A THERETICAL AND EXPERT1'MNT
INVE7STIGATTON 0'P THE "DERFORMANCE
O7 7T,APPED TTThDERS
by
Bohdan W. Orrnenheim
Undergraduate Diploma, Warsaw 7olvtechnic, Poland, 1970
M.S., 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, M1av, 1P74
Certified By: Thesis Supervisor
Accepted By : ARCHIVE9 Chaman, Dent. Comm ttee on Graduate Students JUL 16 .974 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. Effects of rudder geo- metry are discussed in detail, in particular, effects of taner ration, thickness, sween angle, and flan size.
A lifting surface nrogram was written and used in order to
compare and choose the best rudder planform. 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 in nropeller-rudder axial clearance was ex-
amined. 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 and Exrerimen .a1 Data ......
Acknowledgement ...... 61
0. Nomenclature ...... 63 10. References ......
11. List of Figures and Tables ...... 66
12. Appendix (Lifting Surface Program) 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 of flan area and flao balance. Shapes of rudders were tvnical of a high sneed vessel. Asoect ratio was chosen rather arbitratelv to be 1.4. 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 all- movable 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 con- figurations, namely for single screw-single rudder, double screw- single rudder, and double screw-double rudder of a tyrnical high speed vessel.
-2- 2. Rudder Shane
The initial idea was to nreserve the rudder shane from the series (1), decreasing only the flan area to 105 by moving the flan hinge axis towards the trailing edge; and, if needed, scal- ing the model size to suit the geometrical reouirements of mutual nronortions of the tunnel test section, and nroneller and rudder sizes.
The first difficulty encountered during this attemnt, was that the hinge line of the 10 flan, (which was arbitrarely con- strained to be nerpendicular to the root and tin sections), would 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: to increase the rudder sween angle, thus moving the tin section aft relative to the root section, 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 (ratio of the tin chord to the root chord); or to apply all these changes simultaneously In some suitable way.
Here a matter of ontimization 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 ratio of 0.1 following
-3- 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) is that the latter has a uniform thickness ratio of 0.15. 'or this reason, it was de- cided to adopt a uniform thickness ratio of O.15 over the whole rudder span. Asnect ratio, it was decided, would remain the same as in 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, and is nernendicular to the root and tin sections. 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 drag coefficient, and rudder efficiencv. 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 in- stallation of hinge details of sufficient strength. The ontimum sweep angle came out to be 170 aft, comnared to 110 aft used with the series (1). The optimum taner ratio came out to be, coinci- dentally, identical with that of' the original series, namely 0.60..
(Figure 1).
Purthermore, the combination of a taner ratio of' 0.60 and a 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, significant simplification in the geometry of the flan was made nossible. This is described in the following chanter.
Since the nlanform of the 1% flan rudder had been changed
from the original rudders, and one of the objective of the nresent nroject was to comnare this rudder with the 20l flap rudder, another model had to be built, namely one with a 201 flan, and the same planform as the 10"' flan rudder. Tt was also intended to test one
rudder without a flan, and such a rudder could be ma.d.e of one of
the above, by filling the gap between the skeg and the flap with a filler. Span length was determined from geometrical considera-
tions of mutual proportions, between the rudder and the nroneller,
as well as of the tunnel test section blockage limitations. Tvi-
cal values of the clearance between the ship hull and the nro- peller circumference is 0.3 of the propeller diamter D. Rudder
tip is usually 0.25D above the lowest point of the proneller cir-
cumference. With the nroneller of the diameter 7.4p" selected
for these tests, the rudder span came out to be 7.P75". (Pigure 1).
The sectional shape of the 66 series (5) 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. The 63 series shape was therefore
selected for the current series, because it should have a larger
stall angle and higher maximum lift than the 66 series. The 632~
A015 sections have, however, been slightly altered in order to
develop a plain two-dimensional prismatic flan, and flan gap.
This permits the flap section to consist only of a circular arc
leading edge, 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. 562
200% FLAP HINGE
10% FLAP HINGE
-. ______MAC
4.219
Fig. 1 Planform of the 10% and 20% Flap Rudders -7- Table 1: Comparison of Current Rudder with Rudders Tested in the Oriinil rogram (1)
Current Original udders Rudders
Section shape Modified 63 2 A015 6627020 66E10 Taper ratio 0.60
Geom. aspect ratio 1.40 1.4110
Root thickness ratio 0.1507 0.21)
Tip thickness ratio 0. 1562 0.10
Root chord, inches 8.925 Tip chord, inches 4.219 5.955
Span, inches 7. P75 10 20% Flap chord, inches 1.125
10% 'Flap chord 0.562
Sweep angle, degrees 15 11
MAC, inches 5. 711'21 Area, So. inches 144.3 71.4
Tip shape sauared off sauared off
Location of stock axis in 5M2AC aft of leading edge 37.0 variable
Flap size and flap balance, 0 of MAC 0 flap - 0 bal. 0 flan 0 bal. 10% " - 0 20" "? 0 "? "? I 20% - 0 30% 0 30% "' - 30 140% "? - 0 "~ w40%it - 0 "t 140% "? - 78? 50% - 1)
T - .3:;
-8- 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 process, as well as due to biochemical effects of water. Details 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 Table 1. Rudder shanes of the nresent nro- ject are presented in Figure 1.
Modification of the Rudder to Suit Convenient Manufacture
The NCA 63 2A015 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 sec-
tion 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 ul-
timate ease of fabrication. In fulfilling this requirement, the
root and tip sections differ slightly both from each other and
from the NACA 63 2A015 section.
A program outlined in the followirgparagraphs was written
to modify the 632A015 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 machinest-model maker who machined the model from 6061-T6 aluminum alloy using a milling machine. in narticular, 230 soan- wise cuts were specified along straight lines connecting points of constant percentchord at root and tio. The final machine marks were small and were removed by polishing the surfaces by hand. The resultant accuracv of the rudder surfaces is very satisfactory. Roughness does not exceed 0.002" and the toler- ance of the offsets is smaller than 0.0057".
Modification of the tip foil section by numerical methods was accomplished in the following steos: (the referred sketches are shown an 'Figure 2).
(a) The 63 2 A015 foil sections has a finite trailing edge thickness eaual to 0.032 (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: (see Sketch 2)
= ' - 0.032 - x' 0 < x' < 1 (1) 1 0 Y't= non-dimensional offsets of 612A019 section
(c) 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. This foil has a shar trailing
-10- edge (see Sketches I and 5).
(e) Then a wedge was added to bring the trailing edge
thickness on model scale to 0.020" (±0.010"): (see
Sketch 6).
Y= Y + xt (2)
As shown in Sketch 7, root and tip sections differ in the flap region. The root section was therefore next modified to be identical to the tip section between 70% of tin chord and the trailing edge for this model with a 0.60 taner ratio. "hese stens are as follows:
(f) The 6? 2A015 foil trailing edge thickness was removed by
subtracting a wedre then brought to 0.020" thickness
by adding a wedge: (see Sketch P).
