ratio of the boundary
layer thickness to the span equal to 0.05, the decrease in the
lift curve slope varied from 10% at the model ends to 2' at the
midspan.
The resulting decrease in
the overall lift
was
3.5%.
An experimental approach is presented in (11).
Two-dimensional loading tests
NACA
65-012 airfoil
in
were made of a two foot chord
the 2.5 by 6 foot test
section of the
wind tunnel.
This test indicated that only a very small loss (less than
1%)
in
the average
load may be expected.
It was also shown that large changes in the tunnel wall
boundary layer thickness produce small changes in the overall
load.
(10)
The author of (11) concluded that the theory of Reference
strongly over-estimates the effect of the tunnel wall boun-
dary layer.
The same remark, but without any explanation, apnears
in
(12),
Reference
page 383.
The third approach is presented by K'6rner (13).
It is a
lifting surface program for evaluation of an airplane wing performance in the presence of fuselage.
The theory takes into con-
sideration many effects of the fuselage-wing interactions, including the wing twist, relative position of the wing and the fuselage,
and some details of the wing fuselage
-59-
jointing.
It also presents an analysis of the fuselage boundary layer
thickness effect on wing loading.
That approach is essentially
identical with that shown earlier.in this chapter by the author
of the present work, namely, that the control points on the wing
sense the changes of the oncoming velocity field, and the solution for the circulation distribution is a function of the nonuniform velocity field.
The spanwise harmonics are defined in
such a way, that the lift goes to zero at the wing root section.
The author of the present work feels that the lifting surface
theory, when properly formulated, is the best tool in approaching
the analysis of effects on loading caused by non-uniform velocity
stream due to thick boundary layers.
If the boundary layer thick-
ness on the tunnel wall is smaller than 3 -
5% of the wing span,
its effects become negligible, according to (11) and (12),
and a
lifting surface program is not able to detect any difference in
the velocity distribution.
In this case, the nummerical result
obtained for a velocity field which is uniform at all the spanwise stations will have a very satisfactory accuracy.
In view of the discrepancy of opinions expressed by the
authors of (2),
(5),
(8),
(10) and (13),
concerning the quanti-
tative effects of the viscous flow around airfoils, it seems
that this field
is
far from being solved and still
significant amount of research.
-60-
requires a
Acknowledgement
The author participated in this project as a Research Assistant.
Other participents who contributed a significant amount
of effort and time to this project were ProfessorsJ.E. Kerwin
and P. Mandel, Research Engineer D.S Lewis and Technicians
W. Connoly and G. Graham.
In particular, Professor Kerwin is the author of the following parts of this work:
1.
Conceptual Formulation and Description of the Lifting Surface
Program (2).
2.
Data Reduction Program.
3.
Formulation
4
.
5.
of. Modification Method of the Rudder Sections.
Subroutine HSVEL in (2).
Data Concerning the Propeller used for the Tests.
Decision concerning the basic thickness form of the rudders
was conceived by Professors Kerwin and Mandel, based on the Performance
of the rudders described in
(1).
Professor Mandel prepared the first
draft of (3).
Mr.
Lewis
and Mr. Conolly orepared by hand the plots of rudder coefficients
curves and the Figures
testing of the rudders.
7 and
8 and performed some of the actual
Models of rudders were made with very
satisfactory precision by Mr. Kovar.
All other mechanical narts were made by Mr. Conolly, Mr.
Graham and the ME Machine Shop at MIT sunervised by Mr.
R.
Johnson.
I wish to express my gratitude to all these persons for their
-61-
friendly cooperation.
In particular, I would like to thank Professor Kerwin and
Mr. Lewis for their help, time and effort spent in a very friendly manner on many stimulating discussions.
The project was supported by ONR contract Number,
N00014-67-A-0204-0067.
9.
Nomenclature
A
=
total rudder area (flap plus skeg)
a
=
half width of tip chord @ 20% of tip chord forward
of trailing edge (see Figure 3, Sketch 1)
b
=
half width of root chord 9 same longitudinal position
c
=
length of mean aerodynamic chord
CD
=
drag coefficient = D/P/2 AU_2
=
lift
=
rudder moment coefficient = M/ 0 /2 AcU 2
=
flap moment coefficient = MF/0/2 AcU, 2
ct
=
tip chord
cr
=
root chord
D
=
propeller diameter
D
=
total drag of rudder
flap
=
movable after portion of rudder
CM
2
/3(ct
+cr- CTUR)
CT+CR
coefficient = L/P/2 AU_2
flap area =
rudder area between flap hinge location and trailing
edge of rudder
flap gap
=
distance between trailing edge of skeg and leading
edge of flap measured in the rudder plane of symmetry
with zero flap deflection
GHR
=
General Hydrodynamics Research (program)
L
=
total
M =
MAC
=
MF=
rudder
=
lift
of rudder
total moment acting on rudder about the shaft axis
shown in Figure 8
mean aerodynamic chord
moment acting on flap about flap hinge line
flap plus skeg
-63-
Nomenclature
(cont.)
of rudder
skeg
=
forward portion
taper ratio
=
ct/cr
U
=
local velocity near wall or near splitter plate
in water tunnel
UJO=
uniform flow velocity well away from wall
Uw
=
average velocity of flow over rudder in
propeller
x
=
axial clearance between end of propeller hub
and the leading edge of the MAC
=
axial distance along chord of rudder, non-dimensionalized by a rudder local chord
=
transverse clearance between propeller axis and
rudder plane of symmetry P x = 0
x
Y
Y'=
wake of
transverse distance along thickness of rudder,
nondimensionalized by a local rudder chord
z
=
spanwise distance from splitter plate
x
=
angle of attack on skeg = rudder angle
6
=
angle of deflection of flap relative to skeg =
flap angle
p
=
fluid mass density
-64-
10.
References
14,1972
1.
An Experimental Study of a Series of Flapped Rudders
J.E. Kerwin, P. Mandel, S.D. Lewis, JSR Vol. 16, No.
2.
A Lifting Surface Program for Traperzoidal Control Surfaces
with Flaps - J.E. Kerwin, B.W. Oppenheim, MIT report Nov 1973
3.
A Experimental Study of a Series of Rudders with Small Flaps,
Part II (in preparation) - J.E. Kerwin, P. Handel, S.D. Lewis
B.W.Oppenheim.
4.
Free Stream Characteristics of a Family of Low Aspect Ratio
L.F. Whicker, L.E. Fehlner, DTMB Report
Control Surfaces
933, May 1958.
5.
Theory of Wing Sections Dover Publications, N.Y.
6.
The Elements of Airfoil and Airscrew Theory 2nd Edition, Cambridge University Press, 1Q59.
7.
Sailing Yatch Keels HISWA Symp. Amsterdam,
8.
Evaluation of Lifting Surface Programs for Computing the
Pressure Distributions on Planar Foils in Steady Motion T. Langan, H.T. Wang, NSRDC Rep. 41021, 1973.
9.
Boundary Layer Theory
I.H.
Abott, A.E. Von Doenhoff,
H. Glauert,
J.E. Kerwin, H.C. Herreshoff, 3rd
1973.
-
H. Schlichting,
McGraw-Hill,
1955.
10.
