Critical Mach Numbers of Thin Airfoil Sections with Plain Flaps

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~ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ORIGINALLY ISSUED April 1946 as Advance Confidential Report No. 6A30 CRITICAL MACH NUMBERS OF THIN AIRFOIL SECTIONS WITH PLAIN FLAPS Heaslet and Otway O’M. Pardee Ames Aeronautical Laboratory Moff ett Field, cam.

NACA WARTIME REPORTS arereprints ofpapers originally issued toprovide rapid distribution of advance research results toanauthorized group requiring themforthe war effort. Theywerepre- Some ofthese reports werenot tech- viously held under a security status but arenow unclassified.

nically edited. Allhavebeenreproduced without change inorder toexpedite general distribution.

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___ —.. .-——— — 3 1176013544391 NACA ACR No. 6A30 NAT IONAiJ ADVISORY COMli ITTEZ FOR AER03TAUTICS —-— ADVANCII CONFIDENTIAL REPORT’ CRITICAL MACH NUMBERS OF THIN AIRFOIL SECTIONS WITH PLAIN FLAPS By Max A. Heaslet and Otway O’M. Pardee SUMMARY The critical Mach number, as a function of lift coef— ficient, Is determined for certain thin and moderately thick NACA low-drag airfoils. The results, which are given gr~.phi– tally, Include calculations on the same airfoil sections with Curves are presented plain flaps for small flap deflections.

Indicating optimum critical conditions for the airfoils with flaps and are in a form so that they may be compared with corresponding results for zero flap deflections.

The calculations indicate that, through the use of plain flaps, an increase ma,y be realized in the lift-coefficient range for which the critical Mach number is in the region of high Vaiues characteristic of low-drag airfoils.

INTRODUCTION The necessity of attaining higher speeds and higher altitudes in military aircraft has focused increasing atten— tion on the critical speeds of the airfoil sections used and, as a result of investigations concerned with the calcu~atton of these critical speeds, certain prdpert~es of favorable airfoil sections have become known. For example, it is possible to say as a general conclusion that the type of pressure distribution associated with low-drag airfoils is one which is also favorable to the production of hfgh-critical- speed characteristics.

Moreover, as pointed out in refer- ence 1, the thickness ratio and the camber of a wing section play an important role since the maximum value of critical Mach number decreases approximately linearly as the camber and thickness of an airfoil increases.

It has therefore been almost inevitable that the airfoil sections used on recently — NAOA ACR NO. 6A30 designed high-speed aircraft have had the type of pressure .dlgtribution associated with the low-drag airfoil and have been thinner than those used formerly.

such as are given in reference 1, Critical-speed curves, show that the low-drag airfoil has a region of lift coef- ficients, more or less symmetrically disposed with respect in which the critical Mach to the design lift coefficient, number variation is small, the maximum critical Nach number Outside this sector, being achieved within the region.

which corresponds roughly to the lift-coefficient range for which the low-drag properties of the airfoil hold, there is It iS of critical Mach number.

a sharp decrease in the value obvious that a particularly advantageous situation exists if it is possible to design the wing section of an airplane so that the high-critical-speed and low-drag regions of the wing extend beyond the lift-coefficient range for normal operations . The difficulty of achieving this has already been encountered In the design of fighter aircraft where demands on the maneuverability at high speeds are great.

‘l’he problem arises again in the case of lon.~range bombers since, on extended flights with attendant fuel consumption and with the accompanying disposal of bomb loads, the vari- ation of lift coefficients required may be quite large. The situation is particularly acute for jet-propelled bombers since high speeds are possible of attainment over a wide range of altitude.

The theoretical results of reference 1 show quite clearly that , as the thickness of an airfoil section decreases, the maximum critical Mach number is increased but that this in-- crease is brought about at the expense of the lift-coefficient Thus , range for the high-critical-speed region. the high- speed requirements In the design of an airplane may call for a thin wing section while other specifications may be such that the extent of lift coefficients needed at high speeds extends beyond the natural range of the airfoil.

As a conse- quence, it becomes highly desirable to investigate any method whereby an extension of this ra,nge may be effected. One such method which could presumably be used for this purpose is the use of full-span plain flaps, for in this manner the camber may be modified and the load distribution over the air- foil disposed so that the sharp growth of the pressure peak -> near the nose is restricted.