= Y-.032 - x' + 0'010 .x'3crc (g) Compare the root trailing edge wedge with that of the 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 make the trailing edge angles of root and tip identi- cal on model scale. (h) Re-evaluate the root foil with a sharp trailing edge then multiply all ordinates by the ratio N: Y= - 0.010". x, 0 < xt < (h) 4 3 cr (cont. overleaf) -11- 2 4 I I I II 0% 75% 100% 0% 100% 112% ~I~iI~5 0% 100% L. TIP CHORD I00 7 ROOT CHORD". t7~ b +20% of c, FIG. 2 MIODIFICATIONS TO NACA 63 A015 SECTION 2 -12- y = - N 0 < x' < 1 (5) 5 4 (i) The last step is to bring the trailing edge thickness at the root to 0.020". V = Y'/cr + 0.010 X 0 < x' < 1 (6) 5 Cr Table 2 shows the comparison between the unmodified NACA the 632 A015 coordinates and the corresnonding coordinates for tip and root sections on the MIT flanned rudder model, modified in 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. Its dia- meter is 7.48" and the nitch P.03". Design value of J is 0.8. The onen water characteristics of this propeller, as measured in the MIT tunnel, are shown in PigIure 3. Tabulated values of the same data in the region of the design J appear in Table 3. -12- Table 2: Comnarison of Coordinates NACA 63 2 A015 Basic Thickness Form MIT Modified Section Shape x Y Y V (per cent c) (ner cent c) T Root 0 00 0 0.5 1.203 1.139 1.122 0.75 1.448 1.330 1.298 1.25 1.8P14 4 1.707 1.663 2.5 2.570 2.596 2.514 5.0 3.618P.833 3.715 7.5 4.3 2 14.511 .59 10 4.997 5.171 4.993 15 5.942 6.1714 5.056 20 6.619 6.910 6.652 25 7.091 7.307 7.117 30 7.3814 7.704 7.4'20 35 7.496 7.812 7.533 40 7.1435 7.733 7 45 7.215 7.4-7 7.271 50 6.858 7.104 6.915 55 6.397 6.602 6. 450 60 5.R20 6.004 5.93 65 5.173 .339 5.02h 70 4.468 4.616 75 3.731 3. 884 3.813 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- 0 40 0.9 0.8 0.7 0.6 o0 0.5 0 I-a 0.4 0.3 0.2- 0.\- 0.0.- -0.1 - 0.0 0.2 0.4 0.6 0.8 4 1.0 1.2 ADVANCECOEFFICIENTJ Fig. 3 - Open-water characteristics of NSRDC Propeller 4427 as measured in the MIT Tunnel. 15th August 1973 Table 3: Onen Water Characteristics of Proneller h427 as Measured in ITTW'_ater Tunnel MODEL 41127 5-BLAD)E J-COR KT KQ E KT/J**2 0.550 0.283 0.0471 0.526 0.937795 0.560 0.278 0.0465 0.534 0.08984 0.570 0.273 0.0458 0.542 0.843023 0.580 0.269 0.0451 0.550 0.799716 0.590 0.261 0.0444 0.557 0.798886 0.600 0.259 0.0)430 0.565 0.72036P 0.610 0.254 0.0431 0.572 0.680112 0.620 0.219 0.0425 0.579 0.649679 0.630 0.245 0.0418 0.586 0.617242 0.640 0.240 0.0412 0.593 0.586592 0.650 0.235 0.0105 0.600 0.557591 0.660 0.231 0.0399 0.607 0.530162 0.670 0.226 0.0393 0.613 o.504180 0.680 0.221 0.0387 0.620 0.)479574 0.690 0.217 0.0380 0.626 0.456224 0.700 0.212 0.0374 0.632 0.434057 0.710 0.208 0.0368 0.637 0.41204 0.720 0.203 0.0362 0.6)43 0.392965 0.730 0.199 0.0356 0.6)40 0.37901 0.740 0.194 0.03'50 0.6Y4 0.3557412 0.750 0.190 0.0344 O.650 0.)9432 7.760 0.185 0.0330 0.664 0.321915 0.770 0.101 0.0322 0.660 0.2061415 0.780 0.177 0.0326 0.673 0.291073 0.790 0.172 0.0320 0.677 0.276659 0.800 0.168 0.0314 0.61 0.262862 0.810 0.163 0.030p 0.60r1 0.2)406)46 0.820 0.159 0.0301 0.688 0.23675 0.830 0.154 0.0295 0.691 0.22 418 0.8,40 0.150 0.0280 0.6)Q 0.213144 0.850 0.145 0.0283 0.606 0.201925 0.860 0.141 0.0276 0.6(9 0.191136 0.870 0.136 0.0270 0.700 0.190750 0.880 0.132 0.0262 0.702 0.170746 0.890 0.127 0.0256 0.703 0.161102 0.900 0.122 0.0249 0.703 0.151796 0.910 0.118 0.0242 0.704 0.142012 0.920 0.113 0.0235 0.703 0.131130 0.930 0.100 0.0228 0.702 0.125730 0.940 0.103 0.0221 0.700 0.117626 0.950 0.090 0.0214 0.690 0.10079h 0.960 0.094 0.020P 0.69)4 0.102201 0.970 n.o0o 0.010A 0.. 0.004 P70 0.980 0.0)4 0.0191 0.6P)4 0.007790 0.990 0.079 0.0183 0.67P ~0.08002) 1.000 0.074 0.0175 0.660 0.07 4203 -16- 3. Design of Test Equioment The experiments were run in the 20"x20" test section of the water tunnel, 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. The base of it consists of a heavy stainless steel worm gear, set on a tapered plug that rotates in a matching hole, in a plexiglass test section window. The rudder shaft passes through a flexible seal, and is securely clamoed. to the floating structure of the dynamometer (Figure 5). Angular clamp- ing is achieved by a tapered key. The floating structure is then connected to the dynamometer base, by a set of Lebow Model 3345 strain gage load cells. The load cells are attached through slen- der, high strength steel, flextures, in order to orovide as close as possible to pin-ended support. These load cells are easily re- movable for calibration, 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 Fig.20b Propeller-Rudder interaction (Photograph taken with a strobe-light) -20- This plate was rigidly secured to the unner window by an aluminum streamlined foil, which occupied about 2/? of the cross-sectional area left above the plate. Inside this foil was the mechanism connecting the model to the shaft, 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 section was split comnletely above and below the plate. The leading and trailing edges of the plate extended 15 inches upstream and downstream from the rudder stock axis. Both edges were faired in order to minimize the leading edge senaration on the plate. As it later turned out, the fpiring was not sufficient and. caused significant trouble. Discussion of this is presented in the Section 7. It was honed that the solitter olate would develon a much thinner boundary layer, 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 (Figure 5). -21- The bottom side of the covernlate was held flush with the bottom side of the splitter plate. The gap between these two plates was equal to l/P". 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. All these mechanical narts, excent the rudder model, 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 both axial and lateral directions (see Figure 6). 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 in a stainlesshousing, extending -22- 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 coverplate to the turntable. The flan hinge moment sensor 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 amlifi- cation. The outnut was read on a seventh digital Tebow strain indicater. The tiller with the gauges was waternroofed, and onera- ted completely submerged, 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. It would be desir- able to locate the stock at the nosition where it is most likely -23-_ 6 U T JNNEL XI SIDE W'ALL SPLITTE PLATE I (PROPORTIONS EXAGGERATED) = sina y=I-cos a * = 0.133 R R |a=a=30* Fig. 6 Scheme of the mechanism changing rudder angles of attack to be installed in oractice. This nosition is roughlv corres- pondent to zero torque on the rudder stock, at an angle of attack between 100 and 150. According to (1), 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. This reauired a redesign of the sealing and clamping systems. In order to measure the angles of attack more accurately, 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 rota- tion. 