Document
NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
REPORT No. 634
CALCULATION OF THE CHORDWISE LOAD
DISTRIBUTION OVER AIRFOIL SECTIONS WITH PLAIN,
SPLIT, OR SERIALLY HINGED TRAILING-EDGE FLAPS
By H. JULIA_ ALLEN For sale by the Superintendent of Documents, Washington, D. C ..... o • . . • _ ° . Price 10 cents Subscription price, $3 per year AERONAUTIC SYMBOLS 1. FUNDAMENTAL AND DERIVED UNITS l Metric , English Symbol Abbrevia- Abbrevia- Unit Unit tion tion meter .................. ft. (or mi.)
Length ...... m foot (or mile) .........
Time ........ t second .... : ............ sec. (or hr.)
second (or hour) .......
s F / lb.
Force_ ....... weight of 1 kilogram ..... kg weight of 1 pound_ _"___ /!
horsepower ...........
Power .... : _ P hp.
horsepower (metric) ..... r..........
miles per hour ...... __ m.p.h.
)'kilometers per hour ...... ] k.p.h.
v Speed__ ......
feet per,second ........ I f.p.s.
[meters per second ....... _ m.p.s.
I 2. GENERAL SYMBOLS v, Kinematic viscosity W, Weight--rag 0, Density (mass per unit volume) Standard acceleration of gravity--9.80665 g, Standard density of dry air, 0.12497 kg-m'_-s 2 at m/s 2 Or 32.1740 ft./see. 2 W 15 ° C. and 760 mm; or 0.002378 lb:-ft. -4 sec. 2 Mass = m, Specific weight of "standard" air, 1.2255 kg/m s or g 0.07651 lb./cu, ft.
Moment of inertia--ink 2. (Indicate axis of I, radius of gyration k by proper subscript.)
Coefficient of Viscosity 3. AERODYNAMIC SYMBOLS Angle Of setting of wings (relative to thrust iw, Area ,- line) Sw, Area of wing it, Angle of stabilizer setting (relative to thrust
G Gap
line) .b, Span Resultant moment O, Chord _g, Resultant angular velocity b _ Aspect ratio V1 Reynolds Number, where l is a linear dimension
v, True air speed I.t
(e.g., for a model airfoil 3 in. chord, 100 _ . 1 T?- 2 m.p.h, normal pressure at t5 ° C., the cor- q, Dynamic pressure--_p_ responding number is.234,000; or for a model of 10 cm chord, 40 m.p.s., the corresponding L, Lift, absolute •coefficient CL_ J_ namber is 274,000) D Center-of-pressure coefficient (ratio of distance CP_ D, Drag, absolute coefficient CD-_-_-- S of c.p. from leading edge to chord length) Do Angle of attack 0[, Do, Profile drag, absolute coefficient CD0=_ Angle of downwash
G
Angle of attack, infinite aspect ratio Do Olo, Induced drag, absolute coefficient _--qS Angle of attack, induced Angle of attack, absolute (measured from zero- Dr, Parasite drag, absolute coefficient CD_--_ lift position) Flight-path angle c, Cross-wind force, absolute coefficient Cc,--_S Resultant force R,
REPORT No. 634
CALCULATION OF THE CHORDWISE LOAD
DISTRIBUTION OVER AIRFOIL SECTIONS WITH PLAIN,
SPLIT, OR SERIALLY HINGED TRAILING-EDGE FLAPS
By H. JULIAN ALLEN Langley Memorial Aeronautical Laboratory 74938--38----1 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS HEADQUARTERS, NAVY BUILDING, WASHINGTON, D. C.
LABORATORIES, LANGLEY FIELD, VA.
Created by act of Congress approved March 3, 1915, for the supervision and direction of the scientific study of the problems of flight (U. S. Code, Title 50, Sec. 151). Its membership was increased to 15 by act approved March 2, 1929. The members are appointed by the President, and serve as such without compensation.
JOSEPH S. AMES, Ph. D., Chairman, SYDNEY M. KI¢AUS, Captain, United States Navy, Baltimore, Md. Bureau of Aeronautics, Navy Department.
CHARLES A. LINDBERGH, LL. D., DAVID W. TAYLOR, D. Eng., Vice Chairman, Washington, D. C. New York City.
WILLIS RAY GREGG, Sc. D., Chairman, Executive Committee, DENIS MULLIGAN, J. S. D., Director of Air Commerce, Department of Commerce.
Chief, United States Weather Bureau.
WILLIAM P. MACCRACKEN, J. D., Vice Chairman, Executive AUGUSTINE W. ROBINS, Brigadier General, United States Committee, Army, Chief Matdriel Division, Air Corps, Wright Field, Washington, D. C.
Dayton, Ohio.
CHARLES G. ABBOT, Sc. D., EDWARD P. WARNER, So. D., Secretary, Smithsonian Institution.
LYMAN J. BRIGGS, Ph. D., Greenwich, Conn.
OSCAR WESTOVER, Major General, United States Army, Director, National Bureau of Standards.
ARTHUR B. COOK, Rear Admiral, United States Navy, Chief of Air Corps, War Department.
ORVILLE WRIGHT, Se. D., Chief, Bureau of Aeronautics, Navy Department.
Dayton, Ohio.
HARRY F. GUGGENHEIM, M. A., Port Washington, Long Island, N. Y.
GEORGE W. LEWIS, Director of Aeronautical Research JOHN F. VICTORY, Secretary HENRY J. E. REID, Engineer-in-Charge, Langley Memorial Aeronautical Laboratory, Langley Field, Va.
JOHN J. IDE, Technical Assistant in Europe, Paris, France TECHNICAL COMMITTEES AERODYNAMICS AIRCRAFT STRUCTURES POWER PLANTS FOR AIRCRAFT AIRCRAFT ACCIDENTS AIRCRAFT MATERIALS INVENTIONS AND DESIGNS Coordi_mtion of Research Needs of Military arm Civil A i_iation Preparation of Research Progrant,_ Allocation of Problems Prevention of Duplication Consideration of Inventions OFFICE OF AERONAUTICAL INTELLIGENCE LANGLEY MEMORIAL AERONAUTICAL LABORATORY WASHINGTON, D. C.
LANGLEY FIELD, VA.
Unified conduct, for all agencies, ef Collection, classification, compilation, and dissemination of scientific and tech- scientific research on the fundamental nical information on aeronautics.
problems of flight.
ERRATA TECHNICAL REPORT _10. 684 CALCULATI0hT OF THE CHORDWISE LOAD DISTRIBUTI0)T OVER AIRFOIL SECTIONS WITII PLAIN, SPLIT, OR SERIALLY HINGED TRAILING-EDGE FLAPS.