The Interference on Wing Spanning a Closed Tunnel Arising
From the Boundary Layers on the Side Walls, with Special
Reference to the Design of Two-Dimensional Tunnels J.H. Preston, PR Soc., 19 Rep. 1924, March 1944.
11.
Effects of the Tunnel Wall Boundary Layer on Test Results
of a Wing Protruding From a Tunnel Wall - R.A. Mendelsohn,
J.F. Polhamnus, NiACA 1244.
12.
Wind Tunnel Technique Sir Isaac Pitman & Sons,
13.
Berechnung der Potentialtheoretischen StromungUm Flugel-RumpfH. Korner, DPVLR.
Kombinationen und Vergleich mit Messungen Dissertation, 34, GFR.
R.C. Pankhurst, D.W. Holder, London,
Ltd.
-65-
11.
List of Figures and Tables
Page
Figures:
1.
Rudder Planform .............................................
2.
Sketches of section modifications ..........................
3.
Propeller characteristics
..................................
7
12
15
...................
4.
TMIT water tunnel ......................
5.
Model in the tunnel test section ............................
6.
Rudder rotations ............................................
7.
Wake survey .................................................
8.
Boundary layer on the splitterplate .........................
9.
Lift coefficient in uniform flow,
20% flap rudder ...........
10.
Drag coefficient in uniform flow, 20% flap rudder ...........
42
11.
Moment coefficient in uniform flow, 200 flao rudder.........
4{3
12.
Flapmoment coefficient in uniform flow, 200 flap
rudder ......................................................
13.
Lift coefficient for 10% flap rudder, in free
stream and behind oropeller..................................
14.
Drag coefficient for 10% flap rudder, in free
stream and behind propeller .................................
15.
Moment coefficient for 195 flap rudder, in free
stream and behind oropeller .................................
16.
Lift coefficient for 20% flap rudder, in propeller
wake ...................................................-..
17.
Drag coefficient for 20% flap rudder, in propeller
.
wake ..............................................---------
49
18.
Moment coefficient for 20% flap rudder, in propeller
wake .. ......................................................
50
19.
Snanwise Distribution of the Lift Slone Coefficient
56
20.
Photographs of the Rudder Model and Test Tunnel Section
-66-
........
....
47
20
Tables:
1.
Comparison of current rudders to the rudders in
the Program (1) .............................................
2.
Comparison of coordinates ..................................... 14
3.
Propeller characteristics ...................................
4.
Principal hydrodynamic characteristics of
rudders in uniform flow .....................................
5.
Principal hydrodynamic characteristics of
20% flap rudder behind propeller ............................
6.
Principal hydrodynamic characteristics of
10% flap rudder behind propeller .............................. 37
7.
Maximum lift-drag ratio for two lift coefficients
8.
Comparison of theoretical and experimental results
........... .37
.........
52
A P P E N D i X
(Lifting Surface Program Listings)
-68-
1.
Sample Computer Output
The first nage of the comnuted output is reproduced on page
95
The value of the aspect ratio corresponds to the wing and
its
mirror image.
The first three lines of the page describe the geometric para-
meters of the wing, reproduced from the innut data.
nition of the symbols is given in Reference (2).
A detailed defi-
The symbols that
anpear in the output are denoted as follows.
z
-
non-dimensional spanwise distance from the root
to the tio
CLA = 3CL/Ba(z)
-
local lift slope coefficient per radian, due to
no flan deflection
the angle of attack,
CLD = 3CL/3(z)
-
local lift slone coefficient per radian, due to
flan de fle ct ion angle ,
=0
CLAR = ;CL/3a
-
overall lift slope coefficient due to a
CLDR = 3CL/36
-
overall lift slope coefficient due to 6
C-ALPHA
-
mode amplitudes for the kth (snanwise) and
Lth (chordwise) modes due to a
C-DELTA
-
mode amplitudes
for due to 6
The second page of the output includes the matrix of the boundary
the control points,
values at all
performed.
It
is
after the calculation has been
a check of the calculation accuracy.
The closer
the matrix terms are to 1 or 0, the better the computation .
The third page of the output gives the results of the calculations performed. by the subroutine
OPTION.
The definitions of the symbols are as follows:
(Overleaf).
-69-
z
-
nondimensional spanwise coordinate
XA/LC
-
local center of pressure position from the flap hinge
line, o a fraction of the local chord, due to alpha
XD/LC
-
local center of pressure position from the flap hinge
line, as a fraction of the local chord, due to delta
CM-A
-
local moment coefficient per radian, with resnect to
the hinge line, per local chord, due to alpha
-
local moment coefficient per radian, with respect to
the hinge moment, per local chord, due to delta
XALE/LC
-
local center of pressure oosition from the leading
edge, per local chord, due to alpha
XDLE/LC
-
local center of pressure position, from the leading
edge, per local chord, due to delta
ZA
-
spanwise position of the center of pressure, due to
alpha, per unit span, measured from the root section
ZD
-
spanwise position of the center of pressure, due to
delta, per unit span, measured from the root section
ALPHA
-
angle of attack
DELTA
-
flap deflection angle relative to the skeg plane
CL
-
lift
CD
-
drag coefficient
L/D
-
lift-drag ratio
CM
-
moment coefficient
XHL/C
-
center of pressure position, per mean chord., measured
from the hinge line
2.
coefficient
Instructions for preparations of Input Data.
The first card has
nine entries:
Symbol
KDM
Column
4
Limitations
KN
Description
os
recommended)
-70-
m
(g
8
LT
4(LT<6
number of chordwise modes (6
recommended)
IHF
12
0, 1
chordwise precision index (0
recommended)
IV
16
0, 1
spanwise precision index (0
recommended)
AR
17-24
O<AR<00
single wing aspect ratio
AF
25-32
0.l<AP<0.9
flap area as a fraction of
the rudder area
T
33-40
T>0.1
PP
4t1-4'8
quarter chord sweep angle
in deg. (positive if swept
downstream)
if KOPT = 1, the subroutine
OPTTON will be called and will
perform the calculations appearing
in the the third page of the
0 or 1
56
KOPT
taper ratio = tin chord/root
chord
output
The geometrical constrains are that the hinge line must be
perpendicular to the tip and root sections, and it cannot intersect neither the leading nor the trailing edges.
The second card has 10 entries; they are the numbers specifying the chordwise positions of the control points.
ded values are 3,
8,
13, 18,
on the first card, and 4,
10,
23, 28, 33,
16,
22,
28,
38,
34,
42,
40,
The recommen-
48 if the THF = 0
46, 52, 58,
if
IHF = 1.
The third card -blank- will terminate the run.
Tf more than
one wing calculation is desired, the data deck can be composed as
follows:
(overleaf)
-71-
First Card
Second Card'-First Card
repeated as many times as desired
Second Card
Blank Card to terminate the run
On the following pages included are the complete listings of
the lifting
surface program,
as well as the complete
-72-
3 page output .