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. The accuracy of this system is approximately 0.010. A special device had been made for measuring and setting the flan deflection angle in place in the tunnel test section. 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 is 0.50 or better. Tunnel flow sneed was measured by means of pressure taps in the contraction section connected to a Meriam !Iodel 33K A3R5 mano- meter with an indicating fluid with a specific gravity of 1.75. The typical column height at 20 ft/s is 1408 mm. Readings were taken with accuracy of 1 mm. The velocities are corrected for both tunnel and gauge temperatures effects. The calibration of pressure tans was obtained by comparison with a Pitot tube trans- verse of the test section. Column heigcht of fluid was corrected 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 numerically 2.183, is rultiplied by the actual manometer reading 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 the test section at the rudder position to the average velocity at this section has been measured using a Pitot tube. P'igure 7 represents a man of this ratio in the region away from the boun- dary layer . 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. This num- ber stays constant for auite a range of velocities (15 - 25 ft/ sec). Typical value of velocity in the test section during tests is 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 plate, but not as small as it would be expected from the Reynold.s number on the plate. The explanation of this relatively thick boundary layer is given in Section 7. The plate itself is away from the tunnel wall boundary layer. On Figure P there are two olots of boundary layer thickness at the mentioned nositlons. The numbers represent 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. Three corrections were aplied to rudder angle of attack. 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. This correction typically reaches a maximum of 0.6 degrees. The second correction accounts for the tunnel wall inter- ference. 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 (deg.) ACT = 0.01674 cL i 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 uniform flow, aoplied to all measurements and consisting of an aoronriate shift in the abcissa on the plots of force coefficients versus angle of attack. -2 C).... F--MAXMMWU)TH OF FAMRED-j Su. ORT SPITRPLATE - QJ4 O Pn 9V96 40or$ . LO .. O LOO pi .9.0 o .n.K.A eel.00 /*9 PRO I L..C. CONTOUR AKESURVEY ABOUT PROPELLER AXIS AT RUDDERSTATION LOOKINGDOWNSTREAM FIG.7 WAKE SURVEY U U00 0.0 0.2 0.4 0.6 0.8 1.0 0.0. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 z (irnches) BOUNDARY LAYER AT PROPELLER POSITION (UPSTREAM) U 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 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 center- line, 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 dia- meter. Unfortunately, the variations described in (7) were never tested due to the budget and time limitations. 6. Results and Conclusions Results of the nerformance tests are shown in Figures 9 - 18 Figures 9 - 12 show the performance of 200 flap rudder in uniform flow for all combinations of angle of attack and flano deflection angle, whereas Vigures 16, 17, 19 show the comnarison of nerfor- mance between uniform flow condition and three different axial positions behind the proneller location for 20fl% flap rudder. The latter are, however, only for zero flan angle condition. Figures 13, 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 and 350 . Data shown include lift coeffi- cient cL, drag coefficient cD, rudder moment coefficient cq, and flo moment coefficient cgo. The moment data in Figure 11 are re- ferred to a phantom stock axis located 491 of MAC aft of the lead- ing edge at the MAC. At this noint rudder moment becomes essen- tially 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% flap rudder of the original series (1) is given in Table 4. 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 in- dependent of the flap deflections for small deflection angles, and is slightly higher than that of the original rudder in this region, while for larger flao deflection, the onnosite becomes true. Drag coefficient of the current rudder is lower at the maxi- mum lift, as well as at the zero angle of attack at all flap de- flections. 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 - SO earlier for larger flan deflec- tions. 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. When the tunnel oressure was lowered during the tests, thusih4-%v cavitation, it was observed that the cavitation inception on the leading edge of the rudder was uni- form 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), that the 10% flap -3 1-_ 0 0 0 0 0 0 0 0 0 0 0 0 -0 1 CL @ CD @ CD @ L/D @ * ap @ * ac @ max max 60 aCL/9a CL @ CL a=0 a=0 CL CL CM CM CMF CMF max max = max max max max max max 0 2.86 0.97 20.0 0 0.015 0.163 5.95 0.260 21.0 -0.007 30 20% flap 5 2.86 1.00 22.0 0.11 0.018 0.342 2.92 0.200 20.5 -0.008 30 10 2.86 1.04 20.0 0.23 0.025 0.300 3.47 0.251 20.0 -0.007 30 rudder 15 2.86 1.11 16.0 0.34 0.036 0.220 5.04 0.255 20.5 -0.007 30 20 2.86 1.17 16.2 0.41 0.049 0.258 4.53 0.255 21.2 -0.011 30 30 2.96 1.35 18.0 0.53 0.082 0.364 3.71 0.195 19.6 -0.017 30 35 3.05 1.40 18.0 0.58 0,105 0.416 3.37 0.190 19.6 -0.017 30 0 2.75 0.80 17.0 0 0.015 0.124 6.45 -0.110 31 5 2.75 0.92 19.0 0.07 0.017 0.180 5.11 -0.128 31 10% flap 10 2.75 0.95 18.0 0.15 0.022 0.220 4.32 -0.150 31 no no 15 2.75 1.01 19.0 0.22 0.025 0.270 3.74 -0.174 31 rudder 20 2.86 1.07 17.0 0.24 0.031 0.218 4.90 -0.192 31 data data 25 2.95 1.10 17.0 0.25 0.039 0.250 4.40 -0.208 31 30 2.98 1.16 18.0 0.26 0.045 0.326 3.56 -0.222 31 avail, avail. 