03(20&

—._

I

TECHNICAL NOTE 2451

~

FOR C OMPUTTNG POISSOIVIATHEIVIATICAL IMPROVEMENT OF METHOD

INTEGRALS INVOLVED IN DETERMINATION OF VELOCITY

DISTRIBUTION ON AIRFOILS

By I. Fl&ge -Ldz

Stanford University

. . ..= . . ... ... . .. ..—...—.

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NATIONAL ADVISORY C~ FOR AERONAUTICS 00b5L10

TECHNICAL NOTE 2451

MTBZMATICAL IMPROVEMENT OF MEWHOD FOR COMPUTING POISSON

INTEGRALS INVOL~ IN DETERMINATION OF VELCEITY

DISTRIBUTION ON AIRFOILS

By I. Fl~ge-Lotz

SUMMARY

The Poisson integral involved in thein velocity distribution resulting from aparallel incompressible flow is solved.

determinantion of the changechange in airfoil profile h

First, three well-developed numerical methods of evaluating thisintegral, all based on the division of the range of inte~at ion tito.small equal intervals, and the difficulties involved in each method,are discussed. Then a new method, based on the use of unequal titervals,is developed, and compared with the other methods by means of severalexamples. The new method is found to give good results for both thedirect and tiverse airfoil problms and @ easily adaptable to rathercomplicated problems. It is particularly recommended for all ‘thosefunctions where steep slopes in still portions of the region.to beintegrated exist.

INTRODUCTION

The ordinary thin airfoil at small singlesof attack produces onlyslight disturbances in the flow of a parallel incompressible fluid.Hence, the influences of camber and thickness upon the velocity distri-bution may be camputed tidependently and theti effects supertiposed.The effeet of camber may be represented by vortices distributed-alongthe chord line of the airfoil section; the effect of the thickness, bysources and sinks also along the same chord ltie. The velocities pro-duced by these sinsularity distributions enable one to compute thepressure distribution on the airfoil rather quickly.

Allen (reference 1) has presented this singularityy method in aform which has proved to be very practical for common usage. However,in special cases the unavoidable evaluation of the Poisson integral h

—_ -——.. .-. —.. -.— --.————— ——=—— _ — ....— —. _—. _ . .. .

2 NACA TN 2451

the course of the computations has given rise to numerical difficulties.Such inte-b are usually ccxnputedby the application of finite differ-ences using intervals of e-quallentih. However, changes in airfoilshape, which result ip marked changeEIin the function to be integratedh OdY mall portions of the range of ~tepation~ req~e that efiremelYsmall interval sizes be employed h this range, and, consequently overthe enttie range of integration. This leads to a considerable smountof computational work; hence, it appears reasonable to discuEs thepossibility of employing interval-sof varying lengths for the evaluationof the Poisson inte~l.

This investigationwas prompted by the difficulties arising fromthe problem of small changes in the shape of symmetrical airfo~ atthe angle of zero lift. The examples ticluded in the present reportare restricted to this case, but the results obtained are in no wayspecialized and may be applied to all problems wherein the Poissonfcltegraloccurs.

This work was done at Stanford University under the sponsorshipaud with the financial assistaace of the National Advisory Committeefor Aeronatiics.

The author wishes to express her appreciation to Mr. H. NormanAbramson for his intelligent and skillful help in the computationalwork and for his assistance in writing the final report. The authoralso wishes to extend her thanks to Mr. R. E. Daunenberg and thecomputing staff of the 7- by 10-foot wind-tunnel section of the AmesAeronautical Laboratory, Moffett Field, California, for preparing theextended tables of the functions jno and jno*.

DISCUSSION OF PROBLEM

The basic reference proffle may W given by ytr = f(x)~ and its

velocity distribution may be known from an earlier computation. Theproblem at hand is that of determintig the change in the velocity dis-tribtiion resulttig from a change h the shape of the profile (indicatedby the dotted line in the following fig.). The difference of these twoshapes is designated as &t.

-——— —–—. .———-

NACA TN 2451 3

.#

tyo

x~

.

Allen (reference 1, p. 7) gives for the change of velocity the equation

where Vn iE the velocity of the basic parallel

(1)

flow. If, byconfo~ mapping of the outside flow re-@on, the center l~e-of theprofile is transformed into a circle by the relation

.2 (1- Cos e)x=~

then the profile is tramfonned into a curveshown below.

approximatefig

(2)

the circle

.. --————.-—————— —. .—— .-

4 NACATN 2451

The change in velocity due to a change in fom will then be given as

(3)

defintig

This is the form most often used for computation pm’poses.because theinverse problem (that of computing the change in shape due to a changein velocity distribution) tiilizes the analytic form

[W]%=abv’f=’)’e (4) n

deftiimg

which is strilclqly similar. Thex,y coordinate system iE given by

().+Zo fi-e

&corresponding fomnula in the ori 1

(5)W(C - x)

accomplished by any one ofthese methods employ the

The evaluation of equdion (3) may beseveral different methods; howevery all ofdevice of replacing the titegral over the range O to 2fiby a sum oftitegrah over intervals of equal lengthpOtits en have correspondtig values ~

distributed (see following”fig.).

Ae. The equally distributedwhich are not equally

.

.,

.——— — ——— .—. .—— . ...— ——

NACA TN 2451 5

.

,.

t 03 04

y 02

‘1

60x—

This arrangement is eometties favorable,

w

and somethes not, depending

upon the particular form of . (See discussion followingax

equation (43). )

The ube of the angular coordinate 19 Ws the advantage that the

flmctionsw or & are periodic functions in 19,and thtidx o

periodicity facilitates the organization of the numerical computations.The disadvantage arises from the fact thAt these functions are usuallygiven as functionq of x, and, since the analytic form is not usuallyknown, any transformatiofi made

if

to

for

dx “o

en = IIAe,then the computor

the values ‘% ’929and SO

DISCUSSION OF SOME

will lead to small errors. For example,

special points which do not correspond

must obtati the values of these functions

forth by interpolarion.

OF THE EXISTING NUMERICAL

SOLUTIONS OF POISSON INTEGRAL

The difficulty encountered in the solution of the Poisson titegral

arises from the fact that the term cot W or & (e~~.2 x-xo --

tions (1) and (3), e.g.) approaches infinity When e approaches @o or

Xo. The difficulty is of much less consequence when

~<ct~= or “ax

~ is given analytically than when a numericalo

computation is undertaken. As a consequence, any simple integration,.. performed by replacing the integral with a smmation over smaller intervals,always requires that the interval in which 80 or X. is located be given

special consideration (see following fig.)..

—— —.. —. ... —— —..

6 I?ACAl!ll 2451

I X.I

ct t t

o’ 1

%“, , x—

%s1

80

1 I 1 I t

0I I I I I I I 1

mewl

e——

A majority of the solutions currently h use have been developed

to such an extent that, for example, #(eo) ~ given by a sum of

()

d(&t) 0products of stigle values of

axand known factors ~; that is,

n\

/ ()pfid(~t) cot e- ‘O deAl e()

170 0=-%0 ax 2

1

f

2fi-eo~ ~t() **.

=-—2X -e ax

cot ~ deo

1 Z[ ‘“+’*‘(&t) cot $ .8*=-—2YC n 8* ax

n

which leads to (see reference 2, e.g.)

(6)

(7)

6.

—— .—— —.

.

.

.

.

MACATN 2451 7

The coefficients &o depend upon the particular method of

J__ld~t ~

numerical integration which is employed. If, for example,ax

replaced by a step-curve, that is, assumed constant in every interval(see fig. below), one set of values of ~. would be obtained.

t

d(&t)

dx

~—

Greater accuracy wbuld be obtained by the assumption that wdx

replaced by straight-line segments (see fig. below), in which casesecond set of values of ~. would be obtained.

t

q’%)

dx

x/c —

is

a

the accuracy of the

character of the approxi-

Y%LA further refinanent would be that of assuming ~ to be composed

of segnents of parabolas, and so forth. Since

‘~ depends upon both theresulting values ofV.

mate curve and the size of interval taken, it is apparent that the samedegree of accuracy might be achieved from manY different combinations

W@of interval sizes and approximations to the function ~ .