FORTRAN IV GA
0001
0002
0003
0004
0005
0006
0007
2
1
4
C
0009
0010
0011
0012
0013
0014
od15
0016
0020
0021
0022
=
74120
14/37/39
DIMENSI1N JGP(31),ZCP(8),XCP(8,10),XCP(31,70),DWNWSA(8,10),DWNWSD(
18,10)I,V(8,10,6,6) ,Q(8,10) ,E(6) ,P(6) ,CLA(24)),CLD(24)
CIMENSION WW(80,37),CA(38),CD(3P),ZMP(Il),NCP(10),CAA(36),C0D(36)
READ(5,l)KCLTIHFIVER,ASRAF,T,PP,KIPT
FORMAT (414,4FR.3,18)
IF(KDM.EQ.0) STOP
READ(5,4) (NCP(N),N=1,10)
FORMAT (1018)
CALCULATION OF GRID AND CONTROL POINT CCORCINATES ****************
15
14
W2=0.5*TAN(PP*3.14159/180.0)
CZYN=4.0*AF*(T+1.C)
DENR=2.0*AR*(T+1.0)
FRO0T=(C7YN-3.0*(T-1.0))/DENR-W2
SROOT=-((5.0+3.0*T-CZYN)/DENR+W2)
ST IP=-( (5.0*T+3.0-CZYN)/DlENR-W2)
FT IP=(3 .0* (T-1 .0) +CZYN)/DE:-NR+W2
IF(SROOT.LT.0.O.AND.STIP.ILT.0.0.AiD.FROCT.GT.0.0.ANC.FTIP.GT.0.0)G
10 TO 14
WRITE(6,1003)AR,AF,T,PP, IHF,IVER, KM,Li
WRITE (6,15)
FORMAT(///,' ERROR IN THE INPUT DATA: SWEEP ANGLE CR FLAP AREA IS
1 TOO BIG OR TAPER RATIO IS TOO SMALL ')
G TO 2
INDX=KDM*(LT-1)
LIT=LT-1
0023
0024
0025
0026
0027
0028
0029
NUNK=LT*KDM
STR=STIP-SROOT
FTR=FTIP-FROOT
RVER=FLOAT (IVER)
RM=14.0+10.0*RVER
LK2=21+5*IVER
LK3=7+5*IVER
0030
LK4=LK3+1
0031
0032
0033
0034
DATL
AR=2*ASR
0008
0017
0018
0019
MA IN
RELEASE 2.0
99
101
DD. 99 I=1,LK3
ZGP(I)=2.0*(FLOAT (T)-1.0)/RM
DO 101 I=LK4,LK2
ZGP(I)=ZGP(I-1)+2.0/(14.0*RM)
0035
ZCP(1)=1.0/RM
0036
0037
0038
0039
0040
0041
0042
0043
0044
0045
0046
0047
0048
0049
DO 105 K=2,6
FK=FLOAT(K)
ZCP(K)=(2.0*FK-1.0+2.0*RVR*(F-K-1.0))/R
ZCP(7)=2.0*(RM/2.0-1.0)/RM+9.0/(14.0*RM)
ZCP(8)=2.0*(RM/2.0-1.0)/RM+19.0/(14.0*RM)
NF=AF*FLOAT(50+IHF*10)+0.1
NS=50+10*IHF-NF
RNS=FLOAT(NS)
RNF=FLOAT(NF)
DO 17 I=1,LK2
XS=ZGP(I)*STR+SROOT
XF=ZGP( I )*FTR+FRO2T
DO 16 J=lNS
XGP(I,J)=XS *(FLOAT(NS-J+1)-0.50)/FLDAI(NS)
105
16
-73-
PAGE 0001
FORTRAN IV G1
0050
005 1
0052
0053
0054
0055
0C56
0057
0058
0059
0060
0061
0062
0063
0064
0065
0066
0067
17
19
20
1100
0068
759
0079
0080
0081
0082
0083
0084
0085
0086
0087
500
0088
0089
721
1100 KD=1,NUNK
CA(KD)=0.0
CD(KD)=O.0
DO 759 KD=1,KDM
00 759 L=1,LT
00 759 K=1,8
DO 759 N=1,10
V(K,N,KDL)=O.0
ZMS=0.5*(ZGP(1+1)+ZGP(1))
XS=ZMS*STR+SROOT
XF=ZMS*FTR+FRI0'T
CGP T=ZMS* (FTR-STR ) +FROCT-SROOT
ZW IG=ARCOS (-ZMS)
00 500 KD=1,KCM
E(KD)=SIN( (2.0*FLCAT(KD)-1.0)*ZWG)
DO 720 J=1,NT
FJ=FLOAT(J)
DO 721 K=1,8
DD 721 N=1,10
ICOMM=I
0090
0091
0092
JCOMM=J
KCOMM=K
NCOMM=N
Q(K,N)=HSVEL(XGP(I,J),ZGP(I),XGP(I+1,J),ZGP (1+1) ,XCP(KN) ,ZCP(K))
IF (J.GT.NS) GO TO 600
SS2=-XS*FJ/(RNS*CGPT)
SS1=-XS*( FJ-1.0)/(RNS*CGPT)
SSW2=ARCOS(1.0-2.0*SS2)
SSWl=ARCOS(1.0-2.0*SS1)
0093
0094
0095
0100
14/37/39
LW5=LK2-1
DO 720 I=1,LW5
0078
0096
0097
0098
0099
DATE = 74120
NS1=NS+ 1
NT=NS+NF
DO 17 J=NS1,NT
XGP(IJ)=XF *(FLOAT(J-NS-1)+0.50)/FLCAT(NJF)
00 20 K=1,8
XSCP=ZCP(K )*STR+SROOT
XFCP=ZCP( K)*FTR+FRCrT
DO 20 N=1,10
IF(NCP(N).GT.NS) GO TO 19
XCP(K,N)=XSCP*FLOAT(NS-NCP())/FL7AT(NS)
G0 TO 20
XCP(K,N)=XFCP*FLIAT(NCP(N)-NS)/FL0AT(NF)
CONTINUE
CALCULATION OF INDUCED VFLOCITIES ******
DO
0069
0070
0071
0072
0073
0074
0075
0076
0077
MAIN
RELEASE 2.0
609
P(1)=SSW2+SIN(SSW2)-SSW1-SIN(SSW1)
P(2)=SSW2-0.5*SIN(2.0*SSW2)-SSWI+0.5*SIN(2. 0*SSWI)
GC TO 710
IF (LT.LE.3)
DO 609 L=3,LIT
FL=FLOAT(L)
P(L)=(SIN((FL-2.0)*SSW2)-SIN(FL-2.0)*SSW1) )/(FL-2.C)+(SIN(FL*SSW1
1)-SIN(FL*SSW2) )/FL
GO TO 710
PAGE
0002
----------------
FORTRAN IV GI
0101
0102
0103
0104
0105
0106
0107
0108
0109
0110
0111
0112
0113
0114
0115
0116
0117
0118
0119
0120
0121
0122