Page 3, legend for figure 5: Change the flap deflection from "30 o" to "50 °''.
Page 3, legend for figure 6: Change the flap deflection from "50 °" to "_0 °''.
Page 4, column 2, line 58: Change "figures 5 and" to "figures 6 and".
Page 13, table II, third line of heading of last column: Insert "100" before r, Zc/C,,.
REPORT No. 634
CALCULATION OF THE CHORDWISE LOAD DISTRIBUTION OVER AIRFOIL SECTIONS WITH PLAIN, SPLIT, OR SERIALLY HINGED TRAILING-EDGE FLAPS By H. JULIAN"ALLEN- SUMMARY In this report a method is developed for the calcula- tion of the incremental normal-force distribution due A method is presented for the rapid calculation of the to the deflection of the flap based upon the results of incremental chordwise normal-force distribution over an experimental investigations; the theoretical relation- airfoil section due to the deflection of a plain flap or tab, ships are used as a basis for the coordination of the a split flap, or a serially hinged flap. This report is in- experimental observations. Employment of exper- tended as a supplement to N. A. C. A. _l{eport No. 63I, imentally- determined airfoil section characteristics wherein a method is presented for the calculation of the makes it possible, moreover, to obtain a distributio_ chordwise normalzforce distribution over an airfoil without consistent in magnitude with that obtained by exper- a flap or, as it may be considered, an airfoil with flap (or iment. The method has been made applicable to an flaps) neutral.
airfoil section equipped with a plain flap or tab, a split The calculations are made possible through the corre- flap, or a serially hinged flap. This report is intended lation, by means of thin-airfoil theo_'y, o[ numerous exper- as a supplement to reference 1, wherein a method, imental normal-force distributions. The method enables similar in its details of development, is presented for the determination of the form and magnitude of the incre- the calculation of the chordwise normal-force distribu- mental normal-force distribution to be made for an airfoil- tion over an airfoil section without a flap or, as it may flap combination for which the section characteristics have be considered, an airfoil section with flap (or flaps) been determined.
neutral.
A method is included for the calculation of the flap In order to facilitate the employment of this method, normal-force and hinge-moment coefficients without neces- the report has been divided into two sections: sitating a determination of the normal-force distribution.
I. The Derivation of the Method.
INTRODUCTION II. The Application of the Method.
The general importance of airfoils equipped with In the derivation, Glauert's theoretical chordwise lift trailing-edge flaps has promoted both experimental distribution is discussed and the empirical alteration of and theoretical determinations of the chordwise dis- the theory is treated. In addition, the development tribution of normal force over such surfaces in an effort of the requisite equations for the determination of the to increase the structural efficiency of their design.
magnitude of the distribution from force-test results is The theoretical investigations have been made m_der given. In the application, the general procedure to be the assumption that the fluid viscosity is negligibly followed in using this method either for airfoil sections small. This assumption must be made, for the present with plain or split flaps or for airfoil sections with at least, in order that the problem may be analytic:flly serially hinged flaps is given in concise form along with handled. Unfortunately, as experiments have shown, an illustrative example. The mathematical derivation viscosity clearly is not a negligible factor in this problem of the theory is given in the appendix.
and, consequently, the theory is not able to predict adequately either the magnitude of the incremental I. THE DERIVATION OF THE METHOD normal force brought about by the deflection of the Glauert (references 2 and 3) has treated analytically flaps or the nature of the chordwise distribution of this the problem of the symmetrical airfoil with a plain incremental normal force.
flap, assuming the airfoil to be of infinitesimal thickness.
On the other hand, the large number of variables The thin-airfoil theory is treated in the appendix of involved in the problem makes it too difficult to develop this paper. It is shown that the incremental lift dis- an adequate method, applicable in the general c'lse, for tribution or, as it will be regarded, the incremental the calculation of the incremental normal force and normal-force distribution due to the deflection of a the incremental normal-force distribution from the flap may be considered, for convenience, to be com- experimental pressure-distribution measurements that posed of two component distributions: (a) the incre- have been made.
REPORT NO. 634_-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS mental additional distribution Pus, and (b) the incre- normal-force distribution is shown by the dotted lines mental basic distribution Pbs. The incremental addi- of figures 2 to 6. Proceeding rearward from the lead- tional distribution is, in form, independent of the ing edge of the airfoil, the pressure difference, which is flap-chord ratio and does not contribute to the quarter- zero at the leading edge, increases rapidly at first, then chord pitching moment, whereas the incremental basic more slowly and, as the hinge is approached, it increases distribution is, in form, dependent upon the flap-chord more and more rapidly until the pressure difference ratio and is responsible for the entire incremental becomes unlimited at the hinge point, where the airfoil quarter-chord pitching moment due to the deflection radius of curvature is zero. Rearward from the hinge, of the flap.
the pressure difference drops rapidly at first, then more The theoretical additional distribution, as given by slowly, and finally more rapidly again to zero pressure the thin-airfoil theory (appendix, equation (A-17)), is difference at the trailing edge. With a hinge radius of shown by the dotted curve in figure 1. Since the incre- curvature other than zero, the basic pressure difference at the hinge becomes finite.
n m
-----lZ-iZ .... __
/1o
o --il
0 .I .2 .3 .4 .5 .6 .7 .8 .,9 LO x/c 0 ./ .2 .3 .4 .5 .6" .7 .8 .9 /.0 FIGURE 1.--AdditionM normM-force distributions.
z/c tTmUIIE 2,--1}asie incremental normal-force distribution, R.A.F. 30 section; 0.10c mental additional distribution due to the deflection of 1)IMu flap at _=10 °.
the flap is identical in form with the additional dis- Numerous comparisons between experimental (made tribution for the airfoil with flaps neutral, the experi- with 0.10c, 0.20c, and 0.30c plain-flap airfoils with flap mentally determined additional distributions given in reference 1 will be used for this method. The four deflections ranging from 10 ° to 60 °) and theoretical incremental basic normal-force distributions (P_dc_b_) classes of additional distribution presented in reference 1 are given in table I and figure 1 (solid lines) of the for plain-flap airfoils generally showed good agreement present report. A key to the class of distribution to be ahead of the hinge but poor agreement behind the employed for 22 airfoils is given in table II of the present binge, particularly for large flap angles. This result is to be anticipated for ahead of the hinge favorable report. (The letters A, B, C, D, and E in column "Classification PD _' designate the class of distribution.) pressure gradients retard the growth of the boundary The remaining airfoil characteristics for these airfoils layer and, conversely, back of the hinge adverse gradi- ents accelerate the growth of the boundary layer. An are given in table I of reference 1.
