NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE 2213

AERODYNAMIC COEFFICIENTS FOR AN OSCILLATING AIRFOIL WITH

HINGED FLAP, WITH TABLES FOR A MACH NUMBER OF 0.7

By M. J. Turner and S. Rabinowitz

Chance Vought Aircraft Division of United Aircraft Corporation

Washington

'.tober 1950

Reproduced From Best Available Copy 20000816 173 DISTRIBUTION SIATEMENT A

Approved for Public Release igKC QUALITY nsKEBOKBD 4 Distribution Unlimited

4 £}MQ0-11^3534

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE 2213

AERODYNAMIC COEFFICIENTS FOR AN OSCILLATING AIRFOIL WITH

HINGED FLAP, WITH TABLES FOR A MACH NUMBER OF 0.7

By M. J. Turner and S. Rabinowitz

SUMMARY

Dietze's method for the solution of Possio's integral equation has "been used to determine the chordwise distribution of lift on an oscillating airfoil with simple hinged flap in two-dimensional compressible flow (subsonic). The results of these calculations have been used to prepare tables of aerodynamic coefficients for lift, pitching moment (referred to quarter-chord point), and flap hinge moment for a Mach number of 0.7, the motions considered are vertical translation, airfoil rotation about the quarter-chord point, and rotation of the flap about its hinge line.

Aerodynamic coefficients are tabulated for k values of T^ (ratio

of flap chord to total chord) and for 12 values of reduced frequency u)r, covering the range from 0 to 0.7- Results are given for one value of Mach number, M = 0.7« Data of this kind have already been presented by Dietze for one value of the ratio of flap chord to total chord, Tj{ = 0.15j these results have been checked independently, and calculations have been carried out for three additional values, T

R = 0.24, 0.33> and O.ij-2. Certain auxiliary parameters, which will be needed in any further calculations of this type, are presented for future reference.

INTRODUCTION

The fundamental integral"equation for the pressure distribution on an oscillating thin airfoil moving at subsonic speed has been derived by Possio in reference 1. Collocation procedures have been used by Possio, Frazer and Skan, and others to obtain lift and moment on an oscillating flat plate. An important contribution has been made by Dietze (see references 2 and 3)> who has developed an iterative procedure for numerical solution of Possio's integral equation. This procedure is particularly well adapted to the calculation of aerodynamic loading on an oscillating airfoil with hinged flap.

NACA TN 2213

It has "been pointed out correctly by Karp and Weil (reference h, p. 11) that Metze's procedure does not properly account for the logarithmic singularity in pressure distribution at the flap hinge. However, for applications in flutter analysis the principal objective is to determine resultant lift and moments rather than pressure distribution. Mathe- matically exact solutions in closed form are available for the stationary thin airfoil with deflected flap, and it is found that lift and moments obtained by Dietze's iterative procedure are in excellent agreement with theoretically exact values. There is good reason to expect that equally satisfactory results will be obtained for the airfoil with oscillating flap. The existence of the singularity in pressure distribution is a consequence of the sharp corner in the idealized, broken-line profile, and of the associated discontinuity in downwash velocity at the flap hinge. The introduction of a finite cosine series for the downwash is in effect equivalent to a slight modification of the idealized profile by rounding off the corner at the flap hinge.

This work has been performed at Chance Vought Aircraft, under the sponsorship of the Bureau of Aeronautics, Navy Department, in order to provide data for the calculation of compressibility effects in control- surface flutter problems. It has been made available to the National Advisory Committee for Aeronautics for publication because of its general interest.

SYMBOLS

TR ratio of flap chord length to total chord length

Ra airfoil region in x,z-plane

&y(x,t) vertical displacement of point on idealized profile, positive upward

5V(£) nondimensional representation of instantaneous chordwise distribution of vertical displacement

'sy(x,t) = isÔe1^)

(E

I total chord length

CD circular frequency

t time

x chordwise coordinate

NACA TN 2213 3

y vertical coordinate

z spanwise coordinate

i dimensionless chordwise coordinate (2x/z)

vy(x,t) vertical component of fluid velocity adjacent to airfoil

g(|) function representing instantaneous distribution of vertical fluid velocity adjacent to airfoil (vy(x,t) = gU)e^t)

üür reduced frequency (a>Z/2V)

V velocity of flight

j{ i) function representing distribution of dipole lines

7jnc distribution of dipole lines for incompressible flow

K(s,M) kernel of Possio's integral equation

u,v • variables of integration

s auxiliary variable (0^(1 - to))

M Mach number

u = 1 - \Jl - M2

p air density

a(x,t) lift per unit area

T(a>r) function defined by Küssner and Schwarz

AK(s,M) kernel difference (K(S,M) - K(s,0))

AK-L(S,M) singular part of kernel difference

AK2(s,M) nonsingular part of kernel difference

k^j constant occurring in formula for AK-j_

k2n coefficient in polynominal representation for AK2

NACA TN 2213

a^, ßlk, env coefficients in recursion formulas for solution of Possio's integral equation

APS lift force on airfoil strip of width Az

AMD pitching moment on airfoil strip of width Az

AMR hinge moment on flap for strip of width Az

3D

<lR

downward displacement of quarter-chord point divided by

semichord

rotation of airfoil, positive in stalling direction

rotation of flap, positive in stalling direction

cgh> kgh aerodynamic coefficients, where g, h = S, D, R

BASIC THEORY

A very complete digest of the literature on aerodynamic theory of oscillating airfoils has been presented by Karp, Shu, and Weil in reference 5. Consequently, the basic theory is outlined briefly merely to exhibit the essential features of the computational scheme and to point out certain errors which have been discovered in Dietze's formulas.