0123
0124
0125
0126
0127
0128
600
760
720
C
801
14/37/39
1.0-2.0*SF1)
V(K,N,KDL)=V(K ,NKD,L)+0(K,N)*E (KD)*P(L
CDNTINUE
***********
CALCULATION OF BOUNDARY CONDITIONS
DO 801 K=1,8
D0 801 N=1,10
DWNWSA(K,N )=0. 0
DWNWSD(K,N)=0.0
LT5=LT-1
NUNK=KDM*( LT-1)
MUNK=NUNK+1
DO 805 N=1,10
DO 805 K=1,8
IV=(N-1 )*8+K
DO 800 L=1,LT5
800
805
00 800 KD=1,KDM
IH=(L-1)*KDM+KD
WW(IV,IH)=V(K,N,KDL)
WW(IV,MUNK)=1.0
CALL PTLSQ (WWCA,80,NUNKKERRCR)
DO 803 N=1,10
0144
0150
0151
74120
DO 760 N=1,10
DO 760 KD=1,KDM
00 760 L=1,LIT
0140
0141
0142
0143
0145
=
P(1)=SFW2+SIN(SFW2)-SFW1-SI NtSFW1)
P(2)=SFW2-0.5*SIN(2.0*SFW2)-SFWI+0.5*SIN(2.0*SFWI)
IF (LT.LE.3) GO TA 700
IN0 704 L=3,LIT
FL=FLOAT(L)
704
P(L)=(SIN((FL-2.0)*SFW2)-SIN((FL-2.0)*SFW1))/(FL-2 .0) +(SIN(FL*SFWI
I)-SIN(FL*SFW2) )/FL
T2=FJOT/RNF
700
Tl=(FJOT-1.0)/RNF
TW2=ARCOS( 1.0-2.0*T2)
TW1=ARCOS(1.0-2.0*T1)
LIT=LT
P(LT)=TW2+SIN(TW2)-TW1-SIN(TW1)
710
DO 760 K=1,8
0139
0146
0147
0148
0149
DATE
FJOT=FLOAT(J-NS)
SF2=(XF*FJOT-RNIF*XS)/(RNF*CGPI)
SFl=(XF*(FJOT-1.0)-RNF*XS)/(RF*CGPT)
SF W2=ARCOS( 1.0-2. 0*SF2 )
SFW1=ARCOS(
0129
0130
0131
0132
0133
0134
0135
0136
0137
0138
MAIN
RELEASE 2.0
803
DO 803 K=1,8
DO 803 KD=1,KDM
00 803 L=1,LT5
IS=(L-1 )*KCM+KC
DWNWSA(K,N)=V(K,N,KDL)*CA(IS)+CWNWSA(K,J)
'4NUNK=KDM*LT
MU NK=NU NK +1
DO 815 N=1,10
DO 815 K=1,8
IV=(N-1)*9+K
00 810 L=1,LT
-75-
PAGE
0003
FORTRAN
IV G1
MAIN
RELEASE 2.0
0152
CO 810
0153
IH=(L-1)*KDM+KD
0154
0155
0156
0157
0158
810
815
DATF = 74120
KD=1,KDM
WW(IV,IH)=V(K,NKD,L)
WW(IVMUNK)=1.0
WW(IV,MUNK)=0.0
IF(XCP(KN).LT.0.0)
CONTINUE
CALL PTLSQ(WW,CD,80,NUNKKERRDR)
0159
0160
DO 804 N=1,10
DO 804 K=1,8
0161
D
0162
DO 804 L=1,LT
804
KD=1,KDM
0165
0166
IS=(L-1 )*KCM+KP
DWNWSD(KN)=V(K,N,KDL)*CD(IS)+DWNWSC(K,N)
**********************************
CALCULATIONS OF FORCES
CLAG=(3.14159**3)*AR*(CA(1)+CA(1+KDM))
CLDG=(3.14159**3)*AR*(CD{1 )+CD(1+KCP)+CC(i+INDX))
0167
0168
SM1=0.0
SM2=0.0
0163
0164
804
C
0169
ZMP(1)=0.0
0170
DO
0171
0172
0173
0174
IF (I.LE.19) LMP( I)=ZMP(I-1)+0.05
IF (I.GT.19) ZMP(I)=7MP(I-1)+0.025
CONTINUE
ZMP(23)=0.995
910
910 1=2,22
0175
ZMP(24)=1 .0
0176
0177
CLA(24)=0.0
CLD(24)=0.C
0178
0179
0180
00 911 1=1,23
SUM1=0.0
SUM2=0.0
0181
0182
CGPT=ZMP(I)*(FTR-STR)+FROOT-SROOI
ZWIG=ARCOS (-ZMP(I))
0183
DO 912 KD=1,KDM
0184
0185
0186
0187
0188
0189
0190
0191
0192
0193
0194
0195
0196
0197
0198
0199
0200
0201
0202
0203
14/37/39
FD=FLOAT(KC)
E(KD)=SIN((2.0*FD-1.0)*ZWIG)
SUM1=SUM1+E(KD)*(CA(KD)+CA(KD+KDM))*2.0/CGPT
912
SUJM2=SUM2+E(KD)*(CD(KD)+CD(KD+KDM)+CC(KC+INOX))*2.0/CGPT
CLA(1)=39.47842*SJM1
911
CLD(I)=39.47842*SUP2
AAI=CA(1)+CA(1+KDM)
AA2=CD(I)+CC(1+KDM)+CD(1+INDX)
DO 913 KD=2,KCM
FK=FLOAT(KC)
SM1=SMI+(2.0*FK-1.0)*((CA(KD)+CA(KD(+KDM))/AAI)**2
913
SM2=SM2+(2.0*FK-1.0)*((CD(KD)+LD(KD+KDM)+CD(KD+IND4X))/AA2)**2
CDIA=f1.0+SM1)/(3.14159*AR)
CDID=(1.0+SM2)/(3.14159*AR)
EFA=1.0/(1.0+SM1)
EFD=1.0/( 1.0+SM2)
KDML=KDM*LT
00 915 N=1,KDML
CAA(N)=CAtN)*9.86958
915 CDD{N)=CD(N)*9.86958
-76-
PAGE C004
FORTRAN IV GI
0204
0205
0206
0207
0208
0209
0210
0211
0212
0213
0214
0215
0216
0217
0218
0219
0220
0221
0222
0223
0224
0225
0226
0227
0228
0229
0230
0231
0232
0233
0234
0235
0236
0237
0238
0239
0240
0241
0242
0243
0244
0245
MAIN
RELEASE 2.0
DATE ='74120
14/37/39
WRITE(6,1003)ARAF,T,PP, IHF,IVER,KDM,LT
1003 FORMAT(*1',5X,13HASPECT RATIO=F5.2,2XI0HFLAP AREA=F4 .3,2X,12HTAPE
IR RATIO=F4.2,2X,12HSWEEP ANGLE=F6.3,3X,4HDErG.,/,6X,29 HPR. CISION IN