In the present report, calculations have been carried out at the request of the Air Technical Service Command, U. S. Army Air Forces, to determine the critical Mach numbers, XACA ACR No. 6A30 asa function of lift coefficients for the N.4CA 64-, 65-, and 66-8erles low-drag airfoil sections with thlc.kness-chord These sections ratios equal to 0.06, 0.08. O.lot and Oo12~ have constant ideal lift coefficients of 0.2 at which lift Another the load ie distributed uniformly over the chorale.

portion of the theoretical calculations i~ devoted to the determination of the critical Mach numbers of theee same small flap deflections, airfoil sections wtth plaln flaps, for in an attempt to study the effect of such flaps on the criti= cal Mach number curves of the airfoils .

An experimental investigation has also been carried Outt under the same general researoh program, to determine the Mach numbers at which the force and moment characteristics This investiga- of the same eections with flaps are divergent.

tion is to be reported separately and will contain a compari- son ~f the theoretical critical results with the experimentally evaluated divergence Mach numbers.

A complete list Gf symbols, as used throughout this report, may be found in the appendix.

ANALYSIS Computation of Critical Mach Number It is now an established convention to define the critical Mach number of a ?)ody as the Mach number of the free stream for which, at some point on the surface of the body, the fluid first reaches a velocity equal to the local velocity In an analogous manner the critical compressi- of sound.

bility speed is defined as the free-stream speed corresponding to that at which the critical Mach number is attained. Ex- perimental evidence, obtained from the study of airfoil sections, Indicates that compression shocks are formed locally on an airfoil surface soon after, if not coinci- dental with, the attainment of Mach numbers corresponding to the critical speed. This shock, however, is not well defined and, so far as can be observed, no strongly developed shock front exists until the free-stream Mach number has risen some- what above Its critical value.

It thus seems quite reason- able to assume, and this hae been further substantiated by experiment , that for airfoils of limited thickness the critical Mach number furnishes a conservative approxi- mation, for the des”ignero for the occurrence of the flow breakdown which iS associated with super-critical speeds and NACA ACR No, 6A30 in the airfoil characteristics produces the sudden changes that are, In general, inimical to good airplane control and performance.

The critical Mach numbere in this report are calculated Under the assumption in the manner used in reference 1* that the flow is iaentropic, the critical pressure coef- ficient is given by the relation Y P - I?.

cr 2 =.

[(

-1

(1) YIP L zi- Y+ 1

qo

cr wher e static pressure in the free stream pr”essure corresponding to sonic velocity at Mcr and occurring at minimum pressure point velocity of the free stream velocity of sound In the free stream critical Mach number , equal to at critical Vo/ao conditions ratio of specific heata (cp/cv = 1.4) density of the fluid in the free stream z!

dynamic pressure :p v ( ) In order that I!cr be related to the low-speed pressure P-P coefficient it is necessary to express the %=0 = ~ (3 ~f= o left-hand side of equation (1) in terms of PM=O and to this end the K&rm6n-Tsien formula (reference 2) has been used, Equating this result, evaluated at the critical value of N, to the right-hand side of equation (l), gives the requlslte expression NACA ACR ~Oo 6A30 +Y+M2 2 2 >.. - ~: .y=l cr.

[( ) YM~r “ For an airfoil, or possible to determine the pressure distribution for a given low-speed lift coefficient, then the critical Mach number of the airfoil can be found by means of equation (2) together with the value of pressure coefficient at the minimum pressure point . As a result of such calculations, the critical Mach number of an airfoil is found as a function of the low-speed section lift coefficient, For design, however, it ctM=O* is highly desirable to know the actual lift coefficient 0 2~ In reference 3, corresponding to the Mach number of flight.

Glauert has developed the approximation c 2M=0 —.

c~bfl = ------- .

“ ~1-M2 which does relate the low- and high—speed lift coefficients , and in this report the Glauert correction has been applied.

Experimental observations show that the accuracy of this correction Is good for Mach numbers UP through critical vnlues.

Calculation of Pressure Distributions for Airfoils with Flaps The theoretical calculation of airfoil pressure distribu- tions has been outllned in reference 4, and in this reference tabular data are given whereby the pressure distributions may be calculated Immediately for all standard NACA airfoil sec- tions with various types of camber lines. The velocity distribution over the airfoil is considered, in conformity with present theory, to be formed from three separate and independent parts: 1. That part of the velocity distribution associated with the bas ic thickness f- “ - ‘-- ‘“ of attack NACA ACR ~0, 6A30 That part of the distribution associated with the 2.