— -—— . . . . ..— --. .-— -— —.— .. ..-— —---- -—— -.-.--— .——. -

8 ma m 2451

Naiman (reference2) has Wed Stipsonis rule for computing thePoisson integg?al,which corresponds to the replacement of the product

([ 1w co.!$by segments of parabolas. The ‘critical interval”dx *

i.e., where cot ~)

+ ~ waa carefully treated by using differences,2 d(Av.\\

(of higher order includtig the fifih derivative of \ ‘z’ ]. I?aiman

divided the peri$d of 21-tin 20 or 40 intervals andcorresponding sets of values of ~o. Other workers

extended the calculation of these values of ~. to

(unpublished information).1

Obviously, the time required for cmputing #o

dx)calculated theat the NACA have

80 and 160 intervals

increhes with the

number of intervals taken because of the increased number of multipli-cations to be performed. In addition, greater Preparations for theccmputtig proce== are necessarily involyed, particularly since the

values of d(%) needed must us~lly be obtained by interpolation.ax

This titerpolationhas to be done rather carefully as it is often notsufficient simply to take the values of the plotted curv= of

d(Ayt)This curve should be checked by difference tables if the

T“. Y

1-+(d &tvalues

dxnare to represent a smooth curve.

,.

For those functions of Ayt which may be well-represented by a

Fourier series, there exists a simple method of evaluating the Poissonintegral which has apparently been overlooked until the present the.This method has the advantage of leading to a computation which doesnot involve the derivative of Ayt.

lNaiman has also suggested a second method for computing thePoisson integral (see reference 3). In this second method he uses

Fourier polynomials to represent the function W.O ~e”cOQUttigdx

procedure is very simple; however

Q

the results depend “l&gely on theA.Yt

degree of approximation of ~ by such a polynomial. Thus, for

large families of functions results are good; however, cases are knownto the author where results were not satisfactory because regions withsteep gradients may not be represented we~ eno@ bY a Fomi=polynomial of moderate order.

,,

.

_ ..—. .—. .— _——— -. —

NACA m 24z

Equation (6)equation (43)) as

may be written infollows:

.

Av _—— p ‘(’%)V. 7(JO dx

and this may be tewritten as

.

.

9

a different form (reference 1, “

Sin’ed6’

Cos e - Cos 90

deCos e - Cos e.

(8)

\(9)

Equation (9) is strikingly similar to an integral ocurri.ngin thetheory of the lift distribution of a finite wing in ticompressible flow.There, the induced angle ai is.given by

[

n d7 de*ai=~’ (lo)

2Yt (j ~ cos et - cos e

where 7 is the local dimensionless circulation.

Multhopp (reference 4) has given a solution for equation (10). Hedivides the range of integration into

(ml+l

)hrtervals (see fig.

below)

with

1’en= n fi. -5+1

7n = ~(en)

J.,, -.. ,_ .,.,

.-.

(U)

-. .———.—. -— - —-—. —.. --- _ -. ———— ——— — - ———.-

10 NAcA TN 2451

and computes CLi at the points 13n. He assumes that y may be

expanded h the form-

7 .~cv sillPe

or

m2

7 =—Am+ln=l

He then obtains the expression

aiv

The prime on the summationomitted fkom the summation

(I-2)

(13)= bVv7v - * bvnyn I

symbol indicates that “n = v is to bebecause that special term has already been

considered in the first term of the right-hand side (i.e., ~V7~ ).

Reference 4 presents tables for the coefficients ~ and bvn for

ml = 7, Is, and 31. Applied to the problemat hand, ml = 31 would

appea to be rather small; therefore a table for ml = 63 has been

computed and is included ~ the present report (appendixA). As accnnparison: For ml = 63, Ae = 2.8M50; for Naiman’s method with

160 p0int8, Ae = 2.25°.

Utilizing this method of titegration which was developed by meansof Fourier series, an expression may be obtained for the velocitydidribution as follows:

(14)

The great advantage of this method is that of shplicity: (1) The

actual computational procedure is very simple and (2) the derivative

d(%)— is avoided.

axThe shplicity of computation is reflected h the

fact that the the required for ccnnputing F at one v~ue of e.o

is approximately half that required by the method of Naiman when theiutervals have approxtitely the same stie. It should be noted,

M

,

-.. ——. . ..—

NACATN 24z Il.

however, that the accuracy of the method of Naiman will be greater thanthat of Multhopp h those cases where the differentiation of Ayt byFourier expansion (equation (12)) does not give good results.

A third method of evaluating the Poisson integral became knownduring the course of the present investigation. b a paper by Timnan(reference 5), the integral is studied in the form

(15)

Thman assumes that d(~) is not given analytically, but only atequidistant points. ~ interpoktion polynomi~ (referen~e 6) for 6(w)

# is employed, and these polynomials replace the function a(~) ti asingle titerval by a function of third order. The polynomial fumction

thus introduced has a continuous ftist derivative,2 and it is evidentthat this continuity is essential for the attainment of good results.

TimmanW divided the period 2Yr into 36 intervah of equallength and established a computing scheme. The function 6(V) is

separated into its

Then

where the factors

present particular

AJ ~ s~etricdV.

equation (16) does

syxmnetricaland unsymmetrical parts so that

~(~)=s+d (16)

(17)

%7 ‘d % are given in tabular form. In the

Cme d(%) is antisymmetric+ (equation (3)) and

(equatio~(k)). Thus the separation indicated by

not require any additional work.

Tinmsa’s method sho~d give good results provided that a sufficientnumber of interv- are taken - the division of 36 intervals over aperiod of 2Yt(i.e., 18 intervals over the chord of the profile) appearsto be insufficient for sn accurate representation of the function which

occurs, W or ‘vdx VT “

%?he polynomials used in the classical interpolation formukm areless smooth (see reference ~, pp. 7 and10, figs.1 and 2).

.. —.- ...— — ___ ,_ ._ —.— —— —. —— . .—

12 NACA TN 24!51

The the required for computing one petit by the method of Thmanis approximately the same as-for Naiman’s method with the same intervalsize.

Other methods of evaluating the Poisson integral have beensuggested. They will not be discussed here as it is the intention ofthis section to consider only the most pract-icalof the known methods.The three methods already discussed have their own particularadvantages and have been especially developed for rapid and sImple

Mor&computation; however, all three of these methods, when ~V. ‘

change rapidly h magnitude, become cumbersome, and require that verysmall htervab be taken over the entire range ofthe scheme of equal interval size is utilfzed.

EVALUATION OF POISSON INTEGRAL BY A

EMPLOYING UNEQUAL INTERVAIS

Development of Method

inte~ation because

METHOD

As the change in airfoil shape, or the change h velocity distri-bution, isretain theintervaL3.

given originally as a function of x it appears logical tocoordinate x in selecting the size of the differentHence, the Poisson integral may be studied in the form

[

c~

T(xo) = - ~ u(x) :Xo x o

which corresponds to equation (1). Confordng with

‘(@t )meaning u(x) = is assumed to be a functiondx

every point of its range of definition.3

Define

4= %+1-%

with

n=o, l, P,3, . . .

(18)

its physical

which is finite in

(lga)

,

.

his restriction will.be dropped later; see discussion beginningwith the first paragraph after equation (32).

_—. —— .-

NAC.ATN 2451.—. . .-

13

and

,For

%<xo<x~+l (lgb)..

convenience, there is chosen (see following fig.)