2DICES:CHORDWISE =1I1,2X,9HSPANWISE=l 11,/,6X,UNUMBER O F SPANWISE MO
3DES =',12,5X,'NUMBER OF CHORDWISE MCCES =',12,//)
WRITE (6,948)
DISTRIBUTION"IOF LIFT,/,10X,'7 = SPAN COORD
FORMAT (1oX,29H4SPANWISE
948
1INATE
(Z=0 AT THE ROOT,Z =1 AT THE TIP)',///)
WRITE(6,946)
FORMAT(15X,IHZ,14X,3HCLA,12X,3HCLD)
946
DO 950 1=1,24
950
WRITE(6,949)ZMP(I),CLA(I),CLD(I)
949
FORMAT (l0X,3(F10.3,5X))
WRITE(6,991)CLAGCLDG
CLAR=',F8.3,
991
FORMAT(//,9X,'OVERALL LIF1 SLOPE COEFF. PER RADIAN
16Xv'CLDR=' ,F8.3)
WRITE(6,1000)CDIACDID
1000 FDRMAT(9X,'INDUCED DRAG COEFF./UNIT LIFT CCEFF.**? CD IA=',F8.3,6X,
I'CDIDCC =',F8 .3)
WRIT'E(6,961 )EFAEFC
FORMAT(9X,29HEFFICIENCIES ARE RESPECTIVELY,12X,4hEFA=F8.3,6X,5HEE-D
961
1 =F8.3)
WRITE(6,955)
FORMAT (/,30X,'MODE AMPLITUDES C-ALPA')
955
WR ITE(6,1002)
L D ENTES CHORDWISE MODE)
1002
FORMAT(9X,50HK DENOTES SPANWISE MODE,
WRI TF(6 ,1001)
1001 FORMAT (18X,3HK=l,9X,3HK=2,9X,3HK=3,9X,3HK=4,9X,3HK=5,9X,3HK=6)
00 957 L=1,LT5
KW=(L-1)*KCM+1
KWMAX=KW+KDM-1
958
FORMAT (8X,2HL=,1I1,1X,1OF 12.5)
957 WRITE(6,958)L,(CAA(N),N=KW,KWMAX)
WRI TE(6,959)
FORMAT (/,30X,'MODE AMPLITUDES C-D-ELTA')
959
WRITE (6 ,1001)
Dii 960 L=1,LT
KW=(L-1)*KDM+I
KWMAX=KW+K FM-1
960 WRITE(6,9583 )L,(CO(N),N=KW,KWMAX)
ARAF,T,PPIHF,IVER,KDM,LT
WRITE(6,1003)
WRITE (6,981)
FORMAT (///,20X,'MATRIX OF DOWNWASH VELOCITIES AT ALL CONTROL POIN
981
ITS',/,20X,'M DENOTES SPANWISE INDEX, N CENCTES CHORCWISE INDEX',/,
',/,18X,'N=1',9X,'N=2',9X,'N=3',9X,'N=4
230X,'DUE TO ALPHA
3',9X,'N=5',9X,'N=6',9X, 'N=7',9X,'N=8', 9X, 'N=9' ,9X,'N=10')
DO 982 K=1,8
WRITE(6,983)K,(DWNWSA(KN),N=1,10)
982
FORMAT (8X,'M=',11I,1X,10F12.5)
983
WRITE(6,984)
984
FORMAT (/,30X,'DUF TO DELTA',/,18X,'N=1',9X,'N=2',9X,'N=3',9X,'N=4
l',9X,'N=5',9X,'N=6', 9X,'N=7',9X, 'N=P', 9X, 'N=9' ,9X,'N=10')
985.K=1,8
DI
-77-
PAGE 0005
FORTRAN
0246
0247
0248
0249
0250
0251
IV GI.
MAIN
RELEASE 2.0
985
DATE = 74120
14/37/39
WRITE (6,983)K,(DWNWSb(KvN),N=1,10)
WRITE(6,986) (NCP(N),N=, 10)
986
FORMAT (/////,10X,'CtONTRCL POINT COLUMNS ARE LCCATEC AT THE DOWNST
IREAM BOUNDARIES OF THE FOLLOWING PANELS : ,/,1OX,101C0,///)
IF(KOPT.FQ.1) CALL OPT ION (CA,CD,KDM,LF, IHF, IVERARAFTPPSROOT,
ISTIP,FROOT,FTI P)
GO TO 2
END
-78-
PAGE 0006
FORTRAN
IV GI
RELEASE 2.0
PTLSo
DATE
0002
0003
SUBROUTINE PTLSQ (A,R,NEQNUN,KFRROR)
DIMENSICN A(80,37),R(36),B(1444)
MUN=NUN+1
0004
00 1 M=1,NtN
0005
00 1 N=1,NUN
0006
0007
0008
L=N+(M-1)*NUN
B(L)=0.0
O0 1 J=1,NEIQ
CC0
0009
1
0010
0011
0012
0013
0014
0015
C16
B(L)=B(L)+A(Jm)*A(JN)
DO 2 M=1,NUN
R(M)=0.0
UD 2 N=1,NEC
2
R(M)=RfIM)+A(N,MUN)*A(N,M)
CALL SIMQ (B,R,NUN,KERR0R)
RE TURN
END
-79-
=
74120
14/37/39
PAGE C001
FORTRAN IV G1
0001
0002
0003
0004
0005
CC06
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
HSVEL
RELEASE 2.0
DATE = 74120
1,4/37/39
FUNCTION HSVEL(Xl,Z1,X2,Z2,X,Z)
HSVEL=0.0
XA=X1
ZA=ZI
XB=X?
ZB ==72
DO
1 N=1,2
T= (XB-XA)/(ZB-ZA)
A=1 .0+T **2
B=-2. 0* (T* ( X+ T*7A-XA) +Z)
C=(X+T*ZA-XA)**2+Z**2
D=X-X A-T* (Z-ZA)
BAC=-4.0*D**2
RAT=1.0
ZC=ZA
XC=XA
DO 2 M=1,2
HSVEL=HSVEL+RAT*( (X-XCC)/SQRT( (X-XC)**2+(Z-ZC)**2)+.1.0)/(Z-ZC)
GO TO 3
IF(BAC.LT.-1.OF-05)
IF(N.NE.1.CR.Z.LT.Z1.1 R.Z.GT. 72) G0 TO 4
VFL=D/(2.0*SQRT(A**3)*(ZC+B/( 2.0*A))**2)
V=SQRT((72-Z)**2+(X2-X)**2)
W=0.5*SQRT((72-Z1)**2+(X2-XI)
R=SQRT((2.0*D*W)**2-(W**2+D**2-V**2)**2)/(2.0*W)
HSVEL=HSVEL+SIGN(1.0/R-ABS(VEL),1)
GO TO 4
3 HSVEL=HSVEL-RAT*0.5*(2.0*A*ZC+B)/(D*SQRT(A*ZC**2+B*ZC+C))
4 RAT=-1.0
2
1
XC=XB
ZC=ZB
XA=X?