> design load of the camber llne 3. That part of the distribution associated with the additional load distribution and related to the angle of attack of the airfoil As a result of this theory it is possible to express In the form the pressure coefficient ‘M=O (3) PM=O = 1 - Au v &v.

are velocity ratios corresponding This method has been used respectively to parts 1, 2, and 3.

throughout the present report for airfoils with and without flaps, and the airfoil data in reference 4 have been used In all cases.

For an airfoil with plain flap it is necessary to find Au Av ~ the effect of the flap on the values of ~ and ~’ o and this can be best achieved by first calculating the change in the load distribution over the airfoil which is brought In reference 5 this problem about by the flap deflection.

has been treated in a semiempirical fashion for the case of but the theory is not immedi- conventional airfoil sections, ately applicable to low-drag airfoils ‘and, for this reason, a different approach is made modeled on the work of Glauert in reference 6 and Allen in reference 7’.

It is an accepted practice to divide the chordwise lift distribution P of an airfoil into two parts: (a) the so– called “basic” J.ift distribution F’b, which depends on camber- line shape and is independent of the angle of attack; and (b) the additional lift distribution Pa, which is variable with angle of attack and In form is independent l f the camber-line shape. When the flap on an airfoil is deflected, the change in lift distribution is called the incremental lift distri- bution and the two component parts are respectively, pa, incremental basic distribution and incremental ‘b6, additional distribution Pa8. It can be shown that the NACA ACR NO. 6A30 ?

incremental additional distr,lbut ion due to the deflection of the flap is identical in form with the additional distri- The- fnc”remental bution for ‘“the airfoil wi-th fl”aps neutral.

basic distribution must be evaluated, however, from a knowl– edge of the airfoil section, the nature of the flap, and the flap deflection. The determination of this variation is therefore undertaken in the following paragraphs.

In conformity with the assumptions usually made in thin-airfoil theory, the airfoil is replaced by its mean camber line. Figure 1 shows the nssumed camber line distri- bution produced by the deflection of the flap, the hinge point A?l approximating of the flap lying between xl and X2 , the chord--line of the airfoil. If x is measured from point A along AF and y is measured from A along a line normal to AF, then in the i’igur”e << AB for O = X = Xl is linear with equatton y=~ BC is parabolic with equation y = axa+bx+d for << xl = x = X2 CD -tan 8 (X-xa) is linear with eq,uation for Y-Ya = << x= = x= c where a, b, and d are arbitrary coefficients and 6 The parabolic section is the angle the flap is deflected.

is to extend over a very small portion of the camber line, the extent of this section being determined later, and is introduced to avoid the sharp break in slope which theoretically would exist and the subsequent requirement of a singularity in the velocity and lift distribution at the hinge point.

This small portion between 3 and C may be thought of ae a fa’iring of the camber line at the point of the sharp break and Is consistent not only with the existence of a boundary layer on the surface of the airfoil, for the layer has a tendency to iron out such abrupt irregularities, but also with the geometry of the median line between the upper and lower surfaces of the airfoil, NACA ACR N@. 6A3@ If the transformation .

,,, = x AC (1- Cos e) is intro duced~ the expressions for the slopes of the three sections may be written in the forms dy fer O zQ2dl AB:—=0 dx << Q BC: = 2ax + b for xl = x = x~ dx (4) d-Jf= CD: - tan 8 dx << = –k for 82 = e = n where G, H, end A, are introduced for simplicity and are decined by the above equations.

Ileference 7 establishes the relationship 6+00 D-e. de opb\ = cot — - cot — (5) ~Yfi~ )

(–) (

% o dx 2 2 where Op b basic distribution fcr inf~nitesimally thin airfoil () -Z-O at chordwise station corresponding to 8 = 00 e variable of integration d, value of 0 at an arbitrary fixeci point * slope l f camber line dx It is l bvious that this expression can be used in conjunction with the slopes given in equations (4) to determine the incremental basic distribution due to the flap deflection.