T.he shape of the theoretical incremental basic lift examination of these comparisons, however, disclosed distribution (appendix, equation (3_-19)) or, of what that, for all three flap-chord ratios at any one given flap is considered to be its equivalent, the incremental basic deflection, the ratio of the experimental basic normal TtIE CHORDWISE LOAD DISTRIBUTION OVER AIRFOIL SECTIONS WITH FLAPS 3 __ o, ....
/
P
0 ./ .2 .3 ,4 ,5 .6 ,7 .8 .,9 0 ./ .2 .,2 .4 .5 .6 .7 .8 .,9 /.0 X/C x/c FZGURE 5.--B_sie incremental normal-force distribution. R. ±. ]?. a0 section; 0.20c FIGURE 3.--Basic incremental normal-force distribution. 2. A. le. 30 section; 0.20c plain flap at 6=a0 °.
plain flap at 5=10 °.
0 ./ .2 .3 .4 .5 .G .7 .8 .9 /.0 x/C FIGURE 4.--Basic inerementM normal-force distribution. Clark Y section; 0.3@ plain flap at _=_I5 °.
REPORT NO. 634--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS force to the theoretical was practically constant for the incremental basic normal-force distribution then corresponding points along the airfoil. That is, if fl is becomes markedly different from that predicted by defined as theory. This abrupt change in the nature of the flow takes place at or near 20 ° flap deflection. The exact angle at which this stalling occurs is a function of a Cn b_/ exp.
number of variables (angle of attack, Reynolds Num- fl=(Pba_ (1) ber, surface irregularities, hinge leakage) and no single \ c,,_J theor.
/3 curve can apply very near this flap angle.
it has been found that values of 5 computed from experi- Values of Pba/C,_b_ have been computed from numer- mental pressure-distribution measurements made over ous pressure-distribution measurements (references 4 0.10c, 0.20c, and 0.30c plain-flap airfoils with the same and 5) and are plotted in figures 2 to 6 along with the flap deflection (references 4 and 5), when plotted in the theoretical distributions and the computed distributions form of obtained by use of the computed curves of mean curves of fl against both _ (points ahead of the values.
hinge) and 1-- (x/c) (points back of the hinge), lib very In table III (a) to (f), the computed distributions of E " P_a/c,_a (equation (1)) are given for various flap-chord nearly oil the s_me curve. In these expressions, E is ratios and flap deflections.
the flap-chord ratio.
The expansion of the experimental results, taken with plain flaps where the flap-chord ratio never ex- ceeded 0.30, to the much higher values given in table III is justified as follows. When the flap-chord ratio is 1.0 for a symmetrical airfoil, the incremental normal- force distribution becomes the incremental additional normal-force distribution and, when corrected by the mean 5 values, the incremental additional distribution should be expected to agree with the experimentally determined additional distribution if the 5 values are truly independent of the flap-chord ratio. In figure 1, the computed distribution obtained by use of the /_ values for the unstalled flow (i. e., a=5 °, 10 °, and 15 °) is shown along with the additional distributions of ref- erence 1. In figure 7, the computed and experimental distributions for a stalled symmetrical airfoil (a = a = 28 °) are shown. From the close agreement between the ex- perimental and the computed distributions shown in figures 1 to 4 for the unstallcd flap and in figures 5 and 7 for the stalled airfoil, it is concluded that, for design purposes, the _ values may be considered independent of the flap-chord ratio.
The method of correlating experimental pressure dis- tributions for airfoils with plain flaps may be employed for airfoils with split flaps. Consider the airfoils with split flaps to be analogous to the airfoils with plain 0 .I .2 .3 .4 .5 .G .7 .8 .9 1.0 x/c flaps, the boundary-layer displacement thickness at any FIGVm_ 7.--Additional normal-force distribution for a stalled symmetrical airfoil.
point back of the hinge for the airfoils with split flaps R. A. F. 30 section.
being as great as the distance from the lower surface This result is used as a basis for extending the analy- of the flap to the upper surface of the undeflected por- sis to cases where no experimental data are available. tion of the airfoil back of the hinge. Analysis of the Curves of the mean values of fl for flap deflections of problem in this manner permits the values of 5 to be 10 ° to 60 ° were determined. It was found that, for determined from experimental data, provided that some flap deflections of 15 ° or less, a single fl curve applied. assumption is made regarding the lift distribution on the undeflectcd portion of the airfoil back of the hinge.
At these small angles the departure between theory and experiment is dight, which shows that the boundary Assume that over this portion at all points back of the layer is still thin and the flow pattern is still reasonably hinge the pressure differences are negligibly small com- like that predicted by theory. As the flap deflection is pared with the corresponding pressure differences over increased, the adverse pressure gradients back of the the split flap itself. This assumption is consistent with hinge are increased and separation finally takes p]ace; the analogy (pressures are propagated undiminished THE CttORDWISE LOAD DISTRIBUTION OVEI{ AIRFOIL SECTIONS WITH FLAPS 5 deflections. Again the assumption is made that a single through a boundary layer) and is supported fairly well f3 curve applies for all flap-chord ratios for any given by experiment (references 6 and 7), particularly for flap deflection.
positive angles of attack and the larger flap deflections.
The development of the requisite equations to deter- Values of fl were obtained for split flaps using the mine the magnitude of the incremental additional and incremental pressure distributions of reference 6. It incremental basic distribution from wind-tunnel force w'/s found that, for flap defleetions of 40 ° or more, tile tests will now be considered. From force tests of tile airfoil with flap neutral, c,,,_ (quarter-chord pitching- moment coefficient) and c, 1 (normal-force coefficient)
2_
corresponding to the normal-force distribution shown in figure 8 (a) are obtained. Again, from force tests of the airfoil at the same attitude with the flap deflected, (a) Normal-force distribution for airfoil with flap c,, 2 and c,, 2 corresponding to the normal-force distribu- neutra].
tion shown in figure 8 (b) are obtained.
Let ACm_Cm2 --Cml ! I _C,,=%--C,,/ I (_) where c., x' and c,, t' are the pitching-moment and normal-force coefficients corresponding to the normal- force distribution for the airfoil with flap neutral when plotted normal to the chord of the airfoil with flap deflected, as shown in figure 8 (e). Then Ac_ and Ac,, are the pitching-moment and normal-force coefficients of the incremental normal-force distribution when the incremental distribution is plotted normal to the chord (b) Normal-force distribution fop airfoil with flap deflected.
of ,firfoil with flap deflected, as shown in figure 8 (d).
For the commonly used airfoils, the approximation _'.=<'/ (a) Cml _- Cmi" ] is sufficiently exact except in the rare case when the flap-chord ratio E and the flap deflection _i are simul- taneously large. (See figs. 8 (a) and (c).)