ZA=-Z2
XB=X1
ZB=-Z1
RETURN
END
-80-
PAGE
0001
FORTRAN IV iG1
cco
C002
0003
OPTION
RELEASE 2.0
OATE
= 74120
14/37/39
SUBROUTINE CPTTCN(CA,CC,KDM,LT, IHF-, IVERAR,AF,T,PPSROOT,STIPFROCD
IT,FT IP)
PIM'ENS ION CA(30),CD(36),CMM(6),t(6),Z( 12),P(6),SIMPSN(11)
DATA SIMPSN/0.33333,1.33333,0.666f-7, 1.33333,O.666(7,1.33333,0.6666
17,1.33333,0.6,1.5085,0.0/
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
DATA KI/5/,KO/6/
WRITF (KO,1003)ARAFT,PPIHFIVERI,KCM,LT
1003 FORMAT(*1',5X,13HASPFCT RATIO=F5.2,2X,10HFLAP AREA=F4.3,2X,12HTAPE
1R RATIO=F4.2,2X,12HSWEEP ANGLE=F6.3,4HDEG.,2X,29HPRECISION INDICES
2:CHORDWISF =1I1,2X,9HSPANWISF=111,/,6X, 'NUMBER OF SPANWISF MODES =
3',I2,5X,'NUPHER OF CHORDWISE PCDES =',12,//)
CONT=0.0166
CTM=(FT IP-STIP+FROOT-SROOT)*0.5
XTM=tFRO]T+FTIP)*0.5/CTM
INDX=KDM*(LT-1)
WRITE(6,1546)
1546 FORMAT(//,22X,'7',8X,'XA/LC',8X,'CM-A',8X,'XD/LC',PX,'CM-D',5X,'XA
ILE/LC',5X,'XDLE/LC',/)
AA1=CA(1) +CA (1+KDM)
AA2=CD(1)+CC(1+KDM)+CD(1+INDX)
0015
CLAG=(3.14159**3)*AR*AA1
0016
CLCG=(3.14159**3)*AR*AA2
0017
0018
0019
0020
0021
0022
0023
0024
0025
CtvINA=0.0
CMIND=0.0
ZCPA=0.0
ZCPD=0.0
00 1515 IR=1,10
CLA=0.0
CLD=0.0
I=IR-1
ZI=FLOAT(I)
0026
0027
0028
0029
Z( IR)=ZI/10.0
XT=Z(IR)*(FTIP-FROOT)+FROOT
XL=Z(IR)*(STIP-SRCT)+SRO;'T
CT=XT-XL
0030
0031
PRC=XT/CT
ZWIG=ARCOS (-Z(IR))
0032
DO
0033
RD=FLOAT(KD)
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
0044
0045
0046
0047
1520
KD=1,KDM
E(KO)=SIN((2.0*RD-1.0)*ZWI;)
CLA=CLA+E(KD)*(CA(KD)+CA(KD+KCM))*78.9568/CT
1520 CLD=CLD+E(KD)*(CD(KD)+CD(KD+KDM)+CD(KD+INDX))*78.9568/CT
LW8=LT-1
ZCPA=ZCPA+CLA*Z(IR)/(10.0*CLAG)*SIMPSN'(IR)
ZCPD=ZCPD+CLD*Z(IR)/(10.0*CLDG)*SIMPSN(IR)
P(1)=3.14159
P(2)=3.14159
0. 1522 L=3,LWP
1522 P(L)=O.0
P(LT)=3.14159
tMM(1)=3.14159*(CT+4.0*XL)/4.0
CMM(2)=3. 14159*(CT+2.0*XL)/2.0
CMM(3)=-3.14159*CT/4.0
-81-
PAGE 0001
FORTRAN IV GI
0048
0049
OPTION
RELEASE 2.0
CMM(LT)=3.14159*XT/4.0
0051
0052
CMAL=0.0
CMDL=0.0
0053
CHA=0.0
0054
CHD=0.0
0055
DO
0056
0057
0058
0059
0060
CMAK=0.0
CMDK=0.0
CHAK=0.0
CHOK=0.0
DO 1530 L=1,LWR
0061
0062
0Q63
MM=KD+KDM*(L-1)
CHAK=CHAK+CA(MM)*P(L)
CMAK=CMAK+CMM(L )*CA(MM)
1530
1540
MM=KD+KDM*(L-1)
CHDK=CHDK+CD(PM)*P(L)
1535 CMDK=CMDK+CMM(L)*CC(MM)
CMAL=E(KD)*CMAK+CkiAL
CMDL=E(KD)*CMDK+CMDL
CHA=E(KD)*CHAK+CHA
0071
0072
0073
0074
0075
0076
1540 CHC=E(KD)*C4DK+CHO
XAL=CMAL/ (CHA*CT)
XDL=CPDL/(CHD*CT)
XLEA=1.0-(PRC-XAL)
XLEO=1.0-(PRC-XDL)
GMAZ=XAL*CLA
0077
0089
0c90
0091
0092
0093
0094
14/37/39
KD=1,KDM
0070
0088
74120
00 1535 L=1,LT
0064
0078
0079
0080
0081
0082
0C83
0084
0085
0086
0087
=
DO 1525 L=4,LW8
1525 CMM(L)=0.O
0050
0065
0066
0067
0068
0069
DATE
CMDZ=XDL*CLD
WRITE(KO,1550)Z(IR),XAL,CMAZ,XtL,CMOZXLEAXLED
FORMAT(20X,F5.2,2(5X,F7.3,4XF8.4),2(5X,F 7.3))
CMINA=CMINA+CMAZ*CT*SIIMPSN(IR)
CMIND=CMIND+CMDZ*CT*SIMPSN(IR)
1515 CONTINUE
SA=CMINA/(CLAG*CTM*10.0)
SD=CMIND/(CLDG*CTM*10.0)
CMINA=CMINA/(CTM*10.0)
CMIND=CMINC/(CTM*10.0)
XLEAM=1.0-(XTM-SA)
1550
XLEDM=1.0-(XTM-SD)