NACA ACR NO,” 6A30 The integration of the integral is straightforward and the final result is

sin+(6~-eo) sin+(ei+~o)

——- ‘(6) -A(G cos 90 + H) In - 1?

sin~(el-eo) sin#6a+90)

I

For a given flap deflection of & degrees the value of A is known and it merely remains to fix the coefficients G and H and determine 9X e~ so that the theoretical and incremental basic distribution is consistent with experiment.

(0=01) the parabola Impose now the condition that at x = x1 This has zero slope and & radius of curvature equal to r.

requires that Moreover, at x = xa (8 = Qz) the slope of the parabola must thus equal -A, :;(COS 92- Cos el) =-A r and if -, h, and e 1 are known, It is possible to find c 01+02 e z, Letting the hinge point of the flap be at — the 2’ parabola can be oriented so that the values of 0 at lt~ end points are symmetrically disposed with respect to 6 at the hinge point and for small flap deflections el and 02 can be found.

Using equation (6) and the derived values of the various parameters, a comparison was made between calculated values of incremental basic distribution and available experimental data for small flap deflections.

It was found that when r was set equal to the thickness of the airfoil at the hinge point the agreement wa~ quite good over the entire airfotl surface and that at the hinge point, where maximum values of P~~ are attained, the results were reasonably accurate NACA ACR NO. 6A30 ~.

to justify the use of the theory to calculate critical Mach numbers.

By the methods of reference 7 it follows that = 2A sin 82 + G(82--91)+ ~ (sin 28%- sin 2Q~) czb& + 2H(8in ’32 - sin el) (7j and = 2A(lT-e2) -2H(6241) –2G(sine2- sin 61) cl (8) a~ where incremental basic lift coefficient c~b6 incremental additional lift coefficient c ~a6 Since ez–~1 is small, for small deflections of the flap, it is possible to approximate equations (7) and (8) by simpler Under the expressions and this was done in the calculations.

same assumption, the peak point of the incremental basic lift distribution is at the hinge point and can be approximated quickly.

The relation between the velocity distribution and the chordwise lift distribution over the airfoil has been given by Allen (reference 8) in the form (g) .

and (10) NACA ACR No. 6A30 where —,,— .— . . .

,-. ..– v -.

velocity over upper surface of airfoil v () Ou” v velocity l ver lower surface of airfoil

~

()

L pressure distribution over base profile ‘f P lead distribution over airfoil Since the contribution of the flap deflection to P has been calculated, the effect on the velocity distribution can be Substitution in equation (3) will give the low- determined.

speed pressure coefficient at any point along the airfoil; from the minimum pressure, and, the critical speed of the section is determinable. In calculating the contribution of the incremental basic lift distribution to the velocity l n the surface of the airfoil, It is well to bear in mind that the expression for given in equation (6) was derived opb6 under the assumptions of thin airfoil theory. In reference 7 it has been pointed .out,tiatt~ a first order of approximation Hence, it follows that the incremental velocity associated with this portion of the lift distribution is given by ‘opb6 .

The final form l f equation (3) may therefore be written as Au where — is determined from the design load of the basic V.

camber line and ~F is a function of the flap-chord ratio ~ be and the deflection angle of the flap.

NACA ACR ~Om 6A30 12 DISCUSSION OF RXSULTS ,.

In figure 2 are shewn the critical curves Of a typical (NACA 651–21Q.) for various flap defections, airfoil section both positive and negative.

It is to be observed that each critical curve is composed of three distinct parts: a substantially flat top and two steep sides. These three portions correspond to three differ– ent conditions Qn the airfoil determining the minimum pressure peak.

The thin airfoils considered are characterized by having large additional velocities near the nose and but a moderate rise in the basic velocity distribution ap>ro?cking the maxi- The combination results in a. velocity mum—velocity point.

distribution which is double–peaked for lift coefficients differing more than a small amount from the design lift, The velocity peak at the nose appears suddenly, the after velocity peak still remaining; there is no continuous tran— For a certain range of lift coefficient about the sit ion.

design lift the forward velocity peak is less then the rear one; for this range the critical Mach number is determined by the velocities near the maximum–velocity point of the base profile modified by the small additional velocities for this region. This latter velocity peak will be termed the mid- it from velocity pe:tks which peak velocity to distinguish appear at hinge points on flapped airfoils to be mentioned This ran~e of lift coefficient corresponds to the later.

top of the critical curve. The small additional velocities produce a near linear change in Nach number with lift coef- ficient, giving to the top a slight slope as shown in figure 2.