35 3.06 1.20 18.0 0.33 0,052 0,34 4 3.49 -0.238 31 1, 0 2.07 0.78 20.0 0 0.021 0.20 3.90 -0.105 30 0.002 30 original 5 2.74 0.88 20.0 0.05 0.023 0.23 3.82 -0.150 30 0.005 30 10 2.82 0.98 20.0 0.12 0.028 0.27 3.63 -0.180 30 0.007 30 20% flap 15 3.01 1.09 20.0 0.19 0.039 0.31 3.52 -0.227 30 0.010 30 20 3.04 1.19 20.0 0.27 0.050 0.35 3.40 -0.260 30 0.012 30 rudder 25 3.06 1.27 20.0 0.35 0.075 0.39 3.26 -0.290 30 0.014 30 30 3.08 1.32 20.0 0.43 0.085 0.42 3.14 -0.310 30 0.016 30 lin (1) .35 3.24 1.40 20.0 0.47 0.110 10.47 2 -0,345 30 0,017 30 Table 4: Principal Hydrodynamic 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 rudder might exhibit a more desirable performance than similar rudders with larger flaps, the results of the present oroject seem to be somewhat dissapointing. Table 4 shows that the maximum 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. Plan moment coefficient data for the 10, flan rudder is not available, 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. Maximum lift increases 35% and 32 on two tacks resnectively. Stall angles occur at 50 and 9O later. Drag coefficient 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, and its flan is in the neutral -36- 0 0 0 0 0 0 Q Q Table 5: 20? Flap Rudder behind Propeller I 0 CL C CD 0 Lmax CD @ CL X L/ ;a max max MalX -a + - + a a=0n a- a o prop 0 2.86 -0.97 0.97 -20 20 0.015 0.163 0.163 20% flapT 0.5 D 0 3.23 -1.5 1.43 -25 29 0.004 0.264 0.642 rudder 0.75 DJ 0 3.23 -1.5 1.43 -25 29 0.004 0.264 0.642 LL1.0 D. 0 3.23 -1.5 1,43 -25 29 0,004 0,264 0,642 Table f: 10% Flap Rudder behind Propell r 10?/ Flan Rudder behind Pro eller I tA -4 1 Table 7: Maximum Lift-Drag Ratio 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 of attack relative to the free stream direction is zero. 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. The ontimization of this geometrical relationship may be worth some further in- vestigation. Another interesting result of the propeller-rudder configu- ration is that the steady forces on the rudder are comoletely in- denendent 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 undeflec- ted 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 un- deflected 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, and ?00" with flan deflection angle of 350 on the operational tack. MaJor Conclusions The following conclusions anoly to the 7eynolds number of the experiments which were about 0.96 * 10* (based on the MAC), but there is no evidence that they do not ?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% flap rudder of the orig- inal series (1). 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 superior to all rudders considered in both orojects and is 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 desir- able to develor a linkage that incorporates a variable ratio bet- ween the flan angle and the rudder angle. (See Table 7). L. 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_ -to -0 - -o-03 ANL FA IL ERE Fig.9 20% Flap Rudder in Uniform Flow 0 0 0 0 0 0 1.2 L LL (U ANGLE OF ATTACK -- DEGfEES Fig.10 20% Flap Rudder in Uniform Flow 49 W (A. I- I I I STOCKAXIS 49% Cf MAC AFT CMAC LEADING EDGE - - - O.2 ------_ - - -- @0 a 0 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 z .C| - . ------ 0 0 30' 0 FLAP G C : U z woso a 10 .0!FLA 205 -30 -20 -0 0 10 20 30 ANGLE OF ATTACK DESREES Fig.12 20% Flap Rudder in Uniform Flow now A 0 S - '.5 owIle F n 1u -0-4 - - ..- .... -. ------ Z7- * -loo0 U-- ANa.A- OFATIAK D~4 Behind a Propeller Fig.13 10%Flap Rudder In.Uniform Stream and VMvatm 11. .04 . x 1-- 0123 gvce FIg.14 10% Flap Rudder In Uniform Flow and Behind a Propeller a U w- .2 ------ -J-- - ~ -40 -M Flow and Behind a. Propeller Ftg.15 10% Flap Rudder in Uniform 9 0 V PRO- 4x 10 I-o 5 .. 0.z uIAp -3I2 -' O203x ANL O TAK ~ -DECE - 10 .....-.. -30 - 0 -00103 ANGLE OF ATTACK DLCM4LES Fig.16 20% Flap Rudder Behind a Propeller v ------ *X-0.75 aIX-I. u K 4 --- - ~ ------ -30 -zoOA A E 3 AN'.GLEOF ATTACK~ EEE FIg.17 '20% Flap Rudder Behind a Propeller Q3 -I STOCKAXIS = 49% OF MAC AFT OF MAC LEADINGEDGE 02 oL * -0.3- -30 -20 -go 0 0 20 30 ANGLL OF AT lACK ~EGREES Fig.18 20% Flap Rudder Behind a Propeller 7. Comparison of Theoretical and Exnerimental Data Table 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. The values in the last row of the Table overleaf (Table 8) will be described later in this chanter. The table shows that there is quite a strong discrepancv bet- ween the theoretical and experimental predictions. The authors of (2) believe that the theoretical results are correct to with- in 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, oper- ating in a fully separated, turbutent region, where the net velo- city 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 20% flap 3CL/ca aCL/as CL @ CL @ CD @ gCM/aa XCP/MAC @ CD @ rudder 6=00 a=0 0 a=00 6=ii a=1008=10o a=10 6= 6=00 6=00 6=00 a=j theory 3.134 1.771 0.547 0.856 0.0476 -1.893 -60.4 % 0.0085 experiment 2.86 1.17 0.48 0.73 0.049 -1.609 -61.6 % 0.0150 experiment 0.057 -1.725 -58.7 % 0.0152 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. As it was later confirmed, 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. It determined that the velocity increases now by 15.58 more as compared with the previous splitter plate shape, or by 15.57% as compared with the original tunnel test section. New boundary layer thickness measurements showed that the thickness decreased from 1.4" to 0.35". The latter value is in a close agreement with the theoretical prediction, which for the considered Feynolds number on the plate is equal to O.34".The results of the test section of the 20 flan rudder in this new flow condition is oresented in the last row of Tmable P. -53- The agreement between the theoretical and exDerimental re- sults is now much better. The lift curve slope on the rudder with undeflected flap differs by 6% from the theory, which according to (8) is within the reasonable limits. It was expected that the theoretical and new experimental values of the drag co- efficient 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 theo- retically, in other words, to obtain theoretical results of the rudder characteristics for the condition of non-uniform distribu- tion 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 dis- placed 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 distribu- tion 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 coeffi- cient 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 due to -55- 0 00000000 0 0 0 0 9 a'U' S , 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 root sectI on tip section FIG.19 SPAIll SF Di STRIMITio0 OF LIFT SLOPE COEFFICIENT (THEORY) the presence of the boundary layer on the tunnel wall is reason- able, since the same result (3%) was confirmed by the experiment. Since the real rudders operate in the shin hull boundary layer and wake, the author believes that the results presented in 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 displace- ment 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. The lift generated by this negative cumber subtracts 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 lift- ing 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 boun- dary 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. Another assumption made in that work was, that this change in 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, an -58- 