The usual assumptions of thin-airfoil theory are adopted, leading to Possio's integral equation for the chordwise lift distribution on the oscillating airfoil. Rectangular coordinates are employed, with the x-axis alined in the direction of the undisturbed flow. The airfoil is replaced by a deformable sheet of zero thickness which, in its undisturbed position, occupies the region Ra

- 2/2 = x = 2/2, y = 0

of the x,z-plane.

The lifting surface executes sinusoidal oscillations in which each point moves along a line parallel to the vertical y-axis. Displacements are independent of the spanwise coordinate z, and maybe represented in

the form

Sy = 6y(x,t) =|5y(i)ei(Dt

KACA TN 2213

In accordance with the usual convention, it is the real part of equation (l) which has physical significance..

The y-component of velocity of a fluid particle adjacent to the lifting surface is related to 5y by the equation

dSv Ö5V ö-ov 2V öSy vy = 3T+ v o^r= sr + T w (2)

where | = 2x/Z and

(2) it follows that

1=|=1 inside Ra. From equations (l) and

vv= gdJe1^ (3)

where

g(|) = V^Sy + -^ W

where

(Oj. = <oZ/2V

In Dietze's derivation of the basic integral equation the region Ra is covered with dipole lines of density V7(x)eia,t per unit length in the chordwise direction. By equating the vertical velocity induced by the dipole covering to that given by equation (3) the following integral equation is obtained for the determination of y:

"»I

g(|) = cDp ^> 7(lo)K(s,M) d|0 (5)

subject to the condition that r(l) shall be finite; the kernel of the integral equation is given by

K(s,M) =- .isXM

k\jl - M2 Hn(2)(|s|X) - iM-i-H^^dsIX)

. X i(l - M2)e"1SM log

M

« \jl - M2 1 - V1 " M2

I eiuH0(2)(|u|N) du (6)

NACA TN 2213

with

s = aird - lo)

X = M/(l - M2)

The lift a per unit area (positive upward) is given "by

a(x,t) = pVr(x)e icut (T)

NUMERICAL SOLUTION OF POSSIO'S INTEGRAL EQUATION

Dietze's approximate solution of equation (5) (Possio's integral

equation) is of the form

7 * ?inc + 7i + 72 + • • • + 7n (8)

where 7inc and 7V (v = 1, 2,

equations

,, n) are solutions of the integral

g(l) = oir ^ 7inc(ô)K(s,0) d|0 (9)

gv(0 = 0*^ 7v(!-0)K(s,0) &Z0, v = 1, 2, . . ., n (10)

and where Pi

gl(|) =0^) 7inc(ô)[K(s»°) " K(s,M)] d£0

J-l (11)

Pi

gv(|) = ü>r*) 7V.I(|O)[K(B,0) - K(s,M)] d£0, V = 2, 3, • • ., n

J-l (12)

KACA TN 2213

K(s,0) = lim K(s,M) M->0

The required solutions of equations (9) and (lO) are obtained by the methods of reference 6. Convergence of the process has been proved only for the stationary case. However, computational experience furnishes convincing evidence that the process does converge in the more general case.

It will be observed that Dietze's process requires essentially the solution of a succession of integral equations with kernel K(s,0) from the incompressible problem. The functions gv(5) are obtained by direct integration in accordance with equations (ll) and (12).

In case gyd) can be represented by a cosine series of form

gv(l) = V(A0 + 2 X An cos nA 0 ^ 0 £ ir (13)

with

I = -cos 0

then it is known from the work of Küssner and Schwarz (reference 6) that

7VU) = -2VU0 cot £ + 2 ]P a^ sin njZfj (lk)

where

80 = (^T~)(Ao - Ai) + Ai

^ = ^~(An-l " An+1) - An, n ^ 1

(15)

and T(a>r) is the function of reduced frequency defined in reference 6. The T-function is related to Theodorsen's C-function by the equation T = 2C - 1.

In order to facilitate the evaluation of the integral occurring in equation (12) the kernel difference

AK(s,M) = K(s,M) - K(s,0)

8 NACA TN 2213

is expressed in the form

AK(s,M) =AK1(s,M) + AK2(s,M) (16)

where

AKl(S)M) = M + kll + k12 loge |s| + s(k13 + tu loge |s|) (IT)

k10 i(L - N/TT^)

kn = '^vrn^ J 2rt[N/rr^L 2(1 - M2)J

loge 7M

1 + vrn^

l12 2*V \[T^y£)

13 2rt

/M 1 + 10ge 1 Wl - M2 + (1 - *P*

| *. (x - § *) loge ^

1 - -2 M2 -

1 + 3M2 - 2

2(1 - M2) 372

kl^="27 1 +

3M2 - 2

2(1 - M2) 372

loge7 = 0.57722 (Euler's constant)

NACA TN 2213

The norisingular part AK2(s,M) is replaced by a polynomial of ninth degree1

AK2(s,M) * - y- kPnsn

n=2 (18)

whose coefficients are determined separately for each Mach number "by fitting the polynomial to tabulated values in the interval |s| = 1.8.

Equation (12) may be written in the form

Sy(D = -u>r^ 7V_2 AK(s,M) d£0 (19)

If it be assumed that

7v-l = -2V(PO cot I + 2 Z! Pn sin njzA (20)

then it follows that (upon carrying out the required integrals defining gy, in accordance with equations (17), (18), and (19) and applying equations (15) to solve equation (lO)),

?V -2v(q0 cot I + 2 ZI qn sin njZfJ (21)

where

<*n = *Ln + «ten (22)

"TDietze's definition of the approximating polynomial (table 3 of reference 2) differs in sign from that given here. However, it has been found by carefully checking the derivations that the minus sign is required in equation (l8) in order to justify the recursion formulas for calculating yv from 7v_i- It is believed that this is merely an

error of presentation, since Dietze's numerical results are found to be substantially correct.