At some lift coefficient, the velocity at the forward pressure peak becomes equal to the vel{ city at the midpressure peak; then for large increments of lift from the design lift, the peak velocity at the nose is the maximum and changes rapidly with change in lift coefficient due to the large The result is that the critical kfa.ch additional velocities.

number falls rapidly with increasing increment in lift coef– ficient. ‘l?he lift coefficient for which the velocities at the nose and midpressure peaks are eoual is the point of intersection of the top and the side; it is the point at which the absolute maximum velocity jumps from a position aft on the airfoil section to a point in the immediate vicinity of the nose *and will be denoted as the “dt?clfn~tion NACA ACR iiO. 6A30 II on the critical c~r~e.

There are two such declination point ??Or posi– points, for positive and negative lift increments.

tive lift increments the velocity peak at the nose appears on for negative lift increments it appears the upper surface; on the lower surface.

Since airfoils with flaps are ,equivalent to the original airfoil with modified cam b8r, the effect is to change the basic velocity distribution while leaving the additional The critical curve, then, velocity distribution the same.

for the flapped airfoil is very similar to the unflapTed The steep sides are but shifted as a whole by c~br5” for the incremental basic velocity at practically parallel, the nose is negligible. The only appreciable change occurs in the top and the declination points.

~mall enough thnt the velocity peak For flap deflections e.t the hin~e point is less than the midpeak velocity, the “top of the critic?,l curve will have essentially the same slope, since the airfoil has the same additional velocities, but is shifted up or down depending upon whether there is a negative or positive flap deflection. The increment of lift coefficient between the upper and lower declination points will consequently vary.

For flap deflections large enough that the pressure peak at the hing:e point is greater than the midpeak pressure, the top of the critical ourve has a lesser slope than before, the additional velocities at the hinge point being less. It iS to be noted that for negative flap deflections the peak velocity at the hin{:e point appears on the lower surface where the additional velocity increment with lift coefficient is negative; consequently, the top of the critical curve has a reversed slope to that of the unflapped and positively flapped airfoils.

This is clearly shown in figure 2.

The 10CUS of the upper and lower critical Mach number declination points together with the top of the critical CUrVe f02?IlI what may be termed an “OIJttIIIUJJI CritiCal CUrVe.” All points on this curve correspond to flap deflections for which, at a given lift coefficient, the airfoil section achieves the maximum possible critical Mach number, as can be seen in figure 2.

The extension of the region of high critical Mach number by means of flaps is done in two parts.

The first, where the hinge pressure peak is less than the ,midpeak pressure, is a near linear extension of the orl.ginal curve.

In this regi~n NACA ACR No. 6A30 .

is realized without too a reasonable gain in lift coefficient In the second much sacrifice in the critical Mach number.

the hinge peak predominates; the part, after the break, sacrifice in critical Mach number for increased lift is correspondingly greater.

criticfil curves for In figure 3 are shown the optimum and 66-series of airfoils having thickness- the N.ACA 64-, 65–, chord ratios of 0.06, 0.08, O.10S and- 0.12 with O.1OC, O.~OC, The critical curves of the original and 0.30c plain flaps.

(6=0) are in each instance shown as airfoils without flaps The rest of the critical curves for other flap dotted lines.

deflections are not shown, as was done in figure 2, but rather !lb.eseloci, as only the locus of the declination points.

noted previously, form optimum critical curves for the given in figmre 3 has a iiaCliof the curves airfoils with flaps.

form directly analogous to the optimum curve of figure 2, having the typical extsnsion, for small flap deflections, of the hi~h-critical-spe ed. lift—coefficient r%nge.

There are two predomi~ate variations to consider: section thickness and airfoil family. By family, reference is made to the position of the velocity peak on the b?.se pro?ile.

The curves show a general trend to higher critical Mach numbers and smaller lift-coefficient ranges for the thinner sections.

The e:zte~sion of the lift-coefficient range by Leans of flaps The shows this same tendency though to a lesser degree.

decrease in the extension of the lift-coefficient range for the thinner sections is the result of higher velocity peaks at the hinge point, together with a smaller va.riat%on in velocity along the base profile so that the break in the The larger peak velocities for the thin curve occurs sooner.

sections are due to the fact that the value of the peak velocity varies inversely as the section thickness at the hinge point.