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 were made of a two foot chord NACA 65-012 airfoil in 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. The author of (11) concluded that the theory of Reference (10) strongly over-estimates the effect of the tunnel wall boun- dary layer. The same remark, but without any explanation, apnears in Reference (12), page 383. The third approach is presented by K'6rner (13). It is a lifting surface program for evaluation of an airplane wing per- formance in the presence of fuselage. The theory takes into con- sideration many effects of the fuselage-wing interactions, inclu- ding the wing twist, relative position of the wing and the fusel- age, and some details of the wing fuselage jointing. -59- 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 solu- tion for the circulation distribution is a function of the non- uniform 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 span- wise 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 requires a significant amount of research. -60- Acknowledgement The author participated in this project as a Research Assis- tant. 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 follow- ing parts of this work: 1. Conceptual Formulation and Description of the Lifting Surface Program (2). 2. Data Reduction Program. 3. Formulation of. Modification Method of the Rudder Sections. 4 . Subroutine HSVEL in (2). 5. 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 Per- formance 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 7 and 8 and performed some of the actual testing of the rudders. 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 friend- ly 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 posi- tion 2 c = length of mean aerodynamic chord /3(ct +cr- CTUR) CT+CR CD = drag coefficient = D/P/2 AU_2 = lift coefficient = L/P/2 AU_2 CM = 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 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 lift of rudder M = total moment acting on rudder about the shaft axis shown in Figure 8 MAC = mean aerodynamic chord MF= moment acting on flap about flap hinge line rudder = flap plus skeg -63- Nomenclature (cont.) skeg = forward portion of rudder 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 wake of propeller x = axial clearance between end of propeller hub and the leading edge of the MAC x = axial distance along chord of rudder, non-dimen- sionalized by a rudder local chord Y = transverse clearance between propeller axis and rudder plane of symmetry P x = 0 Y'= 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 1. An Experimental Study of a Series of Flapped Rudders - J.E. Kerwin, P. Mandel, S.D. Lewis, JSR Vol. 16, No. 14,1972 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 Control Surfaces - L.F. Whicker, L.E. Fehlner, DTMB Report 933, May 1958. 5. Theory of Wing Sections - I.H. Abott, A.E. Von Doenhoff, Dover Publications, N.Y. 6. The Elements of Airfoil and Airscrew Theory - H. Glauert, 2nd Edition, Cambridge University Press, 1Q59. 7. Sailing Yatch Keels - J.E. Kerwin, H.C. Herreshoff, 3rd HISWA Symp. Amsterdam, 1973. 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 - 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 - R.C. Pankhurst, D.W. Holder, London, Sir Isaac Pitman & Sons, Ltd. 13. Berechnung der Potentialtheoretischen StromungUm Flugel-Rumpf- Kombinationen und Vergleich mit Messungen - H. Korner, DPVLR. Dissertation, 34, GFR. -65- 11. List of Figures and Tables Figures: Page 1. Rudder Planform ...... 7 2. Sketches of section modifications ...... 12 15 3. Propeller characteristics ...... 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 ...... 47 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 .... 20 -66- 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 ...... 37 8. Comparison of theoretical and experimental results ...... 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. A detailed defi- nition of the symbols is given in Reference (2). 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 the angle of attack, no flan deflection 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 values at all the control points, after the calculation has been performed. It is 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 calcu- lations 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 Z A - 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 coefficient 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. Instructions for preparations of Input Data. The first card has nine entries: Symbol Column Limitations Description KDM 4 KN os m (g recommended) -70- LT 8 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 AF 25-32 0.l T 33-40 T>0.1 taper ratio = tin chord/root chord PP 4t1-4'8 quarter chord sweep angle in deg. (positive if swept downstream) KOPT 56 0 or 1 if KOPT = 1, the subroutine OPTTON will be called and will perform the calculations appearing in the the third page of the output The geometrical constrains are that the hinge line must be perpendicular to the tip and root sections, and it cannot inter- sect neither the leading nor the trailing edges. The second card has 10 entries; they are the numbers speci- fying the chordwise positions of the control points. The recommen- ded values are 3, 8, 13, 18, 23, 28, 33, 38, 42, 48 if the THF = 0 on the first card, and 4, 10, 16, 22, 28, 34, 40, 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 3 page output . -72- FORTRAN IV GA RELEASE 2.0 MA IN DATL = 74120 14/37/39 PAGE 0001 0001 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) 0002 CIMENSION WW(80,37),CA(38),CD(3P),ZMP(Il),NCP(10),CAA(36),C0D(36) 0003 2 READ(5,l)KCLTIHFIVER,ASRAF,T,PP,KIPT 0004 1 FORMAT (414,4FR.3,18) 0005 IF(KDM.EQ.0) STOP 0006 READ(5,4) (NCP(N),N=1,10) 0007 4 FORMAT (1018) C CALCULATION OF GRID AND CONTROL POINT CCORCINATES **************** 0008 AR=2*ASR 0009 W2=0.5*TAN(PP*3.14159/180.0) 0010 CZYN=4.0*AF*(T+1.C) 0011 DENR=2.0*AR*(T+1.0) 0012 FRO0T=(C7YN-3.0*(T-1.0))/DENR-W2 0013 SROOT=-((5.0+3.0*T-CZYN)/DENR+W2) 0014 ST IP=-( (5.0*T+3.0-CZYN)/DlENR-W2) od15 FT IP=(3 .0* (T-1 .0) +CZYN)/DE:-NR+W2 0016 IF(SROOT.LT.0.O.AND.STIP.ILT.0.0.AiD.FROCT.GT.0.0.ANC.FTIP.GT.0.0)G 10 TO 14 0017 WRITE(6,1003)AR,AF,T,PP, IHF,IVER, KM,Li 0018 WRITE (6,15) 0019 15 FORMAT(///,' ERROR IN THE INPUT DATA: SWEEP ANGLE CR FLAP AREA IS 1 TOO BIG OR TAPER RATIO IS TOO SMALL ') 0020 G TO 2 0021 14 INDX=KDM*(LT-1) 0022 LIT=LT-1 0023 NUNK=LT*KDM 0024 STR=STIP-SROOT 0025 FTR=FTIP-FROOT 0026 RVER=FLOAT (IVER) 0027 RM=14.0+10.0*RVER 0028 LK2=21+5*IVER 0029 LK3=7+5*IVER 0030 LK4=LK3+1 0031 DD. 99 I=1,LK3 0032 99 ZGP(I)=2.0*(FLOAT (T)-1.0)/RM 0033 DO 101 I=LK4,LK2 0034 101 ZGP(I)=ZGP(I-1)+2.0/(14.0*RM) 0035 ZCP(1)=1.0/RM 0036 DO 105 K=2,6 0037 