10 MCA TN 2213

and

q10 = H(-P! + ßooPo' + ß0lPl' + t&Pz)

«11 = KP1 + P10PO' + ßllPl' **V?2 + ß13P2")

q12 = ^(P2 + ß2oP0' + ß21P2' + ß22P2" + ß23P3")

^ = tn + aiV+a2V,+a3PnM,>n-3

q20 = 2rt^öPV^0V

10-n

^2n

with

= (-l)n+12rt2lp '6nv, n^l V=0

*o' =-^(PO

+ PI)

pl' =-|(P0 + P2)

Pn' =ir(Pn.l -Pn+l)> n'2

Pn"-s(W-W)'n*2

Pn,"=27(Pn-l"-Pn+l">n-3

(23)

The quantities at, ßik, and €nV are defined in the translation

of reference 2 (see table 5, p. 28, and tables 6 and 7, PP- 30-32). It was found that the formula for 6QV is given incorrectly by Dietzej it

should read :0V = P^~ft>OV + Siv) - Biv (2k)

NACA TN 2213 11

Presumably this is merely an error in presentation, since, as already noted, Dietze's numerical results have "been verified by independent calculations. Numerical values of the parameters <XJ_, ß-j^ for M = 0.7

and reduced frequencies ranging from 0 to 0.7 are given in table I.

EVALUATION OF KEENEL AND KERNEL DIFFERENCE

By making use of the relations among Bessel, Hankel, and Neumann functions and by separating the kernel into real and imaginary parts an expression of the following form is obtained:

K(s,M) = K'(s,M) + iX"(s,M) = ~A(s'M) cos (sXK) +B(S,M) sin (sXM) +

k\]l - M2

B(s,M) cos (sAM) - A(s,M) sin (sXM) , . i - (25)

h\Ji - M2

where

A = J0(|s|X) - M-^NÎsIX) - |\/l - M2 loge 2* sin (s £)

1 - \jl - M2 /

rsM (1 - M2) sin (s £j / [cos u J0(|u|M) + sin u N0(|u|M)] du -

SM (1 - M2) COS (S |j / [cos u N0(|u|M) - sin u J0( |u|M)] du

B =; U0(|s|X) + Mi Ji(|s|X) + £\ll - M2 loge M cos (a ±\

|S| * 1 -\[TTW \ M/

(l - M2) cos fs |J I [cos u Jo(lulM) + sin u N0(|u|M)] du -

+

,sA M

(1 - M2) sin (s ^J I [cos u NQ(|U|M) - sin u J0( |u|M)] du

\ (26)

12 NACA TN 2213

By introducing a new variable of integration

M

equations (26) are transformed into

A = Jo(|s|X) -M-jlj-HidslX) -|Vl -M2 loge ^—__ sin (s £)

ln (s M) F°S (VM)J°(,V|X) + sin (vs)No(|v|x)l

DB(s |) LB (V|)H0(|V|X) - sin (v |)j0( |v|X)l

dv

dv (27)

B = N0(|s|\) +MT|T J1(\s\\) +§\/l - M2 loge M

1 -\/l - M2 (4) cos ( s — ) +

ns

cos

sin

(s I) \cos (vs)Jo(|v|x) + sin (vl)No(|v|x)J Jo L

(s s)J [cos H)H°(,V|X) -sin H)jo(iv^)]

dv -

dv (28)

NACA TN 2213 13

Numerical values of the following integrals are required:

I-i(s) =

12(B) =

cos (v — J J0( I v|X) dv

r I sin \v M) JO( I vtx) dv

I^Cs) = I cos

io (v JJ)N0(|V|X.) dv

I4(s) = I sin (V|)N0(|V|X)' dv

(29)

The evaluation of I-, and Ip can be obtained by numerical integration

in a straightforward manner. However, since No(x) has a logarithmic singularity at x = 0, it is necessary to express Io and 1^ in

different form. Upon making the substitution (reference 7, pp. 130, 132)

J N0(x) = Jö(x) loge 22£ _ B0(X)

where 1

1 + ö , 1 1 1 +2+3, w-<$*mJ--iiP@--

Ik NACA TN 2213

and integrating by parts, the following equations are obtained:

£ I3(s) = I^s) loge Il(v)

V

r»s

dv - cos

UO

(v SE)B0(VX) dv (30)

5^(B) ^(s)' loge Z|_

ns Io(v)

dv

ns

JO sin v JJJBO(VX) dv (31)

It follows from equations (29) that Ii and I3 are odd functions, while Io and 1^ are even. The integrals occurring in equations (30)

and (31) can be evaluated numerically without difficulty; it should be noted that

Il(v) lim = 1 v->0 V

I2(v) 11m _£ = 0 v-^0 v

In computing numerical values of the kernel Dietze has used the tables of Bessel and Neumann functions given in reference 7, which do not permit a satisfactory determination of BQ since No is tabulated to only four (in some cases three) places. In recalculating the kernel the seven-place tables given in reference 8 have been used.

For evaluation of the integrals In the formulas given in reference 9, page 227, have been extended to include fifth differences. These formulas have been used with an interval A(sX) = 0.05, and the results have been checked up to sX = 0.30 by using an interval A(s\) = 0.02. Recalculated values of K(s,0.7), K(S,0), AK(S,0.7), AKX(S,0.7), and AK2(s,0.7) are given in table II. These values are found to be in close agreement with those given by Dietze; where differences exist, the new values are believed to be more accurate. In making comparisons it should be noted that Dietze has tabulated -K(s,M).