~ + Xm+l

‘o = 2

0 I I , 1

%/ 4%+1tc

Xo

(lgc)

function u(x) is approximated by straight-line segments (see thirdsketch in preceding section). Then, for ~ < x <~+lj

u(x) = +@ + a(xn+l)- ‘(%)(x-%)%

= an + “n+~an(x- XJu

from which there is obtained

(20)

\___———. —... —— .._ ——

14

sc

()ax

TXO=. A tY(x)—‘0 ‘-%

Also,

1=.—l-r

[/

%+1z %

. . . . . —.—-— .

/

‘n+l1=-—Y’cx

%

NACA TN 2451——

%+*(X-%)ax

x - X.

%+1 - ‘nan +

(x- Xo+xo-xn

% )1ax

1{( )x ‘n+l - an=-—x AXn

X[an +

x - X. 1&+ - .,

-1

r%+1 ~— . jnoX-x.

L/s

by definition. The function jno, in the different

given by tifferent expressions as follows:

rlo%‘n+l - ‘o

%1 - ‘0

Ik = l-o* ‘::1- ~-%

log ‘o - %+1ex

9-%

.

regions of

for X. > %+1 > ~

J

(21)

(22)

x, is

.

.

MICA TN 24s1. ...- -.. — ...

15

Introducing jno into equation (21), there results

T(xo) = . +

[x( ) X[%+1 - % + an+ ‘n’l - ‘n

11Axn ~o-~) ‘no

[

J1 z ‘I&o + ‘x( )( Xo-xn=-—l-c %+l-unl+~ Sno

~(24) .n

Or, defining

1 +.‘o-%AXn & = ilno* (25)

there results, finally,

Stice Xn+l = ~ +%, the functions jno and jno* maybe

written as

j *=l+xO-%’jnono6

1

(27a)

andthisformshows that jno md jno* - 5-+”are functions of —AXn

only.

—. . .._— ___ .. . . . _____ .. . . .__ ._. ._ ~ ..— _ .-—-— .—.—. ——___

16

For %1 -xO +*.Ax~ Jno ~ O

For X. - ~=$~ jno =0 ,jno*=l

J.

% .~- X.For very large

AXnY

jno’ + &

% - X.For very large negative .

A%

1+-—0 . .

3E2”

% - X.with

%

These functions

It is seen

near

.

= E*,

(2i%). . .

are given in figure 1 and in table 1.

that -highabsolute values,of jno and jno* occurx - X.

those values of n%

Figure 1 gives an idea of

and

sufficient

values of

4which characterize the critical interval.

the characteristic q~ities of jno

% - X.as functions of ; however, the representation is not

A./

for picking out values for a computation. Table I gives the

J%

-d jno: for -49.5<no ;Xo< 49.5. ‘IT@ table might

4 %+%1 (If’xo= z equation (19c)) the critical interval is

given by -0.5<xn - ‘< 0.5.A%

\

.

—— —..—— —-— .— —

3

.

NACA TN

be usedmethod.

24x \

for rough computation and for getting acquatited with theIn general, it is advisableto use those tables which are

given in appendix B:

It will prove of benefit to investigate the exactness of thatportion of the integral which contains the stigularityi Recalltigthat the function u(x) was replaced by a straight line in everyinterval (equation (20)), there is obtained:

[

Ax‘o+ 72-

1 u(x) ~ = 1-—,x

.—x-x. IT

(%+1- %)

J Ax‘o- -2-

if ~< Xo< Xm+l. Now, let

the critical interval aromd

a(x),=

Then,

u

an expansion of the function

~ be assumed aa follows:

17

(28)

u(x) in

\

AxXo+—

21-—Yt

Xo-&2

Cka(x- ..)2 +C@) + Crt(Xo)(x - Xo) + **.

CM(X. X)3+5YQ+X-X)4+3: 0 4: o“”” (29)

.

u(x)X-x. rCIX=-* ( )&+a’ X. ~“’(xo) UX3+ -

()-—

3! 32”””L

= - +a’m+,-Q -*(%2- .,-J{1 lQa=-—

It ( )18 ‘+1 - am -

.

* [(%+2 )( 1})‘Um+l+‘m-um-l - (30)

Comparison of formulas (30) and (28) shows that the error in the ‘critical interval is approximately given by .

,“

- ——— -———- -–—–——-—— — ——— —.—...—.—. —- .—.

NACA TN 2451

—.

The error of evalmting the whole integal by finite differences maybe esthated by using two different interval distributions aud comparingthe results for a given ~.

However, in addition to that error of the result produced byreplacing the Poisson integral by a sum there exists another error.This sum cannot be computed exactly, but has a certain error dependingon the accuracy of the given data for an(x) and the tabulated values

*As the function u(x) =

qAYt)‘f jno and jno . usually

ax

error of c1 = 1 x 10-3 it has proved amply satisfactory to

has an

EA= Sno—

ad jno* to fou decimal places, the error betig less than

G2 = 5 x 10-5. The error of

is smaller thsm its upper limit given by

mis formula shows that the tifluence of =1 ~ stronger than the

influence of 62 as long as ‘~l”.l + ~lan+, - “nl~~yrthanl- as it is h our later examples - and the sums jno

ad ~ljn~l are always larger than 1. An increase of subdivisions

makes the sw in the upper limit of the error (32) grow, thus requirl.nga higher accuracy, especially in an and perhaps also in the values

x‘f ilno and jno .

,In establishtig the solution of equation (18) it was assumed

that u(x) is finite throughout its range of “clefhit ion. If it isdesired to compute the change in shape due to a proposed change ofvelocity distribtiion, this restriction must be eklminated, as willbe reco~ized immediately.

.

WA n? 2451

Equation (5) may be written in the form

(5’)

tiitting the factorVxok - ‘O)J

which does not affect the integration

process, the titegral maybe reduced to the form of equation (18) by -deftiing

* . .l(.) (33)xc-x”

However, al(x) will be tifinite at x = O and x = c if()$X +0

and()& #O; therefore, a special consideration of the nei~b~rhoodOc

of x.O and X=C is required.” This is done by splitting theintegral tito the.followtig three parts:

.

with el and ~2 being small compared with c. The integral

(34)

may be treated as was explialnedformerly for

JcU(X)*

o 0

\

. .—— - —.——. —— — -———————— _—— ——— —— --— .—. ———

(see equation (18)) because al(x) is ftiite for El < x < c - e2.

For the ftist and third integals, however, a new integration formfimust be developed. 13y

v c-=

there is obtained

rc

Hence, the method used

third. In most cases

introducing

[1X and Cl(X) = al U(X) = al*(W)

(35)

for the ftist titegdwill albo apply to the

[\AT will he zero and there will be no need.7[’ojc

for a special evaluation in the neighborhood of x = c.

will have an importtit tifluence on the result of equationif X. is near to El. First the general fommla will be

then a simplificationwill be discussed for ~ >> Cl.

The titegral (36) willbe solved assuming that

- = a. + al(~j + a2&~ for O <x <61AvV.

(36)

(34) Odygiven and

(37)

.

.

*

Only the ffnal formula of this procedure is given here; thedetails of the solution will be found in appendix C.

P

—-——z ———

NACA TN 2451

with

with

.21

M. given h figure 2, and

()

Avao=~

00

*.1 1=.+—a.12 1

with

(39) ‘

they

a2=$p)o - 2E)C,+E)2,JJ “

The coefficients ac)Y al, and a, may be determined first, as

do not depend upon the particular value of Xo, and then Fl X.

()

may be computed. The term depending on al and, a, will exert &

influericeonly for small values of xo/c. After a brief trafitigthecomputor should be able to decide rather accurately when the formula

“(40)

is sufficient and when the more exact expression (equation (38)) isrequired (also see fig. 2).