The additional velocity distributions of all the 6-series airfoils considered are substantially the same over the rear- ward 60 percent of the chord. For tjhts re=,son, all the slopes l f the tops of the critical curves in any given airfoil family The slope are essentially the same (cf. fig. 3(a) to Z(d)).

for different airfoil families will be different as indicated, ( cf. fig. 3(d), 3(h), 3(l)) the slope decreasing for a rear- ward movement of the maximum velocity point on the base profile.

For the different families a rearward movement of the maximum velocity peak is indicative of three changes in NACA ACR NO. 6A30 a lower maximum velocity and fletter section characteristics: additional velocities at the nose, profile velocity, greater and a thicker section at the hinge point giving lower velocity Thus, a rearward movement in pressure peak re— peaks there.

suits in slightly higher critical speeds wlthj however, a smaller range of ltft coefficient in the high critical re.yion togetl>er with a e.mall.er extension of this range by the flaps.

The extension of the critical curve of an airfoil,bY means of flaps will vary with the flap-chord ratio, the dif- ferent optimum critical curves being, however, quite sinilaro A study of the curves given in figure 3 indicates that, of the flape considered, the 0.20c gives the best over-all re- sults.

-foot high- Experiments carried out in the Ames 1- by ~ speed tunnel, on airfoils equipped with orifices for the determination of pressure distributions, have given some in— sight into the validity of theoretical calculations of critical Mach numbers. Th,e sections considered had thiekness- chord ratios equal to 0.12 and 0.15, the results being conl- pared with those given in reference 1. It was found that, throughout the high portion of” the critic~.1-speed curves for the low-drag airfoils, excellent agreement was obtnined ex- cept that the points of declination of the curves were at lift coefficients somewhat beyond those predicted, especially for the case in which the peak pressure point was forwa,rd on the upper surface. The source of this discrepancy is probably twofold: some error exists, in,the estimation of the true lift coefficient, and the l~ar~an—Tsien correction forriul.a is erroneous in the imnediate vicinity of the nose.

It is possible to derive a modification of the Glauert correction fol” lift, by means of the K&,rm~n —Ts ien pressure form.UILa, and the change brought about in lift coefficient can be shown to be in the right direction to agree with experiment.

This change, however, is small and somewhat laborious to apply and therefore was not used.

That errors in predicted pressures , should exist neqr the nose of an airfoil follows from the fact th:lt the Karm~.n-Tsfen formula is postulated on the use of a pressure-density relation which holds for conditions not differin~ .grei~.tly from those in the free stream.

Near the nose, in the vicinity of the stagnation point, some disc- repancy would be expected to occur, and this is confirmed by the experimental pressure distributions.

considerations of theory and experimentn,l results both indicate that the theoretical results are conservative. It is to be expected, however, that, for airfoils with flaps, NACA ACR No. 6A30 of the curves should be of the right the predicted extensions order.-of magnitude. ., .– - . .

CONCLUDING RENARKS Theoretical calculations show that the use of plfiiaflaps on airfoil sections such as are considered in the present re- appreciably the range of the high- port will serve to increase critical Mach number region characteristic of low-drag air- foils. It is to be expected, ,judging from what experimental evidence is available, that the results will underestimate the extent of the critical curve lyin~ between the declination points but the predicted extension, in this portion of the curve, should be of the right order of magnitude.

Ames Aeronautical Lat.ci.:.tc::. .

National Advisory Committee for Aeronautics, Moffett Field, Calif.

3.EFEREITCES Heaslet, lfax A.: 1. Critical Mach Numbers of Various Air- foil Sections. NA(7.4ACR ~~0. 4G18, 1944.

C/. von K~r.m~n, Th. : Compressibility Effects in Aerodynamics.

Jour. Aero. SCi., vol. 8, no. 9, Ju.lY 1941, pq. ~<37-356.

3. Glauert, H.: The Effect of Compressibility on t!~e Lift of’ an Aerofoil. R. & 11. No. 1135, British A.R.C., 1927.

Abbott , Ira H., von Iloenhoff, Albert E., and Stivers, 4.