The coefficients k2n are obtained by fitting a ninth-degree polynomial (see equation U-8)) to the tabulated values of AK2 in the

MCA TN 2213 15

Interval |s| = 1.8 by the method of least squares.. The values obtained in this way for M = 0.7 are

k22 = -0.046728 - 0.04597î

k23 = 0.023019 - 0.023^91i

k^ = O.OO9818 + 0.052855i

k25 = -O.O20228 + 0.003i*88i

k2g = -0.001040 - 0.021510i

k27 = 0.007272 - 0.000283i

k2g = 0.000053 + 0.0031171

kgp = -O.OOO997 + 0.0000121

Since | s| = 0^(5 - |0) and || - |0 | = 2 the approximate representa-

tion for AK2 is valid for reduced frequencies in the range 0 = (Oj. = 0.9. Although the values of the coefficients k2n given here differ from those given by Dietze (seö translation of reference 2, p. 26), there is reason- able agreement of coefficients for lower powers of s. Also the algebraic signs agree; this gives further indication that Dietze must have introduced a minus sign in his definition of the approximate representation for AK2.

NOTATION FOR AERODYNAMIC LIFT AND MOMENTS

In presenting his numerical results Dietze has introduced repre- sentations of lift force and moments involving only real quantities. Complex notation is used in the present report in order to conform more nearly to current American practice; however the essential features of Dietze's notation (translation of reference 3) are retained. Lift and moments on an airfoil strip of width Az for an airfoil with simple hinged flap are as follows:

16 MCA TN 2213

APc

AM

H]

D =

AMR =

where

APS

AMD

AMR

jrpV2QjAzf(üir2css - kss)qs + (ü>r2cSD - kSD)qD + (a>r

2cSR - kSR

npV2(l)2 Az[(CDr2cDS " kDs)qS + (mr2cDD " *DT))% + ("V^DR " W^R]

(mr2cRS " kRs)<te + (ü3r2cRD " ^D)^ + (^RR " *BR)<*B^ npV2U) Az

(32)

lift force, positive upward

stalling moment on airfoil plus flap, referred to quarter- chord point

hinge moment on flap, positive in same sense as AMp

(LAq downward displacement of quarter-chord point

rotation of airfoil in stalling direction *D

qR rotation of flap in stalling direction

i = V + iW . g, h = S, D, R

kgh = kgh' + ikgh"

% = (%' + ^h")6 iiot

The quantities cfih may "be expressed in terms of the functions 0^,

$7, and $12 defined in reference 6 as follows:

'gh

>^ S D R

s 1 1/2 $1^/2«

D 1/2 3/8 Qrj/kn

R $il/2jt <Sy/tat $12 A*2

NACA TN 2213 IT

DISCUSSION OF NUMERICAL RESULTS

The coefficients kgg, kpg, kgj), and k[)j), which do not depend on the ratio of flap chord to total chord TR, are presented in table III

and in figures 1 to 8 for M = 0.7 and a range of a)r from 0 to 0.7«

The hinge-moment coefficients kpg and kj^ associated with air-

foil flapping and rotational motions are presented in table IV and figures 9 to 12 for M = 0.7, reduced frequencies from <%. = 0 to 0.7, and ratios TR = 0.15, 0.24, 0.33, a^ 0.42. Coefficients Pn in the series representation for y

7 X -2V(P0 cot £ + 2 Z_ Pn sin n0 )

are presented in table V for both flapping and rotational motions. It is noted that the coefficients kjjg and kj^p may be computed for any

ratio of flap chord to total chord without further iterations by inserting the coefficients Pn for the appropriate type of airfoil motion into the formula

kRh = (WV5 + | Yl P»,Qn) + ^ HPn' ,Qv)> h = S' D

The quantities Qn are expressed as functions of T-^ as follows:

cos 0 = 2TR - 1, 0 Is 0 ^ n

QQ = (ir - 0)(-l + 2 cos 0) + 2 sin 0 - - sin 20

3 1 Qx = 2(« - 0) cos 0 + ^ sin 0 + j- sin 30

Q2 = -(it - 0) - ^ sin 20 + -i- sin he

_ sin (n - 2)0 2 sin n0 sin (n + 2)0 Qn ~ (n - l)(n - 2) " (n - l)(n + l) + (n + l)(n + 2)' n >

18 NACA TN 2213

The aerodynamic coefficients associated with rotational motion of the flap kg-R, kpg, an<^ ^RR are Presented in table VI and figures 13

to 18 for M = 0.7, reduced frequencies ranging from üOJ. = 0 to 0.7, and ratios of flap chord to total chord TR = 0.15, 0.24, 0.33, and 0.42.

From five to eight iterations have been employed in the calculation of 7. In computing the coefficient k^R the accuracy depends on the

number of coefficients used as a starting basis for 7±nc- The number of

coefficients is reduced by 3 in each successive iteration; in the calculations described in this report 22 coefficients have been used as a starting basis. All coefficients available at a given stage of the iteration have been used to calculate the contribution of jn to y

and the contribution of 7±nc nas "been obtained in closed form from the results of reference 6.

In recalculating the coefficients which are independent of Tg and

of the remaining coefficients for TR = 0.15, values have been obtained

which differ in general by less than 1 percent from those given by Dietze. There are a few isolated exceptions, however. An error of 7 percent has been found in the imaginary part of kpg for o>r = 0.10.

Also there are errors in Dietze's values of the imaginary part of k^g

at clip = 0.02 for all Mach numbers tabulated, including M = 0. The entry for M = 0 should read lO^kj^g " = 1.18 instead of 1.03.