——.. —.— ——..

22. ___

Organization of Computational

NIL(IATN 2451

Procedure for Unequal Intervab;.

Transition from One Size of Interval to Another

A thorough understanding of the method is best achieved byfollowing through a rather shple example; h addition various shortcuts to the method will be demonstrated.

Assume afigure.

function a(x) of the type shown in the following

CJ(x)No \

1

Itof

appears reasonable to takex because

arrangement of.

of the form ofinterval sizes

razher sm@l intervah for small valuesthe curve u(x); therefore, the followingis arbitrarily selected:

E= 0.002 for O<X<O.030

~= 0.006 for 0.030 <x <0.096

Compute T(O.009) with the help of eqwtion (26). Note t~t thecri~ical &erv~ extends from 0.008 to 0.010. Tab$ II(a) givesthe values of X/C, an, fJn+l- an, jno, ~d jno for the range

with ~ = 0.002. At X = 0.030 the interval changes to ~ = 0.006and the same functions are given for the range with this size ofinterval titable II(b). Naturally the range above the broken linein table II(b) is no% utilized in the computation stice this portionhas been considered in table II(a).

NACA TN 24~1-— .. —..

23

x - Xo’

Note that ‘N - pro~esses in table II(b) in the same manner

as in table II(a); this is due to the special choice of ~. If—G= 0.006 were used starting with x = O the critical i@erval forx0 = 0.009 would extend from 0.006 to 0.012. Hence, for ~ = 0.006,

jno = O ad jno* = 1 is to be found at x/c = 0.006.

For rapid computation it is best to have jno *and jno as

%functions of &xo on a paper strip and to place this strip adjacent

to the columns headedby an and crn+l- cm. If ‘n - ‘0Ax

progresses -

as indicated h table I, the correct location of jno . 0 and jno* . 1

at the beginning of the critical interval fries the placement of thestrip.

In the example just treated, the transition from one size ofinterval to another is very easy because X. lies at the midpofitOfan interval of the size 0.006 as well as of the size 0.002, if starttigwith X = O.

If ~ had been chosen 0.004, such a desirable arrangement wouldnot have resulted because ‘x. = 0.009 would not

point of an interval of this size (startingwithat x = O).

As a second example compute the value of T

z. - x. X_ - x.

be located at the mid.

such intervals

at X. = O.01~.

Again, “ “ and U “ will progress as in table I. The valuesE =

Ax”3no = O and jno* = 1 will be placed opposite x/c = 0.014 for the

region with ~ = 0.002 md opposite x/c = 0.012 for the region=

with ~= 0.006. AS long as ~= ~, ~= z, and SO forth andif X. is chosen so as to be at the midpoint of the largest size ofinterval, the computation maybe accomplishedby shif’tingthe stripwith jno ad jno* corresponding to table I.

But suppose that the interval sizes are so arranged and it isdesired to compute a petit’where X. does not lie at the midpotit of

the largest size of interval; for example, X. = 0.013. The value

‘o = 0.013 lies at the midpoint of= interval with ~= 0.002; hence,

. . ..—. ———______ _ _,__ -- ._ ————- --- .- —————. ... . . . ..—_—..—_ .. _ .

for the range O < x< 0.030, jno and jno* maY be taken tiectlY

from table I. However, at x/c = 0.030, intervals of the size.~ = 0.006 commence and there is obta&ed

% - X. = 0.030 - 0.013

E 0.006= 2.833

The value of % - ‘0 progresses by 1, that is, 2.833; 3.833,Ax

4.833, . . . . ~w-the ~ctions jno ~d jno* are needed for

%values of - ‘0 which are not given in table I. One might thinkAx

taking them out of an enlarged diagram (seemuch more convenient to take them out of anconveniently arranged for “advancing by 1.1’appendix B.

The example presented by the figure at

fig. 1); however, it isextended table, which isSuch tables are given iu

the beginning of this

of

section suggested starting at x = O with the smallest intervals.However, other examples may suggest another distribution of intervals.The smallest size of intervals may lie at any paft of O <x < c.There are no restrictions in the arrangement of intervals. (See, e.g.,discussion followtig equation (43).)

Accuracy of Method, Examined by Means of

an Analytical Example

The accuracy of the result depends directly upon the size of theinterval taken and the reliability of the data comprising the func-tion a(x). Because the function u(x) will be replaced by a broken line,a glance at the curve wfil quickly $u.ggestan arrangement of intervals.In addition, the error in the critical interval may be used as a firsttest of the choice of intervals..

As a test of the quality of this new method, tivolvtig unequal

‘(%)intervals, a function a(x) = ~ haa been treated which allows

the analytical Computation of T(x) = ~ .v~

.,

__— —_ —

4 WA TN 2451 25

.

.’

The function u(x) b given analytically as

C1-5X5C ‘(%) _ ()

ax

The followtig arbitrary values have been selected:

c1 = 0.35

c = 1.0

D = Somemultiple of B so thatJ

“mti=o

o dx.

‘(f%)The functions Ayt and ~ are given in figures 3(a) and 3(b),

respectively.

—-.. .—- -— —— ..—

26 NACATN 2@l ‘

AT‘J?he~dfiicd computation of ~ for fi~e 3(b) ~ given ~o

figures 4(a) and b(b). The arrangement of the unequal division for the.

numerical computation of $~ is tidicated h figure k(a).5o

5 It was desirable to obtain the value of ‘v at X. = O; hence,E

the first interval has been placed so that -0.061< ~ <0.001 and the

‘(@t) . ~d(Ayt) () forfunction — = -0.ool<x< o. Stice the function —

dx dxis replaced in every interval by a straight line, the error might beexpected to be large. However, g=O at X. =Owillaidin

preventing the error from letig too large.

.

A more exact solution would be obtained by putting

Axg=o -~<x<o

E31E!=— x O<X<*

~2

1 /()42 g~ ax.— —x —=-: glxl

‘0 Ax X.x.-z

For the interval -$<x<~ equation (26) would yield

[

Ax/21 ax

g~=-

[) 1lgl-O Jno*+OXjno-—

3r_ Ax/2 0 n (

andno error is Introduced. l%r X. # O there is a very small error

which may be avoided by respecting the change of size of the intervalnear x = o.

\.

/

.

.

.

NACATN 2451— — ..

27

Also given in figure 4(a) are points of the ~$? curve deterndned by

the method of unequal intervals. Figure 4(b) presents the sameinformation plotted to a larger scale.

For comparative purpo~s the same problem has been treated by thethree methods of computation discussed earlier, namely, those of Naiman,Multhopp, and Timman. Figures 5(a) and 5(b) show the results obtainedby the method of Naiman; obviously, the 40-point solution does not use

a sufficiently accurate representation of the w curve, while theax

80- and 160-point solutions are quite good, with the exception of the

maximm and minimum points of the $ c_eO In order to obtain a

value at approxhately ~ = 0.036 a solution involving 320 potits would

be required. In this respect the method of unequal intervals is moreadaptable to special conditions without involving much new work than isthe method of Naiman.

The results obtained by Multhopp~s method are given in figures 6(a)and 6(b). The 31-point solution (in Multhoppis somewhat odd manner ofdesignation) corresponds to AO = 5.625°; the 63-point solution, toA9 = 2 ● 81250. The computation is very simple and the results of themethod with 63 points are comparable with that of Naiman with 80 points,with the exception of those near the region O< ’x< 0.01 (thisis

WQshown most clearly in fig. 6(b)).- The very steep peak of ---

at x/c = 0.02 requires rather high harmonics for the repres%tationof &t; consequently, good acc~acy ~ the region near the ori@ maYnot be expected. ~is is substantiated by the fact,that for the 63-petitmethod the highest effective harmonic would have three waves in theregion O < x< 0.04; obviously a sufficient degree of accuracy in thedifferentiation process cannot be obtained.