(c)9istribution shown in (a) plotted normal to Let Ac,,,' and Ac,' be the pitching-moment and the flap- deflected chord.
normal-force coefficients of the incremental normal- force distribution plotted normal to the flap-neutral chord, as shown in figure 8 (e). Since the incre- __t //'% mental basic normal-force distribution is responsible /// \ for the entire quarter-chord pitching moment, then, I/ \ if G is the moment arm in terms of the chord of the basic normal force about the quarter-chord point, (d) Increment normal-force distpibutior due to defleotion of flap.
mCn! _ Crag i "@ Cnb_ or ACre ! 1 (4) (e) Distribution shown in (d) plotted normal to flap-neutral chord.
FIGURE 8.--Normal-force distribution and incremental normal-force distribution for The value of G is a function of E and _. Values of flaps netd;ral slid deflected.
G are given in table IV.
The correlation between the fictitious values of values for plain- and split-flap airfoils were the same.
Ac,,/ and Ac,/ and the measured values of ac,_ and Ac,, In table III (d) to (h), the computed distributions of must be established in order to determine c%a and Pb_/c_ are given for various flap-chord ratios and flap REPORT NO. 634--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS c%s from force tests. Let the incremental flap normal- force coefficient for unit span be given by Ac_=c'_2--cm_} (10) ACn= Cn2-- Cn 1 and using the values of _ and r_ from tables V and VI depending on the type of flap, then (equation (9)) where ,nfs is the incremental flap normal force per unit ACm"= TmACra span and q is the dynamic pressure in the air stream.
Ac,/ : Ac,_ + "r_hcm Then The incremental basic and additional normal-force coefficients may be obtained from equations (4), which are and A 'Ac'/=Ac_+ Ec'f'(1--c°s _) (3 E)} (6) cm = Ac.,-- Ec,,f_ (1 -- cos_) -- ACra !
Cnb?_ G The incremental flap normal force may be considered A _ AcmP as a combination of two components due to the incre- CT_a_ = Cn G mental additional and the incremental basic normal- When the appropriate values of the incremental basic force distributions. Let v,_ and w_ be the ratio of the normal-force distribution, Pb_/c,_, from table III are flap normal force to the airfoil normal force for the used, then incremental additional and the incremental basic nor- mal forces, respectively; then Pb_:(l_b_C_b, (11) \ (_nb&] (7) Cn ff, = "g a$Cn a_ @ "y b?,Cn b _ By the use of the proper class of incremental addi- or tional normal-force distribution, P_/C,a_ , from table I ['. , Acre'\, Acre' (8) The contribution of the additional normal-force dis- tribution is small compared with the basic contribution The incremental basic and additional distributions so that, for the purpose of determining Acre' and Ac_', may be added to give the entire incremental normal- the following approximation may be employed: force distribution, ACrn / enf_ =Tb_ G P_=P_+ Po_ (13) and equations (6) become and this incremental normal-force distribution may be added to the distribution for the airfoil section with A ..... Acm_ (1--cos _) Cn =/_Cn-]-.J_b_ _-- undeflected flap, P_ (which distribution may be obtained by the method of reference 1), to give the normal-force A _--_ h" ACm_ distribution for the airfoil wi_h the deflected flap cm--zac,,_--_%_ (1--cos _)(3--E) so that P_=P_+P_ (14) ACm ! = TmAC m The incremental flap normal-force coefficient is given AcJ= Ac_ + _-_Ac.J (9) by equation (7) and the corresponding flap hinge- Where moment coefficient can be written by analogy.
E(1 --cos _) "/_ C __ _n/f_ _ ht_ (15) Chf_--_2C2 rla_Cna_@rlb_C%_ Values of _ and w_ are given in tables VII and VIII.
The values of rn and r,, have been determined and are As the incremental additional and additional distribu- given in tables V and VI. tions are identical in form Then, given c_2, c_, Cm2, and c_ 1 (c_ may be considered Ta_ _ "Ya as cn; c,,_ may be calculated if c ...... the pitching-moment coefficient about the aerodynamic center, and x_._./c, the ehordwise distance of the aerodynamic center from the quarter-chord point of the section in terms of the Values of -y, and _ are given in tables IX and X, re- chord, are given instead of c,_) spectively.
THECHORDWISE LOAD DISTRIBUTION
OVER AIRFOIL SECTIONS WITH FLAPS
TheflapnormM-force andhinge-moment coefficients
This method for the determination of incremental for theairfoilwithflapneutral maybedeter.mined by ehordwise normM-foree distribution for airfoils with
considering thecontributions ofeach of thecomponent
flaps was developed for airfoils of normal profile and
distributions thatmake uptileflap-neutral normM-force
camber, and therefore it cannot be presupposed that
distribution. In reference 1 the flap-neutral normM-
this method might be applied to airfoil sections of ab- normal form.
force distribution is considered to becomposed of four
component distributions: (a) themoment basic(class
The values of P_s/c,,bs for airfoils with. both plain
1),(b) thecamber basic (class 0,1,or 2), (c)the aero-
and split flaps were determined from tests of airfoils dynamic center, and(d) tile additional (classes A, B, having very small gaps between the wing and the lead-
C, D, andE). The classof eachdistribution to be
ing edge of the flap. It has been found (reference 9)
employed for a number of airfoilsis given in tableII
that any gap between the wing and the leading edge
(ortables I andII ofreference 8)in thecolumn "Classi-
of a plain flap has a detrimental effect upon the aero-
fication PD." Theletter(A,B, C,D, or E) designates
dynamic characteristics. It is probable that this gap
theclass oftheadditional distribution; thefirstnumber
effect will also be true for split-flap airfoils. In tlm
(1) designates theclass of themoment basicdistribu-
absence of evidence to the contrary, the method pre-
tion;thesecond number (0,1,or 2)designates theclass
sented cannot be considered applicable to plMn-flap of thecamber basic distribution.
or split-flap airfoils with large gaps.