Apparently the same error has been carried through for all Mach numbers. A similar error occurs in the imaginary part of kj^p at a>T = 0.60; the

entry for M = 0 should be 10^RD" = 131-8 instead of 132.8, and

this error has been carried through for other values of Mach number as well.

Chance Vought Aircraft Division of United Aircraft Corporation

Dallas, Tex., July 19, 19^9

NACA TN 2213 19

REFERENCES

1. Possio, C: L'Azione Aerodinamica sul Profilo Oscillante alle Velocita Ultrasonore. Acta, Pontificia Acad. Scientiarum, vol. I, n. 11, 1937, PP- 93-106.

2. Dietze, [F.]: Die Luftkräfte des harmonisch schwingenden Flügels im kompressiblen Medium bei Unterschallgeschwindigkeit (Ebenes Problem). Teil I: Berechnungsverfahren. Deutsche Luftfahrt- forschung, Forschungsbericht Nr. 1733, Jan. 20, 19*1-3. (Also available from CADO, Wright-Patterson Air Force Base, as AAF Translation No. F-TS-506-RE (ATI 6961), Nov. 1946.)

3. Dietze, CF\]: Die Luftkräfte des harmonisch schwingenden Flügels im kompressiblen Medium bei Unterschallgeschwindigkeit (Ebenes Problem). Teil II: Zahlen- und Kurventafeln. Deutsche Luftfahrt- forschung, Forschungsbericht Nr. 1733/2, Jan. 24, l^kk. (Also available from CADO, Wright-Patterson Air Force Base, as Translation No. F-TS-948-RE (ATI 9876), March I9V7.)

h. Karp, S. N., and Weil, H.: The Oscillating Airfoil in Compressible Flow. Monograph III, Part II - A Review of Graphical and Numerical Data. Tech. Rep. No. F-TR-1195-ND, Air Materiel Command, U. S. Air Force, June 19**$.

5. Karp, S. N., Shu, S. S., and Weil, H.: Monograph III - Aerodynamics of the Oscillating Airfoil in Compressible Flow. Tech. Rep. No. F-TR-II67-ND, Air Materiel Command, Army Air Forces, Oct. 19^7.

6. Küssner, H. G., and Schwarz, L.: Der schwingende Flügel mit aero- dynamisch ausgeglichenem Ruder. Luftfahrtforschung, Bd. 17, Nr. 11/12, Dec. 10, 19^0, pp. 337-35^. (Also available as NACA TM 991, 19^1.)

7. Jahnke, E., and Emde, F.: Tables of Functions with Formulae and Curves. Dover Publications, 1943.

8. Watson, G. N.: A Treatise on the Theory of Bessel Functions. Second ed., The Macmillan Co., 1944.

9- Scarborough, James B.: Numerical Mathematical Analysis. The Johns Hopkins Press (Baltimore), 1930.

20 NACA TN 2213

TABLE I.- VALUES OF a± AND ßik FOR M = 0.?

jai «on' + iai"; ßik = ßlk« +ißik'*]

(Dp -

ai «2 CG-,

V rr « ' 0^ OUp' 02" a3' a3"

0.02 0 -0.04801 -O.OOO94 0 0 0.00001

.Ok 0 -.09601 -.00376 0 0 .00006

.06 0 -.14402 -.00847 0 0 .00021

.08 0 -.19202 -.01506 0 0 .00049

.10 0 -.24003 -.02353 0 0 .00095

.20 0 -.48006 -.09418 0 0 .00762

.30 0 -.72009 -.21179 0 0 .02573

.40 0 -.96012 -.37652 0 0 .06099

•50 0 -1.20015 -.58831 0 0 .11912

.60 0 -1.44018 -.84716 0 0 .20584

.70 0 -1.68021 -1.15308 0 0 .32686

°¥ ßoo ßoi

ßoo' ßoo" ßoi' ßoi"

0.02 -1.82233 0.4o4i5 -0.00347 O.OOI69

.04 -I.63609 .64055 -.01064 .00688

.06 -I.45768 .80639 -.01952 .01519

.08 -I.29IO3 .92767 -.02917 .02624

.10 -I.I367O 1.01832 -.03904 .03961

.20 -.52194 1.23247 -.08348 .13074

• 30 -.07794 1.27472 -.11434 .24736

.40 .27370 1.24893 -.13107 .38037

.50 .57113 1.18299 -.13419 .52669

.60 .83322 1.08608 -.12388 .68538

• 70 I.O6903 .96176 -.09990 .85624

NACA TN 2213 21

TABLE I.- VALUES OF o^ AND ß±^

FOR M = 0.7 - Continued

03r ^02 ho ß02' ß02" ßio' P10

0.02 -0.00001 -0.00001 -o.oo4i6 0.00148 .Ok -.00004 -.00009

1 -.oi4oi .00591 .06 -.00018 -.00024 -.02809 .01331 .08 -.00042 -.00048 -.04561 .02366 .10 -.00079 -.00081 -.06601 .03696 .20 -.00519 -.00360 -.19879 .14786 • 30 -.01438 -.00769 -.36140 .33268 .ko -.02896 -.01257 -.53416 .59143 .50 -.04790 -.01796 -.70335 .92411 .60 -.07225 -.02363 -.85837 1.33071 • 70 -.10161 -.02952 -.99061 1.81125

0)r ßll ßl2

ßll' ßll" ßl2* ßl2"

0.02 -0.00001 -0.04802 0.00047 0 .Ok -.00005 -.09611 .00188 0 .06 -.00016 -.14430 .00424 0 .08 -.00038 -.19262 .00753 0 .10 -.00075 -.24108 .01177 0 .20 -.00599 -.48583 .04706 0 • 30 -.02020 -.73434 .10590 0 .4o -.04790 -.98513 .18826 0 .50 -.09356 -1.23571 .29416 0 .60 -.l6l66 -1.48287 .42358 0 • 70 -.25672 -1.72280 .57654 0