As mentioned earlier, Timmanrs method might be expected to givegood results if the size of titerval is properly chosen. Inasmuch as

only a table for Ae =36oT?

10° was available, the result of the

computation for ‘~ cannot be expected to be good, as is evidenced byVo

observing figure 7. The result obtained is comparable with that ofMulthopp~s 15-point and Naiman’s 40-point solutions.

An excellent method of examining the accuracy of these methodsstill further is simply that of solvhg the ~v~se problem. From the

curves of w~ just discussed, values for ~ have been computed

--- .-. ———-. —.. .— __ ———- —.. —.— -—- --.— - ..—. .—— ——. — .— . ..-—

28 NACATN 2451

andarepresented in figure 8. The method of unequal intervals givesgood results, ind.icating that the arrangement of intervals chosen wasas good for the inverse problem as for the direct problem. It iS

apparent that Naiman’s method requires even smaller divisions than160 points h order to avoid ticcuracies near the point x/c = 0.04.

The reader may wonder that the inverse problem is not given byMulthopp~s method. It must be recalled that Multhoppls method ofsolvhg the direct problem does not tivolve the differentiation of Ayt;

that is, it is particularly fit for this problem and presents, on theother hand, no analogy for the inverse problem:

Jd(fV$ 1 2’b~cote-code=-~ J*AV sin e.=—

axd6’

2X o v; 2 m ov~cose-coseo

Because more extended tables for Thmants methodand the results obtained from the 36-point method forare very poor, no furtherwill be given.

examples of the application

COMPARISONS OF METHODS

UNEQUAL INTERVALS

OF NAIMAN AND MULTHOPP WITH

BASED ONACTU&G EXAMPLES OF

are not available]which tables exis~of this method

METHOD EMPLOYING

CHANGES IN

AIRFOIL SHAPE

intervals has shownthe function u(x)

good qualities whenis known analytically.

The method of unequalapplied to a problem whereHowever, aa mentioned ealier, th~ ~ction ~ not ~ua~Y ~o~ fianalytic form. This section, therefore, will compare the threeprinciple methods, those of Naiman, Multhopp, and unequal intervalf3,on the basis of actual desi~ problem} solvtig the d~ect Problem

for $2 and ustig these results to solve the inverse problem (excludingo

Multhopp for the

Figure 9(a)

figure 9(b), the

inverse problem).

shows the &+ relations

a(%)ax

relations. Note

for examples I

that the slope

for example II is more than twice that of example I near

and II and

r ‘1(%)of —

ax

x/c = o.“

.

—.——.-— . .

NACA TN 2451

The directfigure 10. The

29

problem for example I by Naimants method is given in160-Petit solution does not show any appreciable

()deviation from the 80-point solutions at the region of $~ ;

()

o-however, near the origin, at ‘~ , the influence of the smaller-sized

O m~

intervab (80:Ae . 4.5°; 160:Ae = 2.25°) is q~te prono~cedc

6The solution by Multhopp~s method Is given in figure ll; 31 pointsaround the half circle are not sufficient for a solution comparable withNaiman~s 80-petit solution, and even a solution based on 63 points doesnot offer much tiprovement. The results are poor, aE might be expected,in the region very near the origin (see precetig section).

Figure 12 presents the results obtained by the method of unequalintervab, compared with results obtaimedby Naiman?s 80- and 160-pointsolutions. The method of unequal intervals gives results corresponitlngto those established by Naiman~s 160-petit solution. The subdivisionused is

As

ure 13.

shown in the figure.

before, the tiverse problem was solved, and is given in fig-

In each case the computed curve of ~ _ the one wed ~

obtatitig the values for the W c~ve ‘&hmetho~g~~egood.dx,

results, thus provtig that the chosen number of divisions was sufficientin Nahan’s method and in the method employing unequal titervals.

The value of w“ computed at x/c = 0.17’1 is of some interest.h

This point was comptied by the method of unequal intervals in twodifferent ways: First, the arrangement of intervals shown in figure 13was utilized to compute the lower point. Then a new arrangement ofintervals (Ax = 0.018 for O< x< 0.36) was set up and the same pointcomputed. The idea was to determtie the inaccuracies that would result.One might predict that, since the point x/c . 0.171 lies at a con:siderable distance from the region of rapid changes in ~ errors of

V. ‘only small ma~itude would be introduced; this is fairly well sub-stantiated by the results shown in the figure because the error thusintroduced is approximately that of the deviation of Naimants 160-pointsolution.

% ecall that thismethod does not involve the differentiationof Ayt.

—. .—-— .—

30 NACATN 2451

Now, turntig our attention to example II, which, it will be

recalled, has a slope of w ofapproximately twice thatofdx

example I, the results given in figures 14 to 17 are obtatied.

For the direct problem Naiman$s method of 16o points and themethod emplo~g unequal intervals give results which are in goodagreement. For the inverse problem (fig. 17) it is apparent that the

method ustig unequal intervals is superior

[-](

see the deviation at

d Ayt

dxw)

given by the method of Naiman . Multhopp’s method gives

a rather good result (fig. 15), which may be attained when the Fourierrepresentation of the Ay curve is adequate.

The two examples thus far presented are favorable for Naiman’smethod because the steep slopes of u(x) occur near x = O wh~re the ~’points Naiman uses are close tog&her. However, going to still steeperslopes near x = O would require a rapidly increastig number of points.The new method offers another POSSibility here. Assume that in that .

(critical region xk < x < xk+l xk maybe O) u(x) may be represented

xby u(x) = ~xn. Then the inted

may be split

s

c

o

1c u(x) ~Ox-xo

into three titegrals

/

‘k u(x) ~a(x) & . —X-x. o ‘-xo

Jc

u(x)

‘k+l x - ‘0

ax

The first and third of these titegrals may be solved inmanner using the functions jno and jno*. The second

be solved analytically.

“ (41)

the usual. .integral will

“

.

..— —

.

.

..

NACA TN 24>1—

This simple form, due to the use of the coordinate x in thePoisson integral, allows a rapid integration, because the integral

J‘k+l ~

%,0= —x-‘k

‘o

csn be solved by recurrence as follows:

~,. = ‘k+l”;‘k” + @n-ljo

with

k J()

%-xo0,0 = “o x

k+l - ‘k

dx

for n~l

31

(42)

(43)

Thus even very steep elopes cause no difficulties.

As already mentioned, examples I and II correspond well to thequalities demanded by Naimants method insofar as the rather steepslopes occur in those portions where the points 13n are close together.

If those steep slopes should occur h other portions of the chord, how-ever, a very great number of points in the Naiman method would be neededin order to represent a(x) adequately, and to get reliable results.In such a case the method using unequal intervals shows its advantageby allowing a free subdivision of the chord.

A third example will serve to illustrate this. Figure 18 shows a

%function u(x) = ~ . The essential values of the function lie in

a part of the chord where even Naimants method with 160 points is notsufficient to represent the function accurately. This is forcibly

Avshown by the two curves of

~

(dotted line) so as to eltiinate

unequal intervals can be

16&point solution, thusNaiman~s method with 16o

Table III 3ndicatesby unequal intervals.

made to

If the function a(x) is modified

the high peak, then the $~ curve by

At pyNa-lsagree with the original ~—Vo

definitely proving that, in this example,potits is insufficient.

the computation for the petit X. . 0.065

-- ——..—.—— .—— .-—— .—— . .——— .- . . ..——— — -— -.—

32 NACA TN 2451

CONCLUDINGDISCUSSION

The new method of evaluating the Poisson integal developed hereinis to be recommended for all those functions u(x), where steep slopesin small portions of the region to be integrated exist. Iu theseportions a very small size of interval may be chosen without requiringthat this same size of interval be used throughout the region ofintegration. In this manner, the work required for computation may bemaintained at a reasonable level even for the most complicated problems.