LOU.iS S,, Jr.: Summary of Airfoil Data. NACA F.CR No. L5~705, 1945.

calculation of the ChOrdw~se Load 5. Allen, H. Julian: Distribution over Airfoil Sections with Pla.ln, Split, or Serially Hinged Trailing-Edge Flaps. N-ACA Rep.

N.o. 654, 1938.

6. Glauert, H.: Theoretical Relationships for an Aerofoil with Iiinged Flap.

R. & N, No. 1095, British A.R.C., 1927.

I NACA ACR ~0. 6A30 H. Julian: General Theory of Airfoil Sections 7. Allen, Having Arbitrary Shape or pressure ‘i.s$?..ibuti OnD NACA ACR 11o. 3G29, 1943.

A Simplified Method for the Calculation 8. Allen, E, Julian: NACA TN NO. 708, 1939.

of Airfoil ~ressure Distribution.

APFENGIX List of Symbols velocity of sound in the free stream chord lensth of ~.irfoll s“ection lift coefficleit low-speed section lift coefficient incremental additional lift coefficient incremental basic lift coefficient Mach number (velocity divided by velocity of sound) P–P o pressure coefficient —– qo () pressure coefficient under low-speed conditions P1.f=o Pf pressure-coefficient distribution over base profile P chordwise lift distribution (difference between pressure coefficients on upper and lower surface of airfoil) chordwise lift distribution produced by additional load distribution chordwise lift distribution produced by basic camber- line loading incremental load distribution produced by flap deflection incremental additional load distribution produced by flap defection R NACA ACR No. 6A30 incremental basic load distribution produced by flap pb8 deflection basic load distribution for infinitesimally thin airfoil O* b incremental basic load distribution for infinitesimally opb6 thin airfoil static pressure P ~pva dynamic pressure q (+ ) r radius of curvature Au increment of local velocity produced by basic lift distribution on airfoil v local velocity Ava increment of local velocity produced by additional lift distribution on airfoil v o velocity of the free stream x distance along chord measured from leading edge of airfoil ordinate of camber line measured from chord line Y Y ratio of specific heats (cp/cv = 1.4) 6 angle of flap deflection e variable defined by equation 2X = c(l— COS e) A tangent of angle 8 density P Subscripts o free-stream” conditions cr critical conditions L lower surface of the airfoil u upper surface of the airfoil NACA ACR !iO. 6X3(3 ,el $%3 ,,. .– . , .— t. ,

L’ ——————

A“” - ‘“’-

B ‘i

F;r

Y~ 6, o i 1’ xl ——————+ I X2 ~

ID “

1= .x

Figure l.- Equivalent camber line for airfoil with plain flap.

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS —-----Critical curves .8 —— —Optimum oritical— curve /‘-?--- % 1 1 1 I / > > < \ \, \ \~ k/\ / \ l\ \ .5 , L t .4 ‘. -. 0 . 2 .4 .6 .8 1, ,0 Czu Pigure El.-Critical Mach number Mcr variation with high-s eed lift CZM for various flap deflections NACA .coefficlent E 651-210 tirfoil section, .20c plain flap.

bNACA ACR No. 6A30 Fige. 3a,b ~“ .9‘ ——— — $0 fLap I —.— Tnfi flmvi .-w” -.-p ,8 - /1 I 7\ .20C M / ~—--— ;300 a / \ \ // / / / .7 - \ } \\ \ Mcr \ / 5’ ~ Y \ / / \ \ \ \ .6 \ / \ / \ / \ / / / \ .5 \ .4 .

— -0 -.4 -.2 0 .2 .4 .6 .8 1.0 ezu #’igW!e 3.- Optimum critical M&h number M r variation with high-speed \ ?02 .100, section lift coefficient c M .20c and .30c plain flaps. (a) NACA 641-206 airfoil sect on.

.9s NATION AIh18dRY .8 / / \ .7 / \ ‘~ / M / Cr \\ / t \ \ \ ‘/ \\ .6 / \ / \ \ \ \ .5 P.

I 1 .4 e -.a -. -.4 o .8 .4 .6 .8 1. o czM (b) NACA 641-208 airfail section.

Figure 3.- dontinued.

.

m

1“

NACA AOR NO. 6A30 Fi@. 3c,d .9 , —.— — No flap —.— .1OC f;ap_ ..8 ,, .20C -.

—--— .30C w k F % .