22 NA.CA TN 2213

TABLE I.- VALUES OF <x± AM) ßik

FOR M = 0.7 - Concluded

0) r

ßl3 ß20

ß13' ß13" ß2o' ß2o"

0.02 0 -0.000004 0.000003 0.00001

.04 0 -.00003 .00002 .00005

.06 0 -.00010 .00008 .00014

.08 0 -.00024 .00019 .00030

.10 0 -.00048 .00037 .00053

.20 0 -.00381 .00299 .00288

.30 0 -.01286 .01010 .00713

.4o 0 -.03049 .02395 .01251

• 50 0 -.05956 .04678 .01778

.60 0 -.10292 .08083 .02134

• 70 0 -.16343 .12836 .02130

"V - ß2l ß22 ß23

021* ß21" ß22' ß22" ß23* P23"

0.02 0 -0.04801 -0.00094 0 0 -0.000002

.04 0 -.09602 -.00376 0 0 -.00002

.06 0 -.l44o4 -.00847 0 0 -.00005

.08 0 -.19208 -.01506 0 0 -.00012

.10 . 0 -.24015 -.02353 0 0 -.00024

.20 0 -.1*8101 -.09413 0 0 -.00191

• 30 0 -.72331 -.21179 0 0 -.00643

.4o 0 -,96774 -.37652 0 0 -.01525

.50 0 -1.21504 -.58831 0 0 -.02978

.60 0 -1.46591 -.84716 0 0 -.05146

• 70 0 -1.72106 -I.15308 0 0 -.08171

NACA TN 2213 23

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2k NACA TN 2213

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KACA TN 2213 25

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26 NACA TN 2213

TABLE V.- VALUES OF COEFFICIENTS Pn IN SERIES REPRESENTATION

7. = -2V [P0 cot ^ + 2 ^_ Pn sin n0 J

[pn = Pn- + iPn"]

For flapping of airfoil

ciij, = 0. 02 (üj. = 0.04 COj. = 0.06

n Pn' P ' ' ■ni Pn' p • • rn P ' rn p '' rn

0 0.00405 0 .02580 0.01195 0.04726 0.02102 O.O6523

1 -.00050 .00005 -.00194 .OOO36 -.000413 .00091

2 0 0 0 .00003 .00002 .00009

3 0 0 0 0 0 0

4 0 0 0 0 0 0

Cüj. = 0.08 u>r = 0.10 cDr = 0.20 (üj. = 0.30

n Pn' ^n Pn' Pn" Pn' P •' -tn Pn' P • '