The analytical treatment of special parts of the titegral ispossible (evaluatingthe remainder by the new method; see precedingsection). h those problems where a transition to very small intervabin part of the titegrat,ionrange would require the determination of agreat many values of an, this idea might be used to advantage.

It should be noted that the smoothness of the function U(X) andits accurate representation by single points is essential for goodresults. If, for eqle, Sfigle pOiU_k Un me simplY t*n from a

curve for ~ very close to one another it may be comptiory to check

these values by a table of differences.

.

.

Stanford UniversityStanford, Calif., December 6, 1950

.

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, ., ,.

NACA TN 2451

.

.VALUESOF Jno AND

35

APPENDIXB

jnO* AS FUNCTIONS OF ~

The values of the functions jno and jno* are presented as indi-cated in the following table. The values are tabulated in a form selectedto minimize the necessity for interpolation except for the region contai&ing the singularities of the functions jno and jm*. Tor ease incomputation, tdles B-I=~x~VIII, ticlusivej are arranged so that the

vertical increment of - is unity. Table B-IX gives additional valuesUA

for the region containing the singularities of the fuctio~ jno ~d

jno*.

TABLE No. RAMGEmy ~my

WI –189 to–go 1.0B-II -89.5 to -40.0 .5B-III –39.9 to+o.o .1B-Iv -19.99 to o .01B-P Oto 19.99 .01&w 20.0 to 39.9 .1E-VII 40.0 to 89.5 ●5&VIII 90 to 189 1.0 -

B-n -1.000 to 0.000 0.001

-. .———— –— — —.

-89.5 ip:-40.o

, #

,

I

t

—

TABLE B-III.- VALUES OF Jno MD Jno* USED IN EVALUATRW

-39.92 y 5 +.0

=

x

=b-39.1 .@-33. -.0261

-.CG+7

z: -.

g %J

+: -. 0s9-31. -.0319

+. .0-

-m. -.03M-m. -. 03X-v. -.0367-!26.

X %-J

43.-m. -.CJ47-QL -0467-m.x -.&m

b“ Am Jm’ .k b’

+.ole’r -o.@ -0.COA9 4.W 4.Ols

-.OQ1 -.mtl -.o1.31 -.m6a -.0).s-.ol% --- -.03.?3 -.CQ69 -.0).3!-.QUJ3 -.036 -.ol@ -.CI?* -.Qly

-.olka -.0283 -. Olu -, 02& -. Olk

-.01h6 -. O&2 -.0146 -be -.Olb;

- 01% . OKQ p 01% -, Om . Olyl

-.0136 -.030 -.0136 -, Om -. Oly

-.0163 -.o&J -.0161 -,om -.0161

%: a% ;:: :r~ :g

:;g :~ ::g -.Oq -.:2

-.0191. -.03?0 -.0193. -.O* :01;

-.0193 -.0 m -.u~ -.097 -. 02md d-. m -. M -.0207 p 13 -.@J?

-.mly -.0499 -.cra16 -.c+y. -.cE17

- @2> -0449 -.E26 -. IMs -. W27

:3, :~j -j?JJ -.. ~~

=-m

IJm’ L L*

-o.@@ -5.@57 ~.Qm-.0).32 -.wa -.o13:-.0136 -.0271 -.01 ,

ial :%7 +1

:g~ :% :$;

-.ollfa -.032k -. 01.+

:;J ;;% %$

-.o~ _:o;* ‘“0

2

-.Olal-.0193 -.0 - O1$A-.(021 -. -. m-.m.lo -. CJ+18 -.mll-.W9 -.043’7 -.mlg

-.=3 -.CW ----.CQ40 -.M79 -.cekl

-.0252 -. GW3 -.0%

493

4. C125!-.(txl

-.om

z%-.9-.037-. Ow-.03;-.033f-.03k-.03

:$

-. Mao-.c43g-. *W

-.CM1-.0535

3

4.013[

-.013:-.013:-.OIM-, Olq-.014-.01 4-.01%-. Oti?

-.mf5-.01

2-.0 1. Olm-. 01s

-. ~3-. a?u-. az?l

-.WU-.ozba-.CQ5>

EQUATIOR (26)

2

kolh’

~.w -9.0L?4-.a65 -.olx-.. -.:;

-.11-, C&% -.014,~. ;0;4:

.09-. WY -.om-.0326 -. 01.fa

~;$ ~f&5

“lX-.0361 -.0

Cif

-.03 -.Olm-. 0 -.0196-. -.crd

1-.bka -,aar+-.0441 -.IZ2Z3-.* -.WZ-.. +2& I

L kc.” J-m

-3.0259-’3.0130 -&m-.0265 -.01311 -.cai

:. -.:;;: ~.

:% ::;~ ;g

-.0307 -.01% -.0

X& y; 2:3

-.0335-..:g +UJ -@&

-.0376 -.oI.@ -.03TI-.oB1 -.0196 -.OW-.dm7 -#w -.QW-.Qh24 -.w .&26-.@43 -.CU23 .CW+

. . . . ~~

-.c5m -.m77 -:c513

4’0’

-o. oul-.o13b

-.013a-,0141-.0146-.Olz. Om-.Ol&l-.016y> Oljl

2-01-.0 3-Colw-.W-,am 2-.cG!14

-.=3-.C%?-.m z. C@g

7

Lo-a

TABLE E-IT.- VALOES 07 & Am &*

(a) -19.XZ s ~+ s -on

usm IN E’ViUUkTIXGMwmoN (26)

Tmii B-Iv. - coNlm?oED

(b)-19.XXS ~S 4.XX where * 2 XX 280

w0)

, .

TAB’IEB-IV.– ~

TABLE B-IV.- COiTTI-

(d) -19.Ix = ‘~ ~ A3.XX Where 69 z XX z 150

I

!2

.

TABm B-m.- Ccmcm’mo

I

,.

TABm B-m.- Ccmcm’mo

I

,.

[

-la

$-t-1.-m.-m.-U.

4.

$32:-h

092 XX 200

c ,

{

I

TABLE &V.– VAImG3 QF ho Am

TABLE B-v. - OomXmED

<%Q=O<(b) O.X .= = ~p.= where 10

I 1 I u 1

&-

TAmE &v. - CcmmmED

I /.\. — <‘- ~ lg.n where 2oZxxZ29‘T

i-%-

m I n6

TAmE B-v. - comm’oED

1

c-kT-

[

n

k

1.

~

Y.

6.

Z:Il..

!s.

it

15.

16.

1.d.19=

TABLE B-v.

(e) O.XX 2~519xx

- Comm

where A.05X X549

, I , , , I ,

TAmE B-v.– cOmm

+0 149

TPBIiE I&V. - Ooru’mom -Pcm

I & I 65 I e‘@ l@ f

I

, I I I I

m B-v. - coNTlmJED

* * G----AL-I

i

I

(i) O.xxs

—

(J) O.xxs

%?

TABLEB-v.-

93 I99 159

1

o

Jm*

—

,Lceb

.’=3

:E.0-%.OM

.Ow

.Olal

:~g

. W@

.o153

.o149,o144.o141.0136

.o133

.Ow

.0U?6

TABm Mm.- WmJ’Es cm’j~~ m &J* USED m mm!mm WJATm (26) .

1

&_.ce41.ce30

.Ce.?o

.mlo

.CuX?