WRITE(K0,1551)SA,CMINA,SDCMIND,XLEAMvXLEDf
1551 FORMAT(/,4X,'INTEGRATFC',17XF6.3,5XF7.4,6XF6.3,5XF7.4,2(6X,F6.
13),/)
WRITE{KO,1600) ZCPAZCPC
1600 FORMAT(X,'SPANWISE POSITIONS OF IHE CENTERS OF PRESSURE ARE : ZA=
Jl,F8.2,5X,'ZD=',pF8.2)
WRITE(KO,1705)
1705 FORMAT(/,11X,'ALPHA',8X,'DELTA',6X,'CEL/ALPH',9X,'CL',12X,'CD',12X
1,'L/D',9X, 'C',1OX ,'XHL/C',/)
1650 KAL=1,21,5
0095
D0
0096
0C97
DO 1650 KRA=1,26,5
KALA=KAL-1
-82-
PAGE C002
FORTRAN IV GI
RELEASE 2.0
OPTION
0100
1500 IF(KR)1501,150 ,1502
0101
1501
DEL=0.0
0103
DIV=99999.9
GO TO 1510
0104
1502
ALA=0.0
DEL=FLOAT(KR)
DIV=99999.9
0108
GO TO 1510
0109
0110
1505 ALA=FLOAT(KALA)
R=FLOAT(KR)
DEL=ALA*R/ 10.0
DIV=DEL/ALA
0111
0112
0113
0114
0115
0116
1510 ALPH=ALA/57.296
DELT=OEL/57.296
AA=AA1*ALPH+AA2*DELT
CLALF=CLAG*ALPH
0117
0118
0119
0120
0121
CLCEL=CLDG*CELT
CL=CLALF+CLCDL
SM=0.0
IF(AA) 1512,1516,1512
1512 O 1514 KD=2,KCM
0126
0127
0128
0129
1514
RKD=FLOAT(KC)
KI=KD+KDM
K2=KD+INDX
SM=SM+(2.0*RKD-1.0)*(((CA(KD)+A(K1))*ALPH+(CD(KD)+CD(K1)+CD(K2))*
lDFLT)/AA)**2
CR=(CL**2)*(1.0+SM)/(3.14159*AR)+C.0C85+CCNT*CL**2
GO TO 1517
1516 CR=0.0085+CCNT*CL**2
1517 CLD=CL/CR
0130
1F(CL.EQ.0.C)XPC=99999.9
0131
0132
IF(CL.EQ.0.0)CM=0.0
IF(CL.EQ.0.0) GO TO 1709
0133
0134
0135
0136
0137
0138
0139
14/37/39
ALA=0.0
0102
0122
0123
0124
0125
= 74120
KR=KRA-1
IF(KALA)1500,1500,1505
0098
0099
0105
0106
0107
DATE
CM=(CMtINA*CLALF*ALPH+CMIND*CLDEL*DELT)/CL
XPC=(SA*CLALF+S0*CLDEL )/CL
1709 WRITE(6 ,1710)ALA,DEL,DrIVCL, CR, CLD,CMXPC
1710 FORMAT(9XF6.1,7XF6.1,8XF6.1,7XF8.4,6XF8.4,5XF8.3,6XF8.5,5X,
1F8.4)
1650 CONTINUE
RETURN
END
-83-
PAGE 0003
FORTRAN
0001
0002
0003
CC04
0005
0006
CC07
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
IV G1
SUBROUTINE SI MO(A,B,N,KS)
DIMENSION A(1),8(1)
T1L=0.0
KS=0
iJ=-N
00 65 J=1,N
JJ=JJ+N+1
BIGA=0
IT=JJ-J
DO 30 I=J,N
IJ= IT+I
IF (ABS (BIGA)-AMS
20 BIGA=A(IJ)
0039
0040
0041
0042
0043
PAGE
SIMQ 490
SIMCQ 500
SI MQ 540
SIMQ 550
S IMQ 560
SIMQ 680
SIMQ 690
SI MQ 700
IF (ABS(BIGA)-TnL) 35,35,40
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
35 KS=1
40
RETURN
Il=J+N*(J-2)
IT=IMAX-J
00 50 K=J,N
11=I1+N
12=I1+IT
SAVE=A(Il)
SIMO
A(12)=SAVE
DO 60 JX=JY,N
IXJX=N*( JX-1)+I X
JJX=IXJX+IT
60 A( IXJX) =At IXJX)-( A(IXJ )*A(JJX
65 B(IX)=B(IX)-(B(J)*A(IX J)
70 NY=N-1
IT=N*N
IA=IT-J
0046
0047
0048
0049
11=N-J
J=1,NY
DO 80 K=1,J
BI~B=8)-A(
IA=IA-N
80
SIMQ
930
SIMQ 940
SIMQ 980
SIMQ 990
SIMQ1000G
SIMQ1020
SIMO1030
SIMQ1040
SIMQ1050
SI MQ 1060
SIMQ 1070
SIMQL110
SI MQ1120
SIMQ 1130
SIMQ1140
SIMQ1150
SIMoQ1160
IC=N
0050
SIMQ 870
SIMQ 91C
SIMQ 920
SIMQ1010
IXJ=IQS+'IX
IT=J-IX
0045
840
SIMQ 850
SIMQ 860
50 A(I1)=A(Il)/B1GA
SAVE=B( I M AX)
B( IMAX)=P(J)
B(J)=SAVE/BIGA
IF(J-N) 55,70,55
55 IQS=N*(J-1)
00 65 IX=JY,N
DC 80
750
760
8C0
810
820
SIMQ 830
A(II)=A(12)
0044
0051
0052
0053
14/3-7/39
A( IJ ) ) )20, 30, 30
IMAX=I
30 CONTINUE
0036
0037
0038
D.ATE = 74120
SI MQ 570
SIMQ 580
S IMQ 590
SIMQ 600
SI MQ 61C
SIMQ 620
SIMQ 660
JY=J+1
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
S IMQ
RELEASE 2.0
IC=IC-l
RETURN
E-N0
-84-
SIMQ117C
IA)*BIC)
SIMQ1180
SIMQ1190
SIMQ12C
SIMQ1210
SI MQ 1220
0001
ASPECT
RATIO=
2.80
FLAP
AREA=.200
PRECISION
INDICES:CHORCWISE =0
NUMBER OF
SPANWISE
SPANWISE
MODES = 6
RAIO=0.60
NUMBFR
0.050
0.100
0.150
0.200
3.217
3.268
3.314
0.250
3.353
0.300
3.385
3.409
3.424
3.429
3.424
0.350
0.400
0.450
0.500
0.550
0.600
0.650
OF CHOIRDWISE 10CES = 6
1.816
1.852
1.883
1.908
1.924
1.934
1 . 936
1.933
1.926
1.913
1.891
1.852
1.783
1.661
1.453
1.300
1.094
0.797
0.364
0.0
3.377
3.330
3.259
3.153
2.994
0. 850
0.900
?.751
2.374
0.925
2.107
0.950
0.975
0.995
1.000
1.263
0.570
0.0
LIFT
CLC
1.667
1.702
1.739
1.778
3.407
0.700
0.750
0.800
1.757
SLOPE
COEFF.
PER
RADIAN
3.134
0.114
0 . 996
CLAR=
INDUCED DRAG COEFF./UNIT LIFT COEFF.**2 CDIA=
EFA=
EFFICIENCIES ARE RESPECT IVCLY
SPANWISE
K=1
MODE AMPLITUDFS C-ALIPHA
L CENOTES CHORDWISE MODE
MODE,
K=4
K=3
K=2
0.01512
0.01870
0.05641
-0.01665
-0.04400
-0.01697
L =1
L =2
0.38319
-0.02694
L =3
L =4
L=5
0.00458
-0.01504
0.00466
0.00373
-0.00359
-0.00627
-0.00569
-0.00013
-0.00267
-0.00291
L =1
L =2
L =3
L =4
L =5
L =6
CEG.
(Z=0 AT THE ROOT,Z =1 AT THE TIP)
CLA
3.101
3.161
Z
0.0
K DENOTES
SWEEP ANGLE=15.CC0
DISTRIBUI ICN OF LIFT
Z = SPAN COORDINATE
OVERALL
TAPER
SPANWISE=0
K=1
0.02892
0.12022
-0.01817
0.03882
-0.01352
-0.01204
MODE AMPLITUDS C-CELTA
K=3
K=2
0.00123
C.C0115
-0.00690
0.00056
0.00298
0.01483
0.00483
-0. 00408
0.02269
0.00421
-0.00025
0.05219
0.01288
0.00711
-85-
K=4
0. 00257
0.00005
0.00559
0.00513
0. CC04 1
0.00272
CLDR=
CD I D=
EF
=
K=5
0.00137
1.771
0.116
0. 9F4
-0.00460
-0.00301
-C.00258
K=6
0.00345
-0.00346
-C. 00270
-0.00149
-0.00131
K= 5
C. 00305
-0.00684
0.00020
-0.00648
-0.00161
0.00238
K=6
-C.00133
0. CCC8O
0.00179
0.00285
-0. 00109
0.00099
-C.00237
ASPECT
RATIO= 2.80
NUMBER
OF
SPANWISE
MCDES
NUMBFR
= 6
SWEEP ANGLE=15.CC
RATIO=0.60
M=2
M=3
M=4
M=5
M=6
M=7
M=8
DUE
N=1
TO
N=2
M=1
-0.03388
0.03361
M=2
M=3
M=4
M=5
M=6
-0.00375
0.10217
-0.06883
-0.06031
-0.03471
-0.06120
-0.06922
-0.05646
CONTROL
POINT
3
0.12668
0.12636
0.12378
0.13446
0.13348
0.13186
COLUMNS
8
-86-
CEG.