The moment basic normM-force coefficient may be
The flap hinges of all plain-flap airfoils, from the
obtained from
tests of which the Pb_/Cnb_ values for the plain flap were determined, were midway between the upper and lower cnb,,=--6.30c,,_.c. I for class 1 (16) surfaces of the airfoils; that is, the radius of curvature The camber basic normal-force coefficient may be of the upper surface above the hinge for each airfoil obtained from was half the depth of the airfoil at the hinge. Tests have been conducted to determine the effect of chang- c_bc= 0 for class 0 ing the radius of curvature at this point from zero to cnb =9.70 z_ for class 1 the full depth of the airfoil at the hinge. (The results c (17) of these tests have-not been published.) The airfoil 18.75 z_ for class 2 Cnbc_ employed in the test was equipped with a 0.60c plain flap deflected 12 ° and with a 0.20c plain flap deflected where z_/c is the camber in terms of the chord. Values 15°; the effect of changing the radius of curvature at the of zc/c are given for a small nmnber of airfoils in table II 0.60c-flap hinge alone was determined. The results and for a large number of airfoils in tables I and II of of these tests show only a negligible change in the reference 8. The additional normal-force coefficient aerodynami_ characteristics (and presumably in the may be obtained from normal-force distribution) with a change in the radius Cna=Cnl--Cnbc--Cnbm (18) of curVature. Because of the limited nature of the tests, these results cannot be considered conclusive for The aerodynamic-center distribution coefficient is plain-flap airfoils in general, and the method presented given by must be considered strictly applicable to plMn-flap Xa c = '_ (19) airfoils with upper-surface curvatures not less than half Ca. c. C Cna the airfoil depth at the hinge.
Values of xa._./c are given for a small number of airfoils The Pb_/c,_ values for plain-flap airfoils were de- in table II (column 3) and for a large number in tables I termined front airfoil tests made at an effective Reynolds and II of reference 8.
Number of about 1,000,000. Comparison of these Finally, tile flap normal-force coefficient is given by tests with tests made at an effective Reynolds Number of about 17,000,000 indicaites that the effect of scale is C_='Y_C,_a÷'Y_%_+'ybmC,,b,_÷'y .... C .... (20) unimportant although, it may be mentioned, in the and the flap hinge-moment coefficient is given by critical region of flap deflections (i. e., for _ near 20 °) there is a tendency at higher scales to maintain the Cbl=_,C,_q-_cC_b_+_b,nC,,b,_+_._.C .... (21) unstalled incremental basic distribution (i. e., the 5=5 °, The various _/ and _ values are given in tables IX 10 ° , and 15 ° type of distribution) up to slightly greater and X.
flap deflections. Pressure-distribution measurements on The flap normal-force and hinge-moment coefficients split-flap airfoils made at effective Reynolds Numbers for the Mrfoil section with flap deflected are of 1,700,000 and 3,200,000 showed apparently no effect from this small change of scale.
Cn_, 2 = Cnq @ Cn.r _ [ (22) It is difficult to make any general statement regarding
I
c% = chfl + cb_ the accuracy of this method for the determination of the 74938--38----2
8 gEPORT NO. 634--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
incremental chordwise normal-force distribution.The
so that, for the example cited,
dispersion ofexperimental pressure-measurement results
era2: -- 0.229 + 1.40 (0.012) = -- 0.212 shown in figures 2 to 6 may be considered typical.
C,2 = 1.40 and II. THE APPLICATION OF THE METHOD c_ 1= -- 0.005 + 0.34 (0.012) = -- 0.001 THE GENERAL PROCEDURE FOR AN AIRFOIL WITH A PLAIN OR A SPLIT FLAP c_i=0.34 In order to determine at a given lift coefficient the The quarter-chord pitching-moment and normal- incremental normal-force distribution over a given air- force coefficients for the incremental normal-force dis- foil section due to the deflection of a plain or a split tribution considered normal to the airfoil chord with flap, it _is necessary to have the following experimentally flap deflected are given by determined characteristics for the airfoil section: (a) With flap deflected: ACm=Cm2--Cmll c_2, the section lift coefficient (given). Ac,=c_2__c_l j (10) Cma._.2, the section pitching-moment coefficient about (equations are numbered as in part I of the report) the aerodynamic center.
so that, for the example cited, x,,.c._, the chordwise coordinate of the aerodynamic-, c /2 center position in terms of the chord.
Acre= --0.212-[-0.001= --0.211 (b) With flap neutral (with the airfoil section at the Ac_= 1.40- 0.34 = 1.06 same angle of attack): The pitching-moment and normal-force coefficients %, the section lift coefficient.
for the incremental normal-force distribution, con- c_a.c.1, the section pitching-moment coefficient about sidered normal to the airfoil chord with flap neutral, the aerodynamic center.
are given by xa._._, the chordwise coordinate of the aerodynamic- c I, center position in terms of the chord.
Acre t = "rmACm [ AcJ = Ac" + TnACmJ (9) (c) The class of additional normal-force distribution to be employed.
Values of r. and Tm are given in tables V and VI. For For illustrative purposes, given: an N. A. C. A.
the example cited, by interpolation from the tables, 23012 airfoil section with a 0.20c split flap (E=0.20) set at 45 ° (_=45°). To determine: the incremental wm=l.16 normal-force distribution for this airfoil, when, with the T,_: --0.30 flap deflected, the lift coefficient c_ 2 is 1.40. From so that reference 10, when Ac,/= 1.16 (--0.211) =--0.245 c_= 1.40 Ac,'= 1.06+ (--0.30) (--0.211) = 1.12 then cm_.c.s = -- 0.229 The incremental normal-force distribution is con- sidered to be composed of two component distributions: (a) the incremental additional distribution, and (b) the _=2.0 ° incremental basic distribution. The magnitude of the For the airfoil with flap retracted, when a--2.0 °, from incremental basic normal-force coefficient is given by reference 11, cz1=0.34 Ac,_' (4) CribS-- G c_a._. = --0.005 Values of G are given in table IV for airfoils with plain and from table II of the present report and split flaps.
For the example cited by interpolation from the table x_._.') =0.012 C /'1 G---- --0.412 and the C class of additional distribution is to be and so employed.