0 0.03053 O.O8067 0.03991 0.09420 0.08342 0.14570 0.12386 0.18312

1 -.00700 .00181 -.01048 .00302 -.03610 .01371 -.07419 .03356

2 .00006 .00018 .00013 .00034 .00123 .00218 .00454 .00039

3 0 0 .00011 -.00004 .00013 -.00007 .00057 -.00043

4 0 0 0 0 0 0 -.00004 -.00003

cur = 0.40 o>r = = 0.50 Oij. = 0.60 COj. = 0.70

n Pn* P ' ' Pn' p ' • Pn' p ' ' rn P ' Ml p ' ' rn

0 0.16431 0.21075 0.20492 0.22786 0.24175 0.23334 0.27133 0.22743

1 -.12273 .06574 -.17824 .11402 -.23186 .17927 -.27797 .26203

2 .01189 .01353 .02588 .02355 .04912 .03341 .08434 .03984

3 .OOI67 -.00157 .00378 -.00432 .00634 -.01014 .00848 -.02027

4 -.00018 -.00014 -.00059 -.00044 -.OOI65 -.00096 -.00355 -.00157

MCA TN 2213

TABLE V.- VALUES OF COEFFICIENTS Pn IN SERIES REPRESENTATION

7 = -2V(P0 cot I + 2 51 Pn sin njA - Concluded

27

For rotation of airfoil

0)r = 0.02 a>r = 0.04 o)r = 0.06

n Pn' rn Pn' p ' ' rn Pn' p •' rn

0 1.29^01 -O.I922O I.I929O -0.27984 I.IO671 -0.32806

1 .00272 .03991 .00793 .07696 .01391 .III89

2 .00032 -.00003 .00125 -.00013 .00270 -.00035

3 0 0 0 0 0 0

4 0 0 0 0 0 0

oij. = 0.08 • a>r = 0.10 cür = 0.20 oor = 0.30

n p • p '' n P ' rn p ' * ^n p ' rn P ' ' rn P ' rn P ' * rn

0 1.03781 -O.35794 0.97989 -O.37637 0.80131 -0.41701 0.70475 -O.45044

1 .02012 .14529 .02691 .17790 .O6175 •33405 .10655 .48796

2 .00461 -.OOO69 .OO697 -.00126 .02586 -.006l2 .05649 -.01645

3 0 -.00009 -.00004 -.00018 -.00037 -.00135 -.00159 -.00454

4 0 0 0 0 -.00004 0 -.00029 .00014

(üj. = o.4o u>r = 0.50 CDj. = 0.60 a^ = 0.70

n V P ' ' ^n P ' ^n P ' • ^n P ' ^n P ' ' ^n P ' ^n p '' ^n

0 O.631OI -O.49399 0.55644 -O.54194 0.47233 -O.59052 0.37484 -O.6177O

1 .16912 .64043 .25543 .78449 .36613 .91393 .49812 1.00553

2 .09907 -.03557 .15194 -.06873 .21279 -.II63O .26823 -.19176

3 -.00472 -.01080 -.01190 -.02105 -.02383 -.03601 -.04688 -.05153

4 -.00097 .00055 -.00196 -.00169 .00531 .00378 -.00929 .00848

28 NACA TN 2213

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O S UN

O H

O OJ s

O d A ON

0

H O

*/ IT* H OJ

ro ro s A

H OJ ro ä IA

H OJ CO ro 3

/ 1- O o O o O O O O O 0 O O

a o H

o H

-st o r-f

-st o H

_=t o o

-=t o H

-=t o H O H s O

H

-=J- O H

O H

O H

O H

-5f O O

H O H

0 H

O H

0 H

-4- O H

-4-

S U

o u

X X X X X X X X X X X X X X X X X X X X X X X X

CO M CO

K CO

(X "« J i5 "w

i5 1 J 1 Jf 1 J* Ja £ 1 J^ 1

NACA TN 2213 29

V

T-.b

/

/

/

A /

/ i

/

/

/

q /

.07 /

/ / /

06 / /

/ /

.05 /

'

9. /

r

/ / /

.04 / /

/ /

/

D3 / /

r

/ 1 . ./ /

02 y /

/

/ /

.01 /

/

/

0 / 0 .0 I .0' 1 .Of 3 .0 3 .1C ) .1 .c I .3 A ,s R .7

Figure 1.- Real part of kgg against reduced frequency for M = 0.7.

30

..o.

NACA TN 2213

/ /

•.9< / / /

/

.8

■.7'

p

c; .20 j

/ .18 /

j '

A .16 / / .14 /

/ .12 /

-.J

7 /

W J0 *—

.08 /

."V

/ .06 '

/ —

.04 /

-.1

/ .02 / j—

°C .0

.0 2

4 .0

.0 6

8 .1

.] 0

L .2 • ' 3 1

^ I • 5 .( .7

Figure 2.- Imaginary part of k^ for M = 0.7.

NACA TN 2213 31

SD

1.8 \ \ \ \ 1.6 \

1

\ \ \ 2.4

\ \ \

V 2.2 \ N v \ \

2.0

1.8

L.e

0 .04 .08 .1 .2 .3 .4 .5 .6 .7 .02 .06 .10 <°r •

Figure 3.- Real part of kgD for M = 0.7.

32 NACA TJ I 2 21 3

•H ^-

. /

,**--

—> / .6

/ /

.4

-.2

/

S /

/ 0

/

/ /

■*—

— -.2 /

y-

--1 \- / /

r / /

r / r -.4 /

-.2 f » +

"SD i -*

-.3 1

\- V A 1

-.4 /

^= :—-■ *■—

-.5 0

.c 2 D4 .(

06 38

10* 1 2 • 3

0 r • 4 • 5 . 6 7

Figure 4.- Imaginary part of kgD for M = 0.7.

NACA TN 2213 33

c ^ h

-\- 01

\

-.001 \ .02

. .03

-.002 _ ,04

,05

W --00 3 _ .06

07

-.004 h 08

■

OQ \

\ -.005 - 10

—'■

\ \

-,1 1 ! \

1

-.006 -P 0 .0 1 .0 3 .1 .2 .3 .4 .5 .6 7 .02 .06 .10

Figure 5.- Real part of k for M = 0.7.

3h RA.CA TN 2213

—

—

_

28 .0028

26 .0026

.24 .0024

.0022 'AA 4- /-

.0020 20 -J- L

.0018 .18

.0016 1

.1b

i —t k^-,0014 7 .14

/■

■ / .0012 f .12 /

7 .10 .0010 f- .08 .0008

— i .0006 i .Ob

f y s .0004 f- .04

S '

1 —

.0002 r .02

/ t- 0 D

.0 2 )4

.C .c

6 8

10* 1 2 3 • 4 . * 5 * 6 ;

Figure 6.- Imaginary part of k for M = 0.7.

NACA TN 2213 35

.024 .60

.022

.020 ,50

.018

.016 40 /

.014

kDD ^ 1 .30 /

.010 / /

.008 / .20

.006 \

\

.004 \ 10

.002 1

n I 0 .04 .08 .1

.02 .06 .10 .3 .4 .5 .6

Figure 7.- Real part of kDD for M = 0.7.

36 MCA TW 2213

KDD

..2 /

—, /

-/. /

-0 / * '

/ *

.18 / / .8' / 1—

I '—

V y

.14 y

s

.6- / >

.i-c- i 1 /

/ / .10

/ . / / .4 y y

.Uö 1 ,/

/

/ '

.UO i / 1

.2' / / .U4

/ /

/ .02, / ■

n / 0 .04 .08 .1

.02 .06 .10 .3 .4 .6 .7

Figure 8.- Imaginary part of kDD for M = 0.7.

IACA

0020

TN 2 21: 5 37

0015 /

,i >/

nnm °v

.0005 V.

\ /

/-\ *u -^■ \

o ^ :> —- \

T ? 0. 15- ^±s

^ c ■—. -—.

-s )01 ^ \ ^ n i

\ "•■15

- r no \ \

V 7

-.003 \X N£ < " \ V*

-.004 \

\, >- V \ \£ \

.- nriR \

V A k ° \ ku,'

Y \ RS

-.C 06 \0

\x? \ \

.W (

\ — nno.

< \

\ \ \ \

\ ,uu».

\ -A

-.010 \ —' \

\

-.011 \ \

-.012 \ —N

\

-.013 \ \

_ ni/1 \ \

0 .05

.0' 2

1 .0

.0! 6

3 .1

.1 3

.2 .3 .4 .5 .6 .7

Figure 9.- Real part of kÜC, for M = 0.7.