,, / k F .7 ~ /, / , \ / \ Mcr / ~ \ .’ :/ // \. \ .6 / \.

/ \ .5 .

.4 -. 6 4 -. -.2 0 .2 .4 .6 .8 1.0 C:M (c) NACA 641-210 airfo~~g~:gt~on.

.- Continued.

.9 .?

lf~r .6 \ .

\ .5 \ \ . .

\ \ J .4 -. 6 -.4 -*2 0 .2 .4 .6 .8 1.0 CZM (d) IiACA 641-212 airfoil section.

Figure 3.- continued.

,,, , ,,, ,,, ,, NACA ACR NO. 6A30 FigiB. 3e$f .9 —.. — No flap . .

—.

—:;;: fl~ _ , B \- .8 / / ~~--— ,30C W, k ~ I f /.

IJ \ .7~~ f \ \\ / / \ I / \ \ 7/ Mcr / \ / \.

\ \ .6‘ \ \ / \ \ / / \ \ / \ / .5 \ \ ,4 -.2 0 -.6 -.4 .2 .4 .6 .8 1.0 Czu (e) NACA 651-206 airfbil eection.

Figure 3.- Continued.

:9‘ NAT10N .6 .

{ / / / \ \ .7 / \ \ / \ \ M \ cr . \. \ / ‘ / .6 % % \ / \ / \ / “ \ t .5 I [ :4 -* 6 ~ -.4 -.a’o .a .4 .6 ,8 .1.0 Czu (f) NACA 651-208 alrj~&~egtion.

.- Continued.

.: ,;.

y NACA ACR No. 6A30 Flge. 3g,h ~“” .9 ?.

m —— .

No fla —.— ? .1OC itiap _ .8“ ‘“ ;20C —.- — .30C a / f \ / +.

/ .7 /1 \ \ . / \ \ \ Mc~ [ \ \ .

\ / .6~ .~“ / \ \ / \ \ \ / /$ / \ / .5 \ / \ \ .4 2 .4 .6 .$3 1 .0 -. 6 -.4 -.2 0 ., CZ.M (g) NACA 651-~0 air;;~rgegtion.

.- Continued.

.9’ .8 \ / .?

/ - /; \ ‘cr / / I \ / \ / \ \ .6 / \ \ / \ / \ / / .5 \ \ \ \ \ \ .4 -.6 -.4 -.2 .2 .4 0 .6 * .8 1.0 cl M (h) NACA 651-212 airf:~~;~c;ion.

.- Continued.

Figs. 3i,j NACA ACR HO. 6A30 .9 —.— — No f- .ap —.— .;:: fl:p_ .8 ,..., —-.— :30C H I \ / \ I / .7 \ / / \ I M \ cr / \ b / \ / .— I .6 \ / \ / \ / \ / \ / .5 ~ \ / \ / \ .4 ? ‘. 4 1.0 -*6 -.4“ -.2 0-. .6 ,8 CZM (i) NACA 661-206 airf;;;u;~cgion.

.- Continued.

.9“ A? VISCRY c OV?t~TTEIt RA ERONAUTI Cs .8 / I / / / / I I \ .7 \\ / \ / \ / H \ \ Cr # / \ / / / \ / h ,6 / \ \ \ / \ / \ /“ \ / \ .5 \ /“ ,, / .4- , .— -. 6 -*4 -.2 0 .2 .4 .6 .8 1. 0 (j) NACA 661-208 tirf;:&~~cj~on!zM - Continued.

, l NAGA AOR No. 6A30 Figs. 3k,l .9 l I ——. _ No flap —.— .8 ~ .1OC fl:p *20C , —--— , .30C, a - / 4 / * / A / \ 7 “ l / \ \ \ / \ & \ / / \ Mcr / \h / / \ \ / / ‘ ,6 , \ / [ \ .

\ / \ t \ / / \ .5 / \ / \ \ .4 ‘ -.6 -.4 -.a o .2 .4 .6 .8 1.0 (k) NACA6f3~-210 airf;~&;gc~ion~zM .- Continued.

.9 .8 / ‘ / .7 r ,6 / “ f \ / / \ .5 .4 .-.

-.6 -.2 0 .2 .4 .6 .8 1.0 CZM (1) NACA 661-212 airfoil seotion.

Figure 3.- Concluded.

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