--0.245 The quarter-chord pitching-moment coefficient is =0.60 c%_--_ 0.412 obtained from , fx_._."l The incremental additional normal force is obtained from and the approximation is made c - , (4) naB-- Acn -- Cn b_ Cn _ Cl THE CHORDWISE LOAD DISTRIBUTION OVER AIRFOIL SECTIONS WITH FLAPS 9 COMPUTATION OF INCREMENTAL BASIC DISTRIBU- For the example cited TION L c.a_= 1.12--0.60=0.52 x/c x I-Z/C x Pb_ Pb__2 Pb$ Pb5 FE -_ ¥ en b_ Cnb_ The incremental additional distribution may then be obtained from 0 0 0 0 1. O0 0.80 2.06 1.23 • 05 .04 .16 • 10 • 90 • 82 2.19 1.31 • 10 .08 • 24 • 14 • 80 • 84 2. 21 1.32 • 20 .16 • 35 • 21 • 70 • 86 2.18 1.31 • 30 .24 • 46 • 28 • 60 • 88 2.12 1. 27 • 40 .32 • 57 • 34 • 50 • 90 2. 02 1.21 1.13 Values of P_dc,,a_ at a number of stations along the • 50 .40 • 69 • 41 • 40 • 92 1.89 • 60 .48 • 83 • 50 • 30 • 94 1.70 1.02 • 70 .56 1.02 • 61 • 20 • 96 1.47 • 88 chord are given for the various classes of distributions I .66 • 80 ,64 1. 26 • 75 • 10 • 98 1.10 ,90 .72 1. 59 • 95 • 05 • 99 • 82 .49 in table I and figure 1. The quantity Pat is the pres- 1.00 .80 2. 00 1.23 0 1.00 0 0 sure difference in terms of q, the stream dynamic pres- sore• normal-force distribution is • Finally the incremental For the example cited found by addition: P_ = P_ + Pb_ (13)
paO=( P°qO
\ Cna_/ so that for tile class C distribution (table I) the values of Pa, in the following table are obtained• I I I COMPUTATION OF INCREMENTAL ADDITIONAL ----4-----+-- _ _ DISTRIBUTION J I I I Pu$/C,,a$ Pa_ x/c
I --
0 0 0 • 0125 4.98 2. 59 • 025 4.23 2.20 .050 3. 22 1.67 • 075 2. 68 1.39 • 100 2.32 1.21 • 150 1.85 .96 • 200 1. 54 .80 • 300 1.14 .59 • 400 • 87 .45 ,/,5 _-_--_-- • 500 .68 .35 • 600 • 51 .27 • 700 • 37 .20 • 800 .24 .13 • 900 .12 .06 • 950 .06 .03 1. 000 0 0 /•0 _ The incremental basic distribution is found from Values of Pbdc_ at a number of stations along the chord are given in table III for plain and split flaps.
O ./ .2 .3 4 .8 .9 /.0 For the example cited, FIGURE 9.--Calculated incremental normal-force distribution. N. A. C. A. 23012 airfoil section ata=2 ° with a 0.20c split flap at _=45 °.
Values of P_dc,,_6 are obtained by interpolation from This addition has been made for the example cited• table III and computed values of Pb_ are given in the In figure 9 the distributions of P_, P_, and P_ are given.
following table. Again, Pb_ is the pressure difference in The incremental normal-force distribution may be added to the normal-force distribution for the airfoil terms of q.
l0 REPORT NO. 634--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS with flap neutral (as may be obtained from an experi- Hence, from equations (16) to (19), mental pressure-distribution investigation or by the c,_,_ = - 6.30 (-- 0.005) : 0.03 method of reference 1) to give the normal-force distri- c_b: 18.75 (0.018) = 0.34 bution for the airfoil with flap deflected c_: 0.34 -- 0.34 -- 0.03: -- 0.03 P2 = P, "4- P_ (14) c .... : --0.03 (0.012) : 0.000 The incremental flap section normal-force and flap From tables IX and X section hinge-moment coefficients are found as the sum _&--0.12 _: --0.04 of the contributions from the incremental additional %_=0.09 w_: --0.03 and the incremental basic distributions: _b.,-- 0.32 _;bm: -- 0.11 so that / 7_f6 -- c,: =qEc-- VS,_a_'+Vb:,,b_ ( c,:_ = (0.12) (--0.03) "4- (0.09) (0.34) "4- (0.32) (0.03) =0.04 h:, f (15) c,v= (--0.04) (--0.03) "4- (--0.03) (0.34) ,4- (--0.11) (0.03) = --0.01 where n: and h: are the incremental flap normal force The flap normal-force and flap hinge-moment co- and hinge moment per unit span, respectively. The efficients are, by addition, values of Va_ and Va_ are given in tables IX and X.
Values of "rba and vb_ are given in tables VII and VIII.
cn:_ = %: "4- c_n [ (2 ) ( For the example cited, from tables IX and X c1,:2 = c1,:1 "4- c/,:_ ] _=0.12; w_:--0.04 For the example cited, and by interpolation from tables VII and VIII cni2 = 0.04 + 1.13 = 1.17 _%_= 1.79; vb_=--0.77 chs2= -- 0.01 -- 0.48 = --0.49 so that, using the values of c,,_ and c,b_ already deter- THE GENERAL PROCEDURE FOR AN AIRFOIL WITH mined, A SERIALLY HINGED FLAP G:=0.12 (0.52),4,4,1.79 (0.60):1.13 The incremental distribution for an airfoil with a c_v_=--0.04 (0.52)--0.77 (0.60)=--0.48 serially hinged plain flap is obtained by determining the incremental distribution for the airfoil with each of By a similar method, the flap normal-force and hinge- the several flaps deflected and then by adding the moment coefficients for the airfoil with flap neutral various distributions.
may be determined from This superposition method will always be applicable provided that all flaps of the system are unstalled (i. e., C_I_=')':,_+'Yb:_b_+%,_C_b,,+')'_._C .... (20) no flap is deflected more than 15 °) with the exception of the final (smallest) flap, which may be stalled or un- chs_ = _c_ ,4, _b_c,,_,4, _b_C,_b_ "4- ,_._. C .... (21) stalled. If, in a combination of a large flap and a small where C,a, C,bc , and c,b,,_ are the normal-force coefficients flap (e. g., a tab), the small flap is deflected oppositely of the additional, camber basic, and moment basic to the large flap, experiment has shown that the method is applicable whether either flap is stalled or not.
distributions given in reference 1, and c .... is a measure of the magnitude of the aerodynamic-center distribution It is necessary to integrate the normal-force distribu- tion curve to determine the several flap normal-force given in reference 1. These coefficients may be found and hinge-moment coefficients for the airfoil with from equations (16) to (19) in part I. Values of v and v are given in tables IX and X. serially hinged flaps.
For the example cited, the pressure-distribution classification is given in table II as C12 and LANGLEY MEMORIAL AERONAUTICAL LABORATORY, NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS, z_=0.018 C LANaLEY FIELD, VA., April 12, 1938.