FK=FLOAT(K) 0038 105 ZCP(K)=(2.0*FK-1.0+2.0*RVR*(F-K-1.0))/R 0039 ZCP(7)=2.0*(RM/2.0-1.0)/RM+9.0/(14.0*RM) 0040 ZCP(8)=2.0*(RM/2.0-1.0)/RM+19.0/(14.0*RM) 0041 NF=AF*FLOAT(50+IHF*10)+0.1 0042 NS=50+10*IHF-NF 0043 RNS=FLOAT(NS) 0044 RNF=FLOAT(NF) 0045 DO 17 I=1,LK2 0046 XS=ZGP(I)*STR+SROOT 0047 XF=ZGP( I )*FTR+FRO2T 0048 DO 16 J=lNS 0049 16 XGP(I,J)=XS *(FLOAT(NS-J+1)-0.50)/FLDAI(NS) -73- FORTRAN IV G1 RELEASE 2.0 MAIN DATE = 74120 14/37/39 PAGE 0002 0050 NS1=NS+ 1 005 1 NT=NS+NF 0052 DO 17 J=NS1,NT 0053 17 XGP(IJ)=XF *(FLOAT(J-NS-1)+0.50)/FLCAT(NJF) 0054 00 20 K=1,8 0055 XSCP=ZCP(K )*STR+SROOT 0C56 XFCP=ZCP( K)*FTR+FRCrT 0057 DO 20 N=1,10 0058 IF(NCP(N).GT.NS) GO TO 19 0059 XCP(K,N)=XSCP*FLOAT(NS-NCP())/FL7AT(NS) 0060 G0 TO 20 0061 19 XCP(K,N)=XFCP*FLIAT(NCP(N)-NS)/FL0AT(NF) 0062 20 CONTINUE CALCULATION OF INDUCED VFLOCITIES ****** 0063 DO 1100 KD=1,NUNK 0064 CA(KD)=0.0 0065 1100 CD(KD)=O.0 0066 DO 759 KD=1,KDM 0067 00 759 L=1,LT 0068 00 759 K=1,8 0069 DO 759 N=1,10 0070 759 V(K,N,KDL)=O.0 0071 LW5=LK2-1 0072 DO 720 I=1,LW5 0073 ZMS=0.5*(ZGP(1+1)+ZGP(1)) 0074 XS=ZMS*STR+SROOT 0075 XF=ZMS*FTR+FRI0'T 0076 CGP T=ZMS* (FTR-STR ) +FROCT-SROOT 0077 ZWIG=ARCOS (-ZMS) 0078 00 500 KD=1,KCM 0079 500 E(KD)=SIN( (2.0*FLCAT(KD)-1.0)*ZWG) 0080 DO 720 J=1,NT 0081 FJ=FLOAT(J) 0082 DO 721 K=1,8 0083 DD 721 N=1,10 0084 ICOMM=I 0085 JCOMM=J 0086 KCOMM=K 0087 NCOMM=N 0088 721 Q(K,N)=HSVEL(XGP(I,J),ZGP(I),XGP(I+1,J),ZGP (1+1) ,XCP(KN) ,ZCP(K)) 0089 IF (J.GT.NS) GO TO 600 0090 SS2=-XS*FJ/(RNS*CGPT) 0091 SS1=-XS*( FJ-1.0)/(RNS*CGPT) 0092 SSW2=ARCOS(1.0-2.0*SS2) 0093 SSWl=ARCOS(1.0-2.0*SS1) 0094 P(1)=SSW2+SIN(SSW2)-SSW1-SIN(SSW1) 0095 P(2)=SSW2-0.5*SIN(2.0*SSW2)-SSWI+0.5*SIN(2. 0*SSWI) 0096 IF (LT.LE.3) GC TO 710 0097 DO 609 L=3,LIT 0098 FL=FLOAT(L) 0099 609 P(L)=(SIN((FL-2.0)*SSW2)-SIN(FL-2.0)*SSW1) )/(FL-2.C)+(SIN(FL*SSW1 1)-SIN(FL*SSW2) )/FL 0100 GO TO 710 ------ FORTRAN IV GI RELEASE 2.0 MAIN DATE = 74120 14/37/39 PAGE 0003 0101 600 FJOT=FLOAT(J-NS) 0102 SF2=(XF*FJOT-RNIF*XS)/(RNF*CGPI) 0103 SFl=(XF*(FJOT-1.0)-RNF*XS)/(RF*CGPT) 0104 SF W2=ARCOS(1. 0-2. 0*SF2 ) 0105 SFW1=ARCOS( 1.0-2.0*SF1) 0106 P(1)=SFW2+SIN(SFW2)-SFW1-SI NtSFW1) 0107 P(2)=SFW2-0.5*SIN(2.0*SFW2)-SFWI+0.5*SIN(2.0*SFWI) 0108 IF (LT.LE.3) GO TA 700 0109 IN0 704 L=3,LIT 0110 FL=FLOAT(L) 0111 704 P(L)=(SIN((FL-2.0)*SFW2)-SIN((FL-2.0)*SFW1))/(FL-2 .0) +(SIN(FL*SFWI I)-SIN(FL*SFW2) )/FL 0112 700 T2=FJOT/RNF 0113 Tl=(FJOT-1.0)/RNF 0114 TW2=ARCOS( 1.0-2.0*T2) 0115 TW1=ARCOS(1.0-2.0*T1) 0116 LIT=LT 0117 P(LT)=TW2+SIN(TW2)-TW1-SIN(TW1) 0118 710 DO 760 K=1,8 0119 DO 760 N=1,10 0120 DO 760 KD=1,KDM 0121 00 760 L=1,LIT 0122 760 V(K,N,KDL)=V(K ,NKD,L)+0(K,N)*E (KD)*P(L 0123 720 CDNTINUE C CALCULATION OF BOUNDARY CONDITIONS *********** 0124 DO 801 K=1,8 0125 D0 801 N=1,10 0126 DWNWSA(K,N )=0. 0 0127 801 DWNWSD(K,N)=0.0 0128 LT5=LT-1 0129 NUNK=KDM*( LT-1) 0130 MUNK=NUNK+1 0131 DO 805 N=1,10 0132 DO 805 K=1,8 0133 IV=(N-1 )*8+K 0134 DO 800 L=1,LT5 0135 00 800 KD=1,KDM 0136 IH=(L-1)*KDM+KD 0137 800 WW(IV,IH)=V(K,N,KDL) 0138 805 WW(IV,MUNK)=1.0 0139 CALL PTLSQ (WWCA,80,NUNKKERRCR) 0140 DO 803 N=1,10 0141 DO 803 K=1,8 0142 DO 803 KD=1,KDM 0143 00 803 L=1,LT5 0144 IS=(L-1 )*KCM+KC 0145 803 DWNWSA(K,N)=V(K,N,KDL)*CA(IS)+CWNWSA(K,J) 0146 '4NUNK=KDM*LT 0147 MUNK=NU NK +1 0148 DO 815 N=1,10 0149 DO 815 K=1,8 0150 IV=(N-1)*9+K 0151 00 810 L=1,LT -75- FORTRAN IV G1 RELEASE 2.0 MAIN DATF = 74120 14/37/39 PAGE C004 0152 CO 810 KD=1,KDM 0153 IH=(L-1)*KDM+KD 0154 810 WW(IV,IH)=V(K,NKD,L) 0155 WW(IVMUNK)=1.0 0156 IF(XCP(KN).LT.0.0) WW(IV,MUNK)=0.0 0157 815 CONTINUE 0158 CALL PTLSQ(WW,CD,80,NUNKKERRDR) 0159 DO 804 N=1,10 0160 DO 804 K=1,8 0161 D 804 KD=1,KDM 0162 DO 804 L=1,LT 0163 IS=(L-1 )*KCM+KP 0164 804 DWNWSD(KN)=V(K,N,KDL)*CD(IS)+DWNWSC(K,N) C CALCULATIONS OF FORCES ********************************** 0165 CLAG=(3.14159**3)*AR*(CA(1)+CA(1+KDM)) 0166 CLDG=(3.14159**3)*AR*(CD{1 )+CD(1+KCP)+CC(i+INDX)) 0167 SM1=0.0 0168 SM2=0.0 0169 ZMP(1)=0.0 0170 DO 910 1=2,22 0171 IF (I.LE.19) LMP( I)=ZMP(I-1)+0.05 0172 IF (I.GT.19) ZMP(I)=7MP(I-1)+0.025 0173 910 CONTINUE 0174 ZMP(23)=0.995 0175 ZMP(24)=1 .0 0176 CLA(24)=0.0 0177 CLD(24)=0.C 0178 00 911 1=1,23 0179 SUM1=0.0 0180 SUM2=0.0 0181 CGPT=ZMP(I)*(FTR-STR)+FROOT-SROOI 0182 ZWIG=ARCOS (-ZMP(I)) 0183 DO 912 KD=1,KDM 0184 FD=FLOAT(KC) 0185 E(KD)=SIN((2.0*FD-1.0)*ZWIG) 0186 SUM1=SUM1+E(KD)*(CA(KD)+CA(KD+KDM))*2.0/CGPT 0187 912 SUJM2=SUM2+E(KD)*(CD(KD)+CD(KD+KDM)+CC(KC+INOX))*2.0/CGPT 0188 CLA(1)=39.47842*SJM1 0189 911 CLD(I)=39.47842*SUP2 0190 AAI=CA(1)+CA(1+KDM) 0191 AA2=CD(I)+CC(1+KDM)+CD(1+INDX) 0192 DO 913 KD=2,KCM 0193 FK=FLOAT(KC) 0194 SM1=SMI+(2.0*FK-1.0)*((CA(KD)+CA(KD(+KDM))/AAI)**2 0195 913 SM2=SM2+(2.0*FK-1.0)*((CD(KD)+LD(KD+KDM)+CD(KD+IND4X))/AA2)**2 0196 CDIA=f1.0+SM1)/(3.14159*AR) 0197 CDID=(1.0+SM2)/(3.14159*AR) 0198 EFA=1.0/(1.0+SM1) 0199 EFD=1.0/( 1.0+SM2) 0200 KDML=KDM*LT 0201 00 915 N=1,KDML 0202 CAA(N)=CAtN)*9.86958 0203 915 CDD{N)=CD(N)*9.86958 -76- FORTRAN IV GI RELEASE 2.0 MAIN DATE ='74120 14/37/39 PAGE 0005 0204 WRITE(6,1003)ARAF,T,PP, IHF,IVER,KDM,LT 0205 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 OF SPANWISE MO 3DES =',12,5X,'NUMBER OF CHORDWISE MCCES =',12,//) 0206 WRITE (6,948) 0207 948 FORMAT (1oX,29H4SPANWISE DISTRIBUTION"I OF LIFT,/,10X,'7 = SPAN COORD 1INATE (Z=0 AT THE ROOT,Z =1 AT THE TIP)',///) 0208 WRITE(6,946) 0209 946 FORMAT(15X,IHZ,14X,3HCLA,12X,3HCLD) 0210 DO 950 1=1,24 0211 950 WRITE(6,949)ZMP(I),CLA(I),CLD(I) 0212 949 FORMAT (l0X,3(F10.3,5X)) 0213 WRITE(6,991)CLAGCLDG 0214 991 FORMAT(//,9X,'OVERALL LIF1 SLOPE COEFF. PER RADIAN CLAR=',F8.3, 16Xv'CLDR=' ,F8.3) 0215 WRITE(6,1000)CDIACDID 0216 1000 FDRMAT(9X,'INDUCED DRAG COEFF./UNIT LIFT CCEFF.**? CD IA=',F8.3,6X, I'CDIDCC=',F8 .3) 0217 WRIT'E(6,961 )EFAEFC 0218 961 FORMAT(9X,29HEFFICIENCIES ARE RESPECTIVELY,12X,4hEFA=F8.3,6X,5HEE-D 1 =F8.3) 0219 WRITE(6,955) 0220 955 FORMAT (/,30X,'MODE AMPLITUDES C-ALPA') 0221 WRITE(6,1002) 0222 1002 FORMAT(9X,50HK DENOTES SPANWISE MODE, L D ENTES CHORDWISE MODE) 0223 WRI TF(6 ,1001) 0224 1001 FORMAT (18X,3HK=l,9X,3HK=2,9X,3HK=3,9X,3HK=4,9X,3HK=5,9X,3HK=6) 0225 00 957 L=1,LT5 0226 KW=(L-1)*KCM+1 0227 KWMAX=KW+KDM-1 0228 958 FORMAT (8X,2HL=,1I1,1X,1OF 12.5) 0229 957 WRITE(6,958)L,(CAA(N),N=KW,KWMAX) 0230 WRI TE(6,959) 0231 959 FORMAT (/,30X,'MODE AMPLITUDES C-D-ELTA') 0232 WRITE (6 ,1001) 0233 Dii 960 L=1,LT 0234 KW=(L-1)*KDM+I 0235 KWMAX=KW+KFM-1 0236 960 WRITE(6,9583 )L,(CO(N),N=KW,KWMAX) 0237 WRITE(6,1003) ARAF,T,PPIHF,IVER,KDM,LT 0238 WRITE (6,981) 0239 981 FORMAT (///,20X,'MATRIX OF DOWNWASH VELOCITIES AT ALL CONTROL POIN ITS',/,20X,'M DENOTES SPANWISE INDEX, N CENCTES CHORCWISE INDEX',/, 230X,'DUE TO ALPHA ',/,18X,'N=1',9X,'N=2',9X,'N=3',9X,'N=4 3',9X,'N=5',9X,'N=6',9X, 'N=7',9X,'N=8', 9X, 'N=9' ,9X,'N=10') 0240 DO 982 K=1,8 0241 982 