SPANW ISE=0
OF
CHORDWISE
= 6
MODES
MATRIX OF DOWNWASH VELOCITIES AT ALL CONTRCL
M DENCTES SPANWISE INDEX, N CENIES CHORDWISE
DUE TO ALPHA
N=4
N=3
N=2
N=1
1.00963
1.02070
1.01131
1.-*01435
0.98722
0.98008
0.96721
0.96267
1.01474
1.01765
1.00844
1.02139
1.00416
0.99608
1.00720
1.00655
0.949979
0.97784
0.99452
0.99057
1.00045
1.01033
1.01236
1.01369
I.00632
0.97607
1.00231
1.00130
0.)9788
1.00589
0.99963
1.00328
M=1
M= 8
TAPER
FLAP AREA=.200
PRECISICN INOICES:CHOROWISE =0
POINTS
INDEX
N=5
1.00307
0.98988
1.00784
0.99632
0.99974
1.00186
1. 0CC74
0.99807
N=6
0.99969
0.99182
1.00375
0.99174
0.99943
0.99582
0.99572
1.00080
N=5
-0.01732
-C. 05768
N=6
-0.00874
0.05782
0. 04046
0.03653
0.04295
0.03799
0.02011
0.03526
N=7
0.99993
1.00064
1.00483
0.99457
0.99856
0.99572
1.00260
0.99086
1.00112
1.00546
0.99319
0.99435
0.99654
1.00235
1.00060
N=7
0.03134
0.14758
-0.00522
1.00126
N=9
1.00006
1.00369
1.00595
0.99771
1.00986
1.CO143
1.00979
0. 99593
N=1C
0.99640
1.00138
0.99944
0.99 075
1.00641
0.99183
0.99982
1.00020
CELTA
N=3
0.02758
-0.05518
0.00869
-0.00214
-0.
02325
0.00201
0.00824
0.00335
ARE LOCATED AT
13
N=4
0.00041
-0. 13353
-0. 07988
-0.09686
-0. 12020
-0.10037
-0.10349
-0.10709
-C.04387
-0.05773
-0.06767
-0.06268
-0.07879
-0.07201
THE DOWNSTREAM BOUNDARIES
27
23
18
OF
N=8
-0.00244
-0.01970
0.11529
0.12782
-0.00347
0. 14908
0.00173
0.14074
0.13612
-0.00560
0.01871
-0.00019
0.14436
THE FOLLOWING PANELS
37
32
:
42
N=9
0.95540
0.88485
0.86969
0.86885
0.88790
0.87483
0.89061
0.88214
N= 10
1.02819
1.04328
1.07358
1.05204
1.04648
1.04475
1.03844
1.04285
SWEEP ANGLE=15.000DEG.
TAPER RATI0=0.60
FLAP AREA=.200
ASPECT RATIO= 2.80
NUMBER OF CHORDWISE MODES = 6
NUMBER OF SPANWISE MODES = 6
POSITIONS
XALE/LC
XDLE/L C
0.594
0.592
0.589
0.584
0.577
-0.567
-0.4101
-0.4194
-0.4307
-0.4412
-0.4470
-0.4442
-0.4305
-0.4042
-0.3585
-0.2690
0.251
-1.8303
- 1.6818
-1.3466
-0.246
-0.241
-0.237
-0.234
-0.232
-0.229
-0.224
-0.214
-0. 201
-0.185
0. 2?3
0.216
0.203
0. 183
0.566
0.564
0.564
0.565
-0.575
-1.8027
-0.225
-0.3990
0.225
0.575
L/ D
CM
CM-A
0.0
-0.589
-0.584
-0.583
-0.582
-0.578
-0.573
-0.566
-1.8262
-0.562
-0.562
INTEGRATED
SPANWISE
CM-D
XA/LC
0.30
0.40
0.50
0.60
0.70
0.80
0.90
-1.8802
- 1.9313
-1.9686
-1.9802
-1.9606
-1.9115
OF THE CENTERS OF PRESStJRE ARE
: LA=
ALPHA
DELTA
0.0
0.0
0.0
0.0
5.0
0.0
0.1545
10.0
15.0
20.0
25.0
0.3091
0.0
0.0
0.0
5.0
5.0
5.0
5.0
5.0
5.0
10.0
10.0
0.0
2.5
5.0
7.5
10.0
12.5
0.0
5.0
10.0
10.0
10.0
15.0
10.0
20.0
10.0
25.0
0.0
15.0
15.0
15.0
15.0
15.0
15.0
7.5
15.0
22.5
30.0
37.5
20.0
0.0
20.0
20.0
10.0
20.0
20.0
20.0
20.0
30.0
40.0
50.0
-87-
INDICES:CHORDWISE
XD/LC
z
0. 10
0.20
PRECISION
DEL/ALPH
0.0
0.5
1..0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
0. 0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2. 0
2.5
CL
0.46
CD
C.0085
0.0117
0.0211
0.8204
1.0522
C.1535
1.2840
1.5158
1.7477
1.9795
1.0939
1.4029
1.7120
2.0211
0.2247
0.3101
C.4097
0. 52 35
0.1649
0.2663
0.3928
0.5447
C. 7218
0.9241
1.3197
2.3302
2.6393
C.237
0.231
0.227
0.571
C.47
0.0369
0.0590
C.08*74
C.0183
0.0246
C.0325
0. 042C
0.0531
0.0657
0.0476
0.0-729
0.1046
0.1425
C. 1868
0.2374
0.0965
0.4636
0.6182
0.7727
0.2735
0.3507
0.4280
0.5053
0.5826
0.6598
0.5469
0.7015
0.8560
1.0106
1.1651
0.249
0.243
C. c
13.258
14.630
12.561
10. 4 76
8.839
14.963
14.252
13.161
12.027
1C.975
10.039
11.489
9.617
8.185
7.089
6.236
5.559
8.502
6.855
1.715
4.888
4.265
3.781
6.632
5.269
4.358
3.711
3.228
2.856
XHL/C
0.0
-0.03482
-0.06963
-0.10445
-0. 13926
-0.17408
-0. 15732
-0.12649
-0.11308
-0.10910
-0.11079
-0.11616
-0.31463
-0.25299
-0.22617
-0.21820
-0.22158
-0.23233
-0.47195
-0.37948
-0.33925
-0.32730
-0.33238
-0.34850
-0.62927
-0.50597
-0.45234
-0.43640
-0.44317
-0.46466
-0.2253
-0.2253
-0.2253
-0.2253
-0.2253
-0.5753
-0.4982
-0.4489
-0.4147
-0.3896
-0.370.3
-0.5753
-0.4982
-0.4489
-0.4147
-0.3896
-0.3703
-0.5753
-0.4982
-0.4489
-0.4147
-0.3896
-0.3703
-0.5753
-0.4982
-0.4489
-0.4147
-0.3896
-0.3703
=0
SPANWISE=0