THE CHORDWISE LOAD DISTRIBUTION OVER AIRFOIL SECTIONS WITH FLAPS APPENDIX dL= P V_x dx (A-6) THEORETICAL RELATIONSHIPS FOR THE THIN AIRFOIL WITH PLAIN FLAP (p is the fluid density) and, since by differentiation of equation (A-2) The application of thin-Mrfoil theory to tile problem of the airfoil with a plain flap is detailed in the following dx =-_ c sin OdO section.
then
Dosi n o o, span-
.0 P = PV2¢ [A0 (1 + cos 0) + A, sin n0 sin 0] _
wise circulation at any point x back of the leading edge q I of the airfoil of chord c subjected to the stream velocity V. Glauert has shown that, if a distribution of vortic- 1 + cos ity along tile chord of the airfoil with a plain flap is =4[A0( s_O)+_A_sinnO] (A-7) assumed in the form Substituting the value of the coefficients (equation (A-3))
0/
(A-l)
+ _ A,_ sin n0 sin O]dO p=[4(1-}-COSsin 0 0)][ a' q_(_Oo)_]q__S_sinnOosinnOn_.
where When the flap is neutral (5=0), the lift distribution is (using the subscript _ for the flap-neutrM case) or
p_F4( +cos
¢ _--[. sin 0 0)] a' (A-8) x=_(1--eos 0) (A-2) The incremental lift distribution dtie to flap deflection is then, in order that Kutta's criterion may be satisfied and that the flow across the chord shall be everywhere tangential to the camber line of the airfoil, the coeffi- p_=[4(l +cos O)(Tr--OO)Ts{n O + _8 sin nOo sin nO]_nTr : (A-9) cients of equation (A-l) must be given by Perring (reference 12) has shown that, for airfoils with serially hinged flaps, the elemental chordwise distribution of circulation may be expressed by equa- tion (A-l), provided that the coefficients be given by 7r-- 0ol 7r-- 0o2 7r-- 0o_ 7i" 7i" An=(2 sill n0o_ _ / 2sin00% 2sin00% . . 2sin A1-- I+-_" 2-t- . . 0Or_% 71" 71" 71" where a' is the angle between the direction of stream
(Aq0)
flow and the unflapped portion of the airfoil, _ is the flap angle measured from the unflapped section, and 0o is the value of 0 at the flap hinge, i. e., 2 sin n001 2 sin nOo2 2 sin hoot A_-- _lzr _2q- • C %7r _Tr nTr cos 0o=--(1--2E) ] where 0o, 0o2,. • • 0o_ and /t_, _2,. a_ are the sin 0o=2_/_) }J (A-4) values of 00 and _ for each of the r number of flaps.
where Eis tile flap-chord ratio Hence, the lift distribution over an airfoil with serially hinged flaps may be expressed by E=_ ¢ P2=Plq-Ps_q-P_2q-P_3-_ • • • P_, (A-11) The pressure difference P (in terms of the stream where P_ denotes the distribution with all flaps neutral dynamic head q) at any point x along the airfoil section (given by equation (A-8)) and P_I, Pa_, • Pa, are is tile lift per unit span experienced by the airfoil at that the incremental distributions due to the individual point in terms of q, or deflection of flaps 1, 2, . . . r, respectively.
dL A characteristic feature of the thin-airfoil theory is
p = p 5)
that the incremental distribution due to the deflection q q of one or more flaps is independent of the original shape the lift of an element of chord but, from wing theory, of the mean camber line of an airfoil. An airfoil with a per unit span is curved mean camber line may be considered essentially 12 I_EPOB.T NO. 634--NATIONAL ADVISORY COMMITTEE FOIL AERONAUTICS as a symmetrical airfoil with an infinite system of serially hinged flaps deflected so as to produce that
o) 2 os, 00 0)
czb_--_c s_n 00 - 7 curvature. It has already been seen that the incre- 1 1 mental distribution due to the deflection of any one of But a number of serially hinged flaps is independent of the deflections of any of the other flaps.
2B()S ?l,?'
n -- log_2 --log'_ sin/'-2 Thus the incremental lift distribution as given by t equation (A-9) is equally applicable to every flapped I]ence airfoil of infinitesimal thickness.
" 1 Now consider the incremental lift distribution given P_ 2 r sin2(°0+°)_ by equation (A-9) to be the sum of (1) the incremental (A-19) .... _log+, additional distribution
c o0 ,oOo Din1/00 0/j
p _ [-4(_- 00) (1+cos 0)-]_ (A _2) The moment of the incremental additional lift about ._1 the quarter-chord point of tile airfoil may be shown and (2) the incremental basic distribution to be zero, hence tile incremental basic lift is reponsible for the entire incremental quarter-chord nmment.
(A-13) p_=[2 8 sin nn_r00 sin nO-] _ REFERENCES 1. Jaeobs, Eastman N., and Rhode, R. V.: Airfoil Section Characteristics as Applied to the Prediction of Air Forces The general form of the incremental additional and Their Distribution on Wings. T.R. No. 631, N. A.
distribution, unlike tile incremental basic distribution, C. A., 1938.
is not a function of tile flap-chord ratio, E.
2. Glauert, H.: TheoretieM Relationships for an Aerofoil with Glauert (reference 3) has shown that tile incremental Hinged Flap. R. & M. No. 1095, British A. R. C., 1927.
lift coefficient (i. e., for d=0) is given by 3. Glauert, H.: The Elements of Aerofoil and Airserew Theory.
Cambridge University Press, 1930.
c _ = 2[(_r-- 00)4- sin 0o]6 (A-14) 4. Jaeobs, Eastman N., and Pinkerton, Robert M.: Pressure Distribution over a SymmetrieM Airfoil Section with Trailing Edge Flap. T.R. No. 360, N. A. C. A., 1930.
The incremental additional lift coefficient is given by 5. Wenzinger, Carl J.: Pressure Distribution over an Airfoil Section with a Flap and Tab. T.R. No. 574, N. A. C. A., L_ 1 _0_ 1936.
ct_ qc qc P_qdx 6. Wallace, Rudolf: Investigation of Full-Scale Split Trailing- 2 +cos O)dO Edge Wing Flaps with Various Chord and Itinge Loca- tions. T.R. No. 539, N. A. (L A., 1935.
= 2 Or-- 00)a (A 15) 7. Wenzinger, Carl J., and Harris, Thomas A.: Pressure Dis- tribution over a Rectangular Airfoil with a Partial-Span and, frem equations (A-14) and (A-15) Split Flap. T.R. No. 571, N. A. C. A., 1936.
8. Pinkerton, Robert M., and Greenberg, Harry: Aerodynamic czba-2 sin 0o_ (A-16) Characteristics of a Large Number of Airfoils tested in the Variable-Density Tunneh T. R. No. 628, N. A. C. A., 1938.
Substitution of the wflues of c<e and c_b_ in equations 9. Wenzinger, Cm'l J.: Wind-Tunnel Investigation of Ordinary (A-12) and (A-13), gives and Split Flaps on Airfoils of Different Profile. T.R.
No. 554, N. A. C. A., 1936.
Pa_ 2(1-1-COS0) 10. Abbott, Ira H., and Greenberg, Harry: Tests in the Variable- (A-17) Density Wind Tunnel of the N. A. C. A. 23012 Airfoil with C_a a _r sin 0 Plain and Split Flaps. T.R. (to be published).