WRITE(6,983)K,(DWNWSA(KN),N=1,10) 0242 983 FORMAT (8X,'M=',11I,1X,10F12.5) 0243 WRITE(6,984) 0244 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') 0245 DI 985.K=1,8 -77- FORTRAN IV GI. RELEASE 2.0 MAIN DATE = 74120 14/37/39 PAGE 0006 0246 985 WRITE (6,983)K,(DWNWSb(KvN),N=1,10) 0247 WRITE(6,986) (NCP(N),N=, 10) 0248 986 FORMAT (/////,10X,'CtONTRCL POINT COLUMNS ARE LCCATEC AT THE DOWNST IREAM BOUNDARIES OF THE FOLLOWING PANELS : ,/,1OX,101C0,///) 0249 IF(KOPT.FQ.1) CALL OPT ION (CA,CD,KDM,LF, IHF, IVERARAFTPPSROOT, ISTIP,FROOT,FTI P) 0250 GO TO 2 0251 END -78- FORTRAN IV GI RELEASE 2.0 PTLSo DATE = 74120 14/37/39 PAGE C001 CC0 SUBROUTINE PTLSQ (A,R,NEQNUN,KFRROR) 0002 DIMENSICN A(80,37),R(36),B(1444) 0003 MUN=NUN+1 0004 00 1 M=1,NtN 0005 00 1 N=1,NUN 0006 L=N+(M-1)*NUN 0007 B(L)=0.0 0008 O0 1 J=1,NEIQ 0009 1 B(L)=B(L)+A(Jm)*A(JN) 0010 DO 2 M=1,NUN 0011 R(M)=0.0 0012 UD 2 N=1,NEC 0013 2 R(M)=RfIM)+A(N,MUN)*A(N,M) 0014 CALL SIMQ (B,R,NUN,KERR0R) 0015 RE TURN C16 END -79- FORTRAN IV G1 RELEASE 2.0 HSVEL DATE = 74120 1,4/37/39 PAGE 0001 0001 FUNCTION HSVEL(Xl,Z1,X2,Z2,X,Z) 0002 HSVEL=0.0 0003 XA=X1 0004 ZA=ZI 0005 XB=X? CC06 ZB ==72 0007 DO 1 N=1,2 0008 T= (XB-XA)/(ZB-ZA) 0009 A=1 .0+T **2 0010 B=-2. 0* (T* ( X+ T*7A-XA) +Z) 0011 C=(X+T*ZA-XA)**2+Z**2 0012 D=X-X A-T* (Z-ZA) 0013 BAC=-4.0*D**2 0014 RAT=1.0 0015 ZC=ZA 0016 XC=XA 0017 DO 2 M=1,2 0018 HSVEL=HSVEL+RAT*( (X-XCC)/SQRT( (X-XC)**2+(Z-ZC)**2)+.1.0)/(Z-ZC) 0019 IF(BAC.LT.-1.OF-05) GO TO 3 0020 IF(N.NE.1.CR.Z.LT.Z1.1 R.Z.GT. 72) G0 TO 4 0021 VFL=D/(2.0*SQRT(A**3)*(ZC+B/( 2.0*A))**2) 0022 V=SQRT((72-Z)**2+(X2-X)**2) 0023 W=0.5*SQRT((72-Z1)**2+(X2-XI) 0024 R=SQRT((2.0*D*W)**2-(W**2+D**2-V**2)**2)/(2.0*W) 0025 HSVEL=HSVEL+SIGN(1.0/R-ABS(VEL),1) 0026 GO TO 4 0027 3 HSVEL=HSVEL-RAT*0.5*(2.0*A*ZC+B)/(D*SQRT(A*ZC**2+B*ZC+C)) 0028 4 RAT=-1.0 0029 XC=XB 0030 2 ZC=ZB 0031 XA=X? 0032 ZA=-Z2 0033 XB=X1 0034 1 ZB=-Z1 0035 RETURN 0036 END -80- FORTRAN IV iG1 RELEASE 2.0 OPTION OATE = 74120 14/37/39 PAGE 0001 cco SUBROUTINE CPTTCN(CA,CC,KDM,LT, IHF-,IVERAR,AF,T,PPSROOT,STIPFROCD IT,FT IP) C002 PIM'ENS ION CA(30),CD(36),CMM(6),t(6),Z( 12),P(6),SIMPSN(11) 0003 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 DATA KI/5/,KO/6/ 0005 WRITF (KO,1003)ARAFT,PPIHFIVERI,KCM,LT 0006 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,//) 0007 CONT=0.0166 0008 CTM=(FT IP-STIP+FROOT-SROOT)*0.5 0009 XTM=tFRO]T+FTIP)*0.5/CTM 0010 INDX=KDM*(LT-1) 0011 WRITE(6,1546) 0012 1546 FORMAT(//,22X,'7',8X,'XA/LC',8X,'CM-A',8X,'XD/LC',PX,'CM-D',5X,'XA ILE/LC',5X,'XDLE/LC',/) 0013 AA1=CA(1) +CA (1+KDM) 0014 AA2=CD(1)+CC(1+KDM)+CD(1+INDX) 0015 CLAG=(3.14159**3)*AR*AA1 0016 CLCG=(3.14159**3)*AR*AA2 0017 CtvINA=0.0 0018 CMIND=0.0 0019 ZCPA=0.0 0020 ZCPD=0.0 0021 00 1515 IR=1,10 0022 CLA=0.0 0023 CLD=0.0 0024 I=IR-1 0025 ZI=FLOAT(I) 0026 Z( IR)=ZI/10.0 0027 XT=Z(IR)*(FTIP-FROOT)+FROOT 0028 XL=Z(IR)*(STIP-SRCT)+SRO;'T 0029 CT=XT-XL 0030 PRC=XT/CT 0031 ZWIG=ARCOS (-Z(IR)) 0032 DO 1520 KD=1,KDM 0033 RD=FLOAT(KD) 0034 E(KO)=SIN((2.0*RD-1.0)*ZWI;) 0035 CLA=CLA+E(KD)*(CA(KD)+CA(KD+KCM))*78.9568/CT 0036 1520 CLD=CLD+E(KD)*(CD(KD)+CD(KD+KDM)+CD(KD+INDX))*78.9568/CT 0037 LW8=LT-1 0038 ZCPA=ZCPA+CLA*Z(IR)/(10.0*CLAG)*SIMPSN' (IR) 0039 ZCPD=ZCPD+CLD*Z(IR)/(10.0*CLDG)*SIMPSN(IR) 0040 P(1)=3.14159 0041 P(2)=3.14159 0042 0. 1522 L=3,LWP 0043 1522 P(L)=O.0 0044 P(LT)=3.14159 0045 tMM(1)=3.14159*(CT+4.0*XL)/4.0 0046 CMM(2)=3. 14159*(CT+2.0*XL)/2.0 0047 CMM(3)=-3.14159*CT/4.0 -81- FORTRAN IV GI RELEASE 2.0 OPTION DATE = 74120 14/37/39 PAGE C002 0048 DO 1525 L=4,LW8 0049 1525 CMM(L)=0.O 0050 CMM(LT)=3.14159*XT/4.0 0051 CMAL=0.0 0052 CMDL=0.0 0053 CHA=0.0 0054 CHD=0.0 0055 DO 1540 KD=1,KDM 0056 CMAK=0.0 0057 CMDK=0.0 0058 CHAK=0.0 0059 CHOK=0.0 0060 DO 1530 L=1,LWR 0061 MM=KD+KDM*(L-1) 0062 CHAK=CHAK+CA(MM)*P(L) 0Q63 1530 CMAK=CMAK+CMM(L )*CA(MM) 0064 00 1535 L=1,LT 0065 MM=KD+KDM*(L-1) 0066 CHDK=CHDK+CD(PM)*P(L) 0067 1535 CMDK=CMDK+CMM(L)*CC(MM) 0068 CMAL=E(KD)*CMAK+CkiAL 0069 CMDL=E(KD)*CMDK+CMDL 0070 CHA=E(KD)*CHAK+CHA 0071 1540 CHC=E(KD)*C4DK+CHO 0072 XAL=CMAL/ (CHA*CT) 0073 XDL=CPDL/(CHD*CT) 0074 XLEA=1.0-(PRC-XAL) 0075 XLEO=1.0-(PRC-XDL) 0076 GMAZ=XAL*CLA 0077 CMDZ=XDL*CLD 0078 WRITE(KO,1550)Z(IR),XAL,CMAZ,XtL,CMOZXLEAXLED 0079 1550 FORMAT(20X,F5.2,2(5X,F7.3,4XF8.4),2(5X,F 7.3)) 0080 CMINA=CMINA+CMAZ*CT*SIIMPSN(IR) 0081 CMIND=CMIND+CMDZ*CT*SIMPSN(IR) 0082 1515 CONTINUE 0C83 SA=CMINA/(CLAG*CTM*10.0) 0084 SD=CMIND/(CLDG*CTM*10.0) 0085 CMINA=CMINA/(CTM*10.0) 0086 CMIND=CMINC/(CTM*10.0) 0087 XLEAM=1.0-(XTM-SA) 0088 XLEDM=1.0-(XTM-SD) 0089 WRITE(K0,1551)SA,CMINA,SDCMIND,XLEAMvXLEDf 0c90 1551 FORMAT(/,4X,'INTEGRATFC',17XF6.3,5XF7.4,6XF6.3,5XF7.4,2(6X,F6. 13),/) 0091 WRITE{KO,1600) ZCPAZCPC 0092 1600 FORMAT(X,'SPANWISE POSITIONS OF IHE CENTERS OF PRESSURE ARE : ZA= Jl,F8.2,5X,'ZD=',pF8.2) 0093 WRITE(KO,1705) 0094 1705 FORMAT(/,11X,'ALPHA',8X,'DELTA',6X,'CEL/ALPH',9X,'CL',12X,'CD',12X 1,'L/D',9X, 'C',1OX ,'XHL/C',/) 0095 D0 1650 KAL=1,21,5 0096 DO 1650 KRA=1,26,5 0C97 KALA=KAL-1 -82- FORTRAN IV GI RELEASE 2.0 OPTION DATE = 74120 14/37/39 PAGE 0003 0098 KR=KRA-1 0099 IF(KALA)1500,1500,1505 0100 1500 IF(KR)1501,150 ,1502 0101 1501 ALA=0.0 0102 DEL=0.0 0103 DIV=99999.9 0104 GO TO 1510 0105 1502 ALA=0.0 0106 DEL=FLOAT(KR) 0107 DIV=99999.9 0108 GO TO 1510 0109 1505 ALA=FLOAT(KALA) 0110 R=FLOAT(KR) 0111 DEL=ALA*R/ 10.0 0112 DIV=DEL/ALA 0113 1510 ALPH=ALA/57.296 0114 DELT=OEL/57.296 0115 AA=AA1*ALPH+AA2*DELT 0116 CLALF=CLAG*ALPH 0117 CLCEL=CLDG*CELT 0118 CL=CLALF+CLCDL 0119 SM=0.0 0120 IF(AA) 1512,1516,1512 0121 1512 O 1514 KD=2,KCM 0122 RKD=FLOAT(KC) 0123 KI=KD+KDM 0124 K2=KD+INDX 0125 1514 SM=SM+(2.0*RKD-1.0)*(((CA(KD)+A(K1))*ALPH+(CD(KD)+CD(K1)+CD(K2))* lDFLT)/AA)**2 0126 CR=(CL**2)*(1.0+SM)/(3.14159*AR)+C.0C85+CCNT*CL**2 0127 GO TO 1517 0128 1516 CR=0.0085+CCNT*CL**2 0129 1517 CLD=CL/CR 0130 1F(CL.EQ.0.C)XPC=99999.9 0131 IF(CL.EQ.0.0)CM=0.0 0132 IF(CL.EQ.0.0) GO TO 1709 0133 CM=(CMtINA*CLALF*ALPH+CMIND*CLDEL*DELT)/CL 0134 XPC=(SA*CLALF+S0*CLDEL )/CL 0135 1709 WRITE(6 ,1710)ALA,DEL,DrIVCL, CR, CLD,CMXPC 0136 1710 FORMAT(9XF6.1,7XF6.1,8XF6.1,7XF8.4,6XF8.4,5XF8.3,6XF8.5,5X, 1F8.4) 0137 1650 CONTINUE 0138 RETURN 0139 END -83- FORTRAN IV G1 RELEASE 2.0 S IMQ D.ATE = 74120 14/3-7/39 PAGE 0001 0001 SUBROUTINE SI MO(A,B,N,KS) SIMQ 490 0002 DIMENSION A(1),8(1) SIMCQ500 0003 T1L=0.0 SI MQ 540 CC04 KS=0 SIMQ 550 0005 iJ=-N S IMQ 560 0006 00 65 J=1,N SIMQ 570 CC07 JY=J+1 SIMQ 580 0008 JJ=JJ+N+1 S IMQ 590 0009 BIGA=0 SIMQ 600 0010 IT=JJ-J SI MQ 61C 0011 DO 30 I=J,N SIMQ 620 0012 IJ= IT+I SIMQ 660 0013 IF (ABS (BIGA)-AMS A( IJ ) ) )20, 30, 30 0014 20 BIGA=A(IJ) SIMQ 680 0015 IMAX=I SIMQ 690 0016 30 CONTINUE SI MQ 700 0017 IF (ABS(BIGA)-TnL) 35,35,40 0018 35 KS=1 SIMQ 750 0019 RETURN SIMQ 760 0020 40 Il=J+N*(J-2) SIMQ 8C0 0021 IT=IMAX-J SIMQ 810 0022 00 50 