38 NACA TN 2213

08

—i

07- / —t

i

■4 >

06 / /

/

/ 05' /

.010 /

.009 '—

/ 04 .008 / -V-

/ / -A / .007 / y / -4 '

/ 03 y / /

' .006

1

'— y ' /

/ /

./ / ".005

/ i «<; r "1 / ^

/ <v /1 02 ,• y

• ^ .004

o-/1 aV1 , 0 #> / £% -—* ^

.003 -// ■^ / u

f7" 01- / /l

.002 / s ^^

O.t

.001 = 0.15 T* 0.15

^ö-n

°c .02

.0 4 .0

.0 6

8 .1 0

1 2 • 3 1 A .7

Figure 10.- Imaginary part of kpg for M = 0.7.

NACA TN 2213 39

RD

0 .02 .04 .06 .08 .10 .1 •4 .5 .6 .7

Figure 11.- Real part of k for M = 0.7. RD

-0 KACA T N S 521 L3

■ 30

28 -^

■ -y

26 —> /

/

/ 24 /

22 -J. -V

/

/ .2U

.018 (

18

- / .016

" 16 *

V K. V .014 /•

v; / | .012 7—.la

r> #

"1 J> f-

.010 J.U

TF = o.: 3f ,08

/

W -008 /

r^ 1/ . 0> .006 °T T "P^

k< » A— .004 .U4

%*■

02 i i-y 1 0 TR - 0.15 .002 i '<<?• y i s

0 V -7 T \- -1 r -.002 \: *-

—

\~ i -.004 r T A: *- -.006

3 .C

.c )2

)4 .c 36

)8 10'

1 2 3

]

• 4 • b " 6 .7

Figure 12.- Imaginary part of kRD for M = 0.7

NACA TN 2213 kl

•

1.2 V \ 2.0

\ \ 1 \

\ \ L.8 SR

\ \

\ \ \ 1.6 A

\ \ \ \ \ \ 1.4

\ \ \ \| \

\ \ 1.2 \,

X r = 0.4; R

;

1.0 1 1 R = 0.33

s 0 T

* = 0.24

ft T , = n it | t J-O

-A

o

0 .04 .08 .1 .02 .06 .10

.2 .4 .6

Figur3 13.- Real part of k for M = 0.7.

k2

w

NACA TN 2213

45 >~jv. \

V x\

s\ ^ft / c

O >-

—^ ■■Q

_o"_ 4U

\ f^< ̂ \> \ •

& / / / >*— \ ;\

'1 7 35 \ ^ \

1 r < ~A \ ... ^

^ \

T r, °"/i /■—

i KW \

V 1 30 \ TR = 0.1 S—

! 1

V v -=

-h L- Â \

'R U.i 4-

.rr 25 \

IK \ \ r^ * 0..9O- V \

ir .20 \ ■

|

ill "ff Tl .15 » > _ j

0 •1 r-r- f v

r ■ r

.10 ■p

.05

■

0 .02

34 .( 06

38 10*

1 • 2 • 3 ur

• 4 • 5 , d •

Figure 14.- Imaginary part of kgR for M - 0.7.

NAC; i TN 2 213 h3

8 5 / r

//

£ o — V /

/• ■—

// /-

7 5 // r- ^

>, / <*■/

If ' DR

y-

j n A

/ / /

S 0* b/

6 5 X '/

^ <^ A yt

/ 0.' Lb / ^

s 0.^-6 ~*r^5>—

"l-^=

~3> = vJ.°~^- '

~A A A

l^r- -r-\,.51 L_L_

1$r ^!^T

. , 0 .04

.02 . .08

36 .10 1 .2 .3 A .5 .6 .7

Figure 15.- Real part of k for M = 0.7.

1* MCA TN 2213

-.5

•

-.4

* /

-.3

*3 ■#•

LDR / '

f ̂ i -.2 / "^

/ /

/ 7^

/ *— i rP

= L .2.4

-.1 *—

—

/ :> ** " »>-"*

T 0. 1 r-

-u

( 3 .0

.0 2

4 .0

.c 6

8 J L0

1 • 2 3

r

i! 1 • 5 X .7

Figure 16.- Imaginary part of k^ for M = 0.7.

NACA TN 2213 ^

.17 /

16 \

\ .15 \

.14 TT o 0.^

13 1

12 -

11

10

W ng * = 0 .33

i 08

B.

07

06

DR

_U T* = 0.5 54

C\A

03

02 0 1^ B

0 .0

.Ü 4 .0

.0 6

3 .1

.1 0

.2 .3 CO

.4 .5 .6 .7

Figure 17.- Real part of k_ for M = 0.7. KJrC

k6 HACA 11 I 2 213

/

-^ /

1 /

15 /

/

14 /

/ /

13 -V /

—

—/ /

12 / —

/

11 /

/ >

10 j /

/

09 /

®A )

"PR 08 / *s

K*- V 07 /

r

/ X ^

06 / ^ S

/ ^ > . y

05 / /

«1 * J*

/ Kj '

010 ,04 / ^y

v / / I

" „ ° c ti j\ / s y

T uo

/ , y _ c }4

7 1 ^^ \ ,02 / s s T-r = <J.

.005 <N L V ̂ i—.

/ / y "^

\ N s / .01 / y y ' TR = 0.15

74_

^0 u /

7 .01 I /

- 005 V / ,02

C ) .0

.0 2

4 .C

.0 )6

8 .1 0

1 £ I \

A I .< .7

Figure 18.- Imaginary part of kRR for M = 0.7.

NACA-Langley - 10-26-50 -1100

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

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