11. Jaeobs, Eastman N., and Pinkerton, Robert M.: Tests in the Variable-Density Wind Tunnel of Related Airfoils (A-18) P_a 4 2 sin hoe sin nO czar- _r sin _ n Having the Maximmn Camber Unusually Far Forward.
T. R. No. 537, N. A. C. A., 1935.
12. Perring, W. G. A.: The Theoretical Relationships for an Equation (A-18) may be rewritten {_s the stun of two Aerofoil with a Multiply Hinged Flap System. R. & M.
series No. 1171, British A. R. C., 1928.
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........ , , , , , , _lI f I_II I : I II I I °11111111 II _ I I [ [ I I ( I I _lllllIfiii!!i ; T c'i I I i I i I I I I : i I : z _l'l'f'_'l'l'l'Jl!!il '_ I" I" I" I" 1" I" I" I" ]" © ¢k _lliIl[ifllll :I _IIIIll_lll _ I" [ I" I" I" I" [" I" I" _11111111111_I ° "l'l'l'l'l']'_'l']'l'l'l'l" , , , , , , , , , , 'I'I'L'['I'I'J'I"I'I'I'I" / / 16 REPORT NO. 634--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TABLE VIII TABLE IX FLAP NORMAL-FORCE PARAMETERS HINGE-MOMENT PARAMETER vb_ _,_ (deg.)
The 7be "Ybm 'Y a.¢.
E Ta or _ Class 1 Class 2 Class 1 5, I0, 15 20 30 40 50 60 O 0 0 0 0 0 .05 .03 --.13 .02 .09 --4. 40 (a) PLAIN FLAPS --4• 95 ,10 .06 --. 25 .05 .17 • 07 .24 --5. 05 .15 .09 --.34 .20 .12 --.40 • 09 • 32 --4. 95 --1.23 --1.36 --1.48 --1.59 --1.63 O. 05 --0.86 --4. 78 .25 .15 --.43 .12 • 38 .10 --. 61 --.81 --.96 --1.04 --1.12 --1.16 .14 • 45 --4. 52 .30 .18 --. 42 .15 --. 50 --.66 --.79 --.86 --.9I --.95 .35 .21 --.37 .17 • 51 --4, 24 .20 --. 43 "-.57 --.68 --.75 --.80 --.83 --3. 93 .40 .24 --. 28 • 20 .58 --. 39 .25 --.52 --.6I --.67 --.72 --.74 .24 .63 --3. 61 .45 .28 --,16 .30 --. 36 --.47 --.57 --.62 --.66 --.68 .50 .31 --.01 .28 • 69 --3. 29 .35 --. 33 --.44 --.53 --.57 --.62 --.64 .55 .35 .14 .32 • 74 --2. 95 --. 32 --.42 --.49 --.54 --.58 --.60 .40 --2. 62 .60 .39 .30 .38 • 79 .45 --. 30 --.40 --.47 ..............................
.42 • 84 --2. 27 .65 .43 .45 .50 --. 29 --.38 --.45 ..............................
.70 .47 .60 .47 .89 --1. 93 --.36 ........................................
.55 --. 28 .60 --. 26 --.35 .........................................
.65 --. 25 ................................................
m. 24 .................................................
.70 TABLE X.--FLAP HINGE-MOMENT PARAMETERS (b) SPLIT FLAPS _/bo _bo _Tbm Ya or _/a_ _/a ,¢.
.E Class I Class 2 Class 1 0.05 ....... --1.27 --1.38 --1.48 --1.59 --1.63 .I0 ........... --.91 --.99 --1.04 --1.12 --1.16 .15 ........ --.75 --.81 --.86 --,91 --.95 O 0 0 0 O .20 ......... --.64 --.70 --.75 --.80 --.83 o. 05 -.01 --. 03 1.95 --. 01 • 04 .25 ......... --.58 --.63 --.67 --.72 --.74 --. 02 .09 --. 92 --. 06 2. 22 • 10 .30 .......... --.53 --.58 --.62 --.66 --.68 2. 33 • 15 --. 03 • 13 -. 02 --. 08 .35 .......... --.50 --.54 --.57 --.62 --.64 -. 03 --.J1 2.37 .20 --. 04 .16 .40 .......... --.47 --.51 --.54 --.58 --.60 --. 05 .18 --. 04 --. 13 2.39 • 25 .45 .......... --.44 --.48 ..............................
2. 36 .30 --. 06 .19 -. 95 --.15 .59 .......... --.42 --.46 ..............................
--. 18 2.31 .35 --. 07 .19 -, 06 --. 08 . 19 --. 06 --. 29 2. 25 .40 .45 --. 09 • 17 -. 07 --. 22 2. 17 .50 --. 10 .14 --. 98 --. 24 2. 09 --. 26 2. O0 .55 --.11 • 11 --. 99 --.13 .08 --.11 --. 28 1.91 .60 • 65 --.14 .04 --. 12 --. 39 1.81 --.32 1.71 .70 --. 15 • O0 --. 13 _fls _'_" _'_
A
\ Z Positive directions of axes and angles (forces and moments) are shown by arrows , / Moment about axis Angle Velocities Axis Force r/ Linear (parallel Positive Designa- Sym- Sym- to axis) (compo- Angular Designation Designation bol tion bol nent along direction symbol • axis ) X L Y---->Z " Roll .... "_ u p Rolling .....
Longitudinal ...... X M v ' q Y Z-)X Pitch .... t_ Pitching ....
Lateral .......... I 'r Z N X _ Y Yaw ..... ¢_ W Yawing ....
. Normal .......... I Z Absolute coefficients of moment Angle of set of control surface (relative to neutral position), _. (Indicate surface by proper subscript.)
C_=q__ S M N (rolling) (pitching) (yawing) 4. PROPELLER SYMBOLS P D, Diameter P_ Power, absolute coefficient Cp_p_)_-_D5 p, Geometric pitch 5 /_Z5 p/D, Pitch ratio Cs, Speed-power._ coefficient = _/_n2 V', Inflow velocity Efficiency V_, Slipstream velocity Revolutions per second, r.p.s.
7b, T T, Thrust, absolute coefficient (JT_pn2D 4 Effective helix angle-- tan-_(2--_-V-rn) Q, Torque, absolute coefficient CQ=pn_--_D5 5. NUMERICAL RELATIONS ] 1b.--0.4536 kg.
1 hp.--76.04 kg-m/s--550 ft-lb./sec.
1 kg----2.2046 lb.
I metric horsepower--l.0132 hp.
1 mi._1,609.35 m--5,280 ft.
1 m.p.h.--0.4470 m.p.s.
1 m_3.2808 ft.
1 m.p.s.--2.2369 m p.h.