K=J,N SIMQ 820 0023 11=I1+N SIMQ 830 0024 12=I1+IT SIMO 840 0025 SAVE=A(Il) SIMQ 850 0026 A(II)=A(12) SIMQ 860 0027 A(12)=SAVE SIMQ 870 0028 50 A(I1)=A(Il)/B1GA SIMQ 91C 0029 SAVE=B( I MAX) SIMQ 920 0030 B( IMAX)=P(J) SIMQ 930 0031 B(J)=SAVE/BIGA SIMQ 940 0032 IF(J-N) 55,70,55 SIMQ 980 0033 55 IQS=N*(J-1) SIMQ 990 0034 00 65 IX=JY,N SIMQ1000G 0035 IXJ=IQS+'IX SIMQ1010 0036 IT=J-IX SIMQ1020 0037 DO 60 JX=JY,N SIMO1030 0038 IXJX=N*( JX-1)+I X SIMQ1040 0039 JJX=IXJX+IT SIMQ1050 0040 60 A( IXJX) =At IXJX)-( A(IXJ )*A(JJX SI MQ1060 0041 65 B(IX)=B(IX)-(B(J)*A(IX J) SIMQ 1070 0042 70 NY=N-1 SIMQL110 0043 IT=N*N SI MQ1120 0044 DC 80 J=1,NY SIMQ 1130 0045 IA=IT-J SIMQ1140 0046 11=N-J SIMQ1150 0047 IC=N SIMoQ1160 0048 DO 80 K=1,J SIMQ117C 0049 BI~B=8)-A( IA)*BIC) SIMQ1180 0050 IA=IA-N SIMQ1190 0051 80 IC=IC-l SIMQ12C 0052 RETURN SIMQ1210 0053 E-N0 SI MQ1220 -84- ASPECT RATIO= 2.80 FLAP AREA=.200 TAPER RAIO=0.60 SWEEP ANGLE=15.CC0 CEG. PRECISION INDICES:CHORCWISE =0 SPANWISE=0 NUMBER OF SPANWISE MODES = 6 NUMBFR OF CHOIRDWISE 10CES = 6 SPANWISE DISTRIBUI ICN OF LIFT Z = SPAN COORDINATE (Z=0 AT THE ROOT,Z =1 AT THE TIP) Z CLA CLC 0.0 3.101 1.667 0.050 3.161 1.702 0.100 3.217 1.739 0.150 3.268 1.778 0.200 3.314 1.816 0.250 3.353 1.852 0.300 3.385 1.883 0.350 3.409 1.908 0.400 3.424 1.924 0.450 3.429 1.934 0.500 3.424 1 . 936 0.550 3.407 1.933 0.600 3.377 1.926 0.650 3.330 1.913 0.700 3.259 1.891 0.750 3.153 1.852 0.800 2.994 1.783 0. 850 ?.751 1.661 0.900 2.374 1.453 0.925 2.107 1.300 0.950 1.757 1.094 0.975 1.263 0.797 0.995 0.570 0.364 1.000 0.0 0.0 OVERALL LIFT SLOPE COEFF. PER RADIAN CLAR= 3.134 CLDR= 1.771 INDUCED DRAG COEFF./UNIT LIFT COEFF.**2 CDIA= 0.114 CD I D= 0.116 EFFICIENCIES ARE RESPECT IVCLY EFA= 0 . 996 EF = 0. 9F4 MODE AMPLITUDFS C-ALIPHA K DENOTES SPANWISE MODE, L CENOTES CHORDWISE MODE K=1 K=2 K=3 K=4 K=5 K=6 L =1 0.38319 0.05641 0.01870 0.01512 0.00137 0.00345 L =2 -0.02694 -0.04400 -0.01697 -0.01665 -C.00237 -0.00346 L =3 0.00458 -0.01504 -0.01352 -0.01204 -0.00460 -C. 00270 L=4 0.00466 -0.00359 -0.00627 -0.00569 -0.00301 -0.00149 L=5 0.00373 -0.00013 -0.00267 -0.00291 -C.00258 -0.00131 MODE AMPLITUDS C-CELTA K=1 K=2 K=3 K=4 K= 5 K=6 L=1 0.02892 0.00123 C.C0115 - 0. 00257 C. 00305 -C.00133 L=2 0.12022 0.00056 -0.00690 - 0.00005 -0.00684 0. CCC8O L=3 -0.01817 0.01483 0.00298 0.00559 0.00020 0.00179 L=4 0.03882 0.00483 -0. 00408 0.00513 -0.00648 0.00285 L=5 0.02269 0.00421 -0.00025 0. CC04 1 -0.00161 -0. 00109 L =6 0.05219 0.01288 0.00711 0.00272 0.00238 0.00099 -85- ASPECT RATIO= 2.80 FLAP AREA=.200 TAPER RATIO=0.60 SWEEP ANGLE=15.CC CEG. PRECISICN INOICES:CHOROWISE =0 SPANW ISE=0 NUMBER OF SPANWISE MCDES = 6 NUMBFR OF CHORDWISE MODES = 6 MATRIX OF DOWNWASH VELOCITIES AT ALL CONTRCL POINTS M DENCTES SPANWISE INDEX, N CENIES CHORDWISE INDEX DUE TO ALPHA N=1 N=2 N=3 N=4 N=5 N=6 N=7 N=9 N=1C M=1 1.02070 1.01131 1.-*01435 1.00963 1.00307 0.99969 0.99856 0.99993 1.00006 0.99640 M=2 0.96721 0.96267 0.98008 0.98722 0.98988 0.99182 0.99572 1.00064 1.00369 1.00138 M=3 1.02139 1.00844 1.01765 1.01474 1.00784 1.00375 1.00260 1.00483 1.00595 0.99944 M=4 1.00655 0.99608 1.00720 1.00416 0.99632 0.99174 0.99086 0.99457 0.99771 0.99 075 M=5 0.99057 0.97784 0.99452 0.949979 0.99974 0.99943 1.00112 1.00546 1.00986 1.00641 M=6 1.01369 1.00045 1.01236 1.01033 1.00186 0.99582 0.99319 0.99654 1.CO143 0.99183 M=7 1.00130 0.97607 1.00231 I.00632 1. 0CC74 0.99572 0.99435 1.00126 1.00979 0.99982 M=8 0.99963 1.00589 1.00328 0.)9788 0.99807 1.00080 1.00235 1.00060 0. 99593 1.00020 DUE TO CELTA N=1 N=2 N=3 N=4 N=5 N=6 N=7 N=8 N=9 N= 10 M=1 -0.03388 0.03361 0.02758 0.00041 -0.01732 -0.00874 0.03134 -0.00522 0.95540 1.02819 M=2 -0.00375 0.10217 -0.05518 -0. 13353 -C. 05768 0.05782 0.14758 -0.00244 0.88485 1.04328 M=3 -0.06883 0.12668 0.00869 -0. 07988 -C.04387 0. 04046 0.11529 -0.01970 0.86969 1.07358 M=4 -0.06031 0.12636 -0.00214 -0.09686 -0.05773 0.03653 0.12782 -0.00347 0.86885 1.05204 M=5 -0.03471 0.12378 -0. 02325 -0. 12020 -0.06767 0.04295 0. 14908 0.00173 0.88790 1.04648 M=6 -0.06120 0.13446 0.00201 -0.10037 -0.06268 0.03799 0.14074 -0.00560 0.87483 1.04475 -0.06922 0.13348 0.00824 -0.10349 -0.07879 0.02011 0.13612 0.01871 0.89061 1.03844 M= 8 -0.05646 0.13186 0.00335 -0.10709 -0.07201 0.03526 0.14436 -0.00019 0.88214 1.04285 CONTROL POINT COLUMNS ARE LOCATED AT THE DOWNSTREAM BOUNDARIES OF THE FOLLOWING PANELS : 3 8 13 18 23 27 32 37 42 -86- ASPECT RATIO= 2.80 FLAP AREA=.200 TAPER RATI0=0.60 SWEEP ANGLE=15.000DEG. PRECISION INDICES:CHORDWISE =0 SPANWISE=0 NUMBER OF SPANWISE MODES = 6 NUMBER OF CHORDWISE MODES = 6 z XA/LC CM-A XD/LC CM-D XALE/LC XDLE/L C 0.0 -0.589 -1.8262 -0.246 -0.4101 0.251 0.594 0. 10 -0.584 -1.8802 -0.241 -0.4194 0.249 0.592 0.20 -0.583 - 1.9313 -0.237 -0.4307 0.243 0.589 0.30 -0.582 -1.9686 -0.234 -0.4412 C.237 0.584 0.40 -0.578 -1.9802 -0.232 -0.4470 0.231 0.577 0.50 -0.573 -1.9606 -0.229 -0.4442 0.227 0.571 0.60 -0.566 -1.9115 -0.224 -0.4305 0. 2?3 0.566 0.70 -0.562 -1.8303 -0.214 -0.4042 0.216 0.564 0.80 -0.562 - 1.6818 -0. 201 -0.3585 0.203 0.564 0.90 -0.567 -1.3466 -0.185 -0.2690 0. 183 0.565 INTEGRATED -0.575 -1.8027 -0.225 -0.3990 0.225 0.575 SPANWISE POSITIONS OF THE CENTERS OF PRESStJRE ARE : LA= 0.46 C.47 ALPHA DELTA DEL/ALPH CL CD L/ D CM XHL/C 0.0 0.0 0.0 C.0085 C. c 0.0 0.0 5.0 0.1545 0.0117 13.258 -0.03482 -0.2253 0.0 10.0 0.3091 0.0211 14.630 -0.06963 -0.2253 0.0 15.0 0.4636 0.0369 12.561 -0.10445 -0.2253 0.0 20.0 0.6182 0.0590 10. 4 76 -0. 13926 -0.2253 0.0 25.0 0.7727 C.08*74 8.839 -0.17408 -0.2253 5.0 0.0 0.0 0.2735 C.0183 14.963 -0. 15732 -0.5753 5.0 2.5 0.5 0.3507 0.0246 14.252 -0.12649 -0.4982 5.0 5.0 1..0 0.4280 C.0325 13.161 -0.11308 -0.4489 5.0 7.5 1.5 0.5053 0. 042C 12.027 -0.10910 -0.4147 5.0 10.0 2.0 0.5826 0.0531 1C.975 -0.11079 -0.3896 5.0 12.5 2.5 0.6598 0.0657 10.039 -0.11616 -0.370.3 10.0 0.0 0.0 0.5469 0.0476 11.489 -0.31463 -0.5753 10.0 5.0 0.5 0.7015 0.0-729 9.617 -0.25299 -0.4982 10.0 10.0 1.0 0.8560 0.1046 8.185 -0.22617 -0.4489 10.0 15.0 1.5 1.0106 0.1425 7.089 -0.21820 -0.4147 10.0 20.0 2.0 1.1651 C. 1868 6.236 -0.22158 -0.3896 10.0 25.0 2.5 1.3197 0.2374 5.559 -0.23233 -0.3703 15.0 0.0 0. 0 0.8204 0.0965 8.502 -0.47195 -0.5753 15.0 7.5 0.5 1.0522 C.1535 6.855 -0.37948 -0.4982 15.0 15.0 1.0 1.2840 0.2247 1.715 -0.33925 -0.4489 15.0 22.5 1.5 1.5158 0.3101 4.888 -0.32730 -0.4147 15.0 30.0 2.0 1.7477 C.4097 4.265 -0.33238 -0.3896 15.0 37.5 2.5 1.9795 0. 52 35 3.781 -0.34850 -0.3703 20.0 0.0 0.0 1.0939 0.1649 6.632 -0.62927 -0.5753 20.0 10.0 0.5 1.4029 0.2663 5.269 -0.50597 -0.4982 20.0 20.0 1.0 1.7120 0.3928 4.358 -0.45234 -0.4489 20.0 30.0 1.5 2.0211 0.5447 3.711 -0.43640 -0.4147 20.0 40.0 2. 0 2.3302 C. 7218 3.228 -0.44317 -0.3896 20.0 50.0 2.5 2.6393 0.9241 2.856 -0.46466 -0.3703 -87-