.Q13A

;%

.o174

.aa

.ol&

.Olm

:x

. cab

$113W6

. Ow

.mm—Tb %0’.@@ O.OA1

.0461 .Cm9

.OM1 . cc?19

.C&2 .W!lo

.0k5 .aml

.@9 .0193

:W? ::%

.qm .ol’f3

.0337 .m67,Cy26 .o162

:2 ;;#

.C2

8

::;3

. .Ow.Wz9

:0252 . 0L?6

3

1

4 5

1

6

g).ckp

. ‘@

.ck31

.cA13

.0357

.O*

.q6a

.W5,Cgti,C531

m=

:%

:%

:%

.CQ >i.02 9

1Jm’

l.w~,CrA.Ccw

.’=?

.W-l

.01$0

.afh

. 0LT6

.0111

.C@

.W9

. Olp

.a otlmlo15

. ~w

.a34

.03.30

:W.0124

8

1

9

.-

. .

!cABm B=m.- VAmE3 w & m &O* mm m _T~ I!?ammm (26)

40.0S ~ 589.5&

J&o I k~ I S.O I 5=.5 fko la &o

&Ol$aD’l&Ol$r#l$ml&’Y’l& I$m’ JnO

IIh h’

‘.M23 0.W61..0u22 .OM1.Ou?o .0%3.0U9 .-

:%

:$ ;%

.mu .alz1O.cdrf.Mkl.’=35.0230.-.a2xJ

:%’.azQ3.&n?

1.0244.=34.@33:~

.~7

.=3

.W

.CmY1

.Cr2a

o.ol&

;%

.’W5,01%

.Qm

.o14!3

.mh3

.CQM

1.olQ4.M=3.Cnul.0120

;%J.0u3. One

:%.Cr61SW%.Cm9

%!

VilLum C1FJno AMD&o*mE.Dm WUaWTNG DWATIONTABm EQIII.- (26)

i

I.&.o I 17X.O M?Im.o I 131.o ls!x.o

* b“ G---~.(JM?.Iml.O%l.Crxl.@51.03s4:%1—

b’

T.@l o.J& ..%3

J’@.W J@.@ .W7.Wo .Om.Ccgo.m30 :3.Cu30 S@

.Ww .W

I.ctgaO.cql.Cw .WJ..ag8 .OqoJ@.-7 :%m-l -@9.W .00$9.w6 .00s9

:% :%

:%.m3.@33

.@

:Ze,cqR

:%I I -

I

I

(26)

I

.

I

I

I

TAB123E-ix.– CONI!INUED

(b) -0.749~ ‘~ ~ -0.XKI

9 8 I 7 I 6 I 9 I 4 I 3 I 2 I 1 I 0

I 1 I

-=-h- * b’—M??&.9739

$3

.M T

.Z3Y:%=.5976. 63IE

. 681’I

.’@$

.W71

‘f!ii

7

:%

.W

.9134

.*3

.9*

.9*

.m

.’

I

TABLE Km.- cONrImED

9 a 7

Jm &#lkl$m*l Jro I$ro*

I=ilh

&*lklb*l Jm l&l*

3 Iell o

JlbYIlrm”

1 , 1 #

WIm

I

I

.

I

I

I

TABIE E-n.- ccmmJDED

(d) -0.2hg S ~+ S 0.003

9 I 8 I 7 I 6 s I 4 I 3

$m’l&ol Jnn

I I 1 I I I 1 1- 1 I I 1

L? 11101

$m*lJrml

I I ! I I J

54 NACA ‘TN2451

DETAILS OF SOLUTION OF INTEGRAL (36)

With the expansion (see equation (37))

= a.

Hence,

1=—

[

a.c

*

al

52*

(c’)

(C3)

(C4)

.

.

,

NACA TN 2451

As the occurring integrals are

s

‘l/c~=

o

55

all of the same type, define

These integrals ~ are easily solved by recurrence.

with

and

J.E1“- $Lo=& lo& 1 for x. < ~1

F o 1+

r

X.q

The function

is given in figure 2 in order to provide a more

the event that ‘o or 1~ is not very small..T1 0

.

.

(C5)

(C7)

(c8)

rapid computation in

.-— —. —--- .—- ——~ —- ——. .— .——.-,— —— ...

56 N4CA TN 2451

. .

(C9)

(Clo)

——

8 NACA TN 2451

Withtheseexpressionsthe integralF1 is as follows:

=

.,

(~aoLo + al*L1 + a *Lc 22 )

+ 2$$)+ a~(,3&+

57

J.

The coefficients aoj al) and .2 Avof the expansion of — are ‘givenVo

by

()Ava. =

:~)x=o+4R)x=,,-E)x=2Jal=+

;

(cl’)

[

‘2 ‘~ (%).=O-2(%).=61+($3X=2EJJ.

—-— -—— .– —.__—_ -—-——— —.—— ____ . . .

!% I?AC!ATN 2451

1. Allen, H. Julian:Arbitrary Shape

REFERENcm

“General Theory of Airfoilor Pressure Distribution.

Sections HavingNACA Rep. 833, 1945.

2. Naiman,Irven: Numerical Evaluation of the e-Integal Occurring inthe Theodorsen Arbitrary Airfoil Potential Theory. NACAARRLhD2741944.

3. Naiman,Irven: Numerical Evaluation by Harmonic Aualyais of thee-Function of the Theodorsen Arbitrary-Airfoil Potential Theory.NACA ARR L5H18, 1945.

4. Multhopp, H.: Die Berechnung der Auftriebsvertei-lungvon Tragfl&eln.Luftfahrtforschung, Bd. 15, Nr. k, April 6, 1938, pp. 153-169.

5. Thnan, R.: The Numerical Evaluation of the Poisson Integral.Rapport F.32, Nationaal Luchtvaartlaboratorimn,June 16, 1948.(E@lish) ‘

6. Schoenberg, I. J.: Contributions to the Problem of Approximateionof Equidistant Data by Analytic Functions. Part A.- On theProblem of Smoothing or Graduation. A First Class of AnalyticApproximateion Formulae. Quart. Appl. Math., vol. IV, no. 1,April 1946, pp. 45-99; Part B.- On the Problem of OsculatoryInterpolarion. A Second Class of Analytic Approximation Formulae,Quart. APpl. Math., VO1. IV, no. 2, July 19!-6,pp. 1L2-141.

.

,.

— .——. — —

I?ACATN 2151 59

xTAHLEI.- -%?VALUESOF jno AND jno* FOR -49.5< ‘N <49.5

%-%3&o 3*

%l-%m& no Ax ho &o*

-49.5 -0.0204 -0.0102 0.5-48.5

1.09% 0.4507-.0208 -.0104 1.5

-47.5.5108 .2338

-.0213 -.0107 2.5-46.5

.3365 .15etl-.0217 -.0109

-45.5.2513 .1204

-.0222 -.OI.12 ::;-44.5

.2007 .Ogo-.0227 -.012.4 5.5 .16TL .O&?

-43.5 -.0233 -.0117 6.5-42.5

.1431 .0698-.0238 -.0120

-41.57.5 .3252 .0613

-.0244 -.0122 8.5-40.5

.l.lw .O*-.0250 -.0125 9=5 .1001 .0492

-39.5 -.0256 -.0129 10.5 .0910 .0448-38.5 -.0263 -.0132 11.5 .0834 .0411-37.5 -.0270 -.0136 1.2.5 .Ono .03&?a-36.5 -.0278 -.0139 13.5 .0715 .0353-35.5 -.0286 -.0143 14.5 .0667 .0330-34.5 -.02g4 -.0148 15.5 .0625 .0309-33.5 -.0303 -.0153 16.5 .0588 .0291-32.5 -.0313 -.0157 17.5 .0556 .o~-31.5 -.0323 -.0162 18.5 .0526 .0261-30.5 -.0333 -.0167 19.5 .@oQ’. .0248-29.5 -.0345 -.0173 20.5 .ok76 .0236-28.5 -.0357 -.0180 a. 5 .0455 .0226-27.5 -.0370 -.0186 22.5 .0435 .M6-26.5 -.0385 -.0193 23.5 ;o&l:

-.0400.0207

-25.5 -.0201 24.5 .0199-24.5 -.0437 -.0210 25.5 .0385 ●0191-23.5 -.0435 -.0219 26.5 .0370 .0184-22.5 -.0455 -.0229 27.5 .0357

-.0476.0178

-21.5 -.0240 28.5 .0345 .0172-20.5 -.@oo -.0252. 29.5 ●0333 .0166-19.5 -.0526 -.0266 30.5 .0323 .0160-18.5 -.0556 -.0280 31.5 .031.3 ●0155-17.5 -.058$ -.02W 32.5 .0303 .0151-16.5 -*o@ -.0316 33.5 .02g4 .0146-15.5 -.0667 -.0337 34.5 .0285 .0142-14.5 -.0715 -.0362 35.5 .0278 .0138-13.5 -.o~o -.0390 36.5 .0270 .0134-12.5 -.0834 -:0423 37.5 .0263 .01.31-U.5 -.0910 -.0462 38.5 .0256 .0127-10.5 -.1001 -.0509 39.5 .0250 ‘ .0125-9.5 -.13J2 -.0567 40.5-8.5

.0244 .0122-.3252 -.0639 41.5 .0238

-7.5.ou8

-.1431-6.5

-.0733 .0233 .0U6-.161cL -.0859 g:; .0227

-5.5.0113

-.2w7-4.5

-.1037-.2513

.0111-.1309 g:; :E7

-3.5 -.3365.01_08

-.lm .021.3-2.5

.0106-.51o8 ,..2771 ::; .0208

-1.5.0103

-1.0985 -.6479 48.5 .0204 .0101-.5 0 1.0 49.5 .0200 .0100

————— ——...—. ——-- -——-——-— --------- -——- —. ._ ______ .___________

60

TAHLE 11.- coMKm!rImBY UmgJAL Im!EKvm,TRANSITION FROM

OH EINTER’VALKCZETOAI?OT!HER

(a)E = 0.002.

x< % %+1 - ‘n

%-%.k ?ua ho’

o Uo “al- Uo -4.5 -0.2513 -0.1309.002 Kl - al -3.5 -.3365 -.1-/7’7.004

: 5- U2 -2.5 -.51o8 -.2773.

.cm)6 - a3 -1.5 -1.0986 -.6479

.008 ~ q-a -. 0 1.0

.010 U5 U6- U5 .; 1.0!385 .4507

.012 U6 CY7- (J6 1.5 ;;:; .2338

.014 . . 2.5 .1588

.016 . . .2513 .1204

.018 . . ::; .2c07 .og70

.020 5.5 .1671 .Om6.5 .1431 .0698

:%% . . 7.5 .1252 .0613.026 . 8.5 .I.132 .0546.028 a14 “ u14U15- 9.5 .1001 .049.030 015 1o.5

(b)~“ = 0.006.

x u~T ‘%+1 - ‘%

%-XC?Ax ho Jno*

o ao. ‘3- Uo -1.5 -1.0986 -0.6479.0Q6 a3 Cf6- C3 -. 0 1.0.01.2 U6 ‘9- U6 .; 1.0986 .4507.018 U9 u~ - ‘9 1.5 .5108 .2338.024 a= U15- au 2.5 .3365 .1588

---- -- ---- -- ---- ---- - ---- ---- ---- ---- ---- ----.030 Kl_5 U16 - S15 .2513 .1204.036 ’16 ‘J17- IS16 ::; .2007 .Oq’o

al ’18- a17:% i

5..5 .16TL .0812~ alg - a18 6.5 .1431 .0698

.@ . . 7.5 .1252 .0613

.060 . . 8.5 .13J.2 .0546

.066 . . 9.5 .1001 .0492

.072 m. 5 .0910 .0448

.078 . . .0834 .0411

.084 . . % .o~o .0380

.O$KI a~ ‘ apfj- U= 13.5 .0715 .0353

.og6 ‘J26 %27- a26 14.5 .06s7 .0330

,

.

.

——— ~— -—.—— .–. —

0.060.0633.0667.070.075.O&l.085.ox.100.110.120

TKBIE III.- CCMKMMTION FOR ~ -0.065 BY UIIWJAL IIWERVKGS

%

o

.m~

.015

.036

.0925

.2000

.1000

.006-.0027-.00160

[_le, fig. 18]

‘%+1 - %

xn

k=”

0.005.010.02J..0565.1075

-.moo-.og4-.m87.OOIJ..00160

-1.3

-.5.5

G = 0.0033

% - X.

%

1.02.0

:::

z = 0.005

-1.09 -0.6480 1.0l.ogg .451.693 .307.406 .U3g.288 .137.223 .107.336 .159.251 .120.201 .097

.167 .o&l

I

62 NACA TN 2451

.

4.0

300 I

,2.0-

/

1.0

%

$Q;1

/ -..- _

;b o t

;\-1 I

\

-1oo ~\ 1

\ L I\ /\ I

\\\ I

\\ J-2.0

c

-3,0 I

-4.0-2 -1,5. -1 -.5 0 .s I M

zn - %AZ

Figure 1.- Characteristic qualities of jno and jno* as functionsXn - Xo

of —..

ax

!.

—— — _————_. ——

NACA TN 2431 63

,.

.

4M*--2X*

1+--

f’

-2

o“ .2 .4 .6 .6 1.0

!

,4

.8

.2

I I I l\l =.1 I D —1 I

. . -, . .

Figure 2.- Function ~ ‘or Computation ‘hen xo/~1 or ~1/xo isnot small.

.—. — ——-——..-— .——. .——. ———. —.— —...— _ .. —--—. —. . . .. . —- .------

m-r=

.4I

/

! I \

.2 \

T\

o0 .04 ,08 ,12 .b6 ,20 .Z4 ,20 2.2 .36

:

(a) Function &t.

Figure 3.- &ical example for testing accumcy of II&&L

t .

.,

,,

,,

,..,-;

,...,.

,,

,.

.,

. .

;..,.

.

,.

.02s .

/ \

,020 / \

I ‘,02,.

I,.

,015 .0\

d~

dx,,

.010 10(

.

.00s,-,-

1$@@ :.,

dxlo\

.,, ,,,

,~ =

*O50’ .04 .00 .1z J(

:.I

\ .2 / .4 .6 .8 ! W

.K

,,

;

.

-r

. . .

T

.20, .24 .2e .32 .36

,, . . x‘c

. . . .

qf%)(b) Function ~

66 NACA TN 2451

.OZOJ A

.Ole

.o\zI

.W)8 Y~

*

I x

.

+V\

e I

o

A

-.

AX = O.OO6 0.045< X<OC426S-,008

AX = 0.018 13.063 < X< O.351

-.012

H-.0}6 ,

0 .04 .08 .12 ~ ,16 .Zo .24

c

—

(a) Plot for O< X< O.25.

A.v-w 4“- Analytical computation of — for figure 3(b) and comparison

Vo

with results by computation with unequal intervals.

.

.

———. .— -——

.

.

.020 AF I, IA Unaqud

/\ I Intarwls

/ j.0\6 —And ticol

#ccmpu a~ion} {.

ala ~ \

/ /\

me r 1 \I

A

~w00

-.004r“t

-.000

-.ola,

‘.016-/

1 ,

~

0 .01 302 .03 ~ .04 .05 .06E

(b) Part of figure k(a) plotted to larger scale.

Figure 4.- concluded.

. . . . ——-—.— .—.-—— ..______ ._. .——.——.. .— -—.—— ______ _

68 NACA TN 2451

.020

[ )~ - Solution, ,. @ Fk21man 40-pmt

o Nalman fw-pint.016.

v● Nalman 160-pmt $020..

t .

.012 ~ \“- ~

~

.008 ,010

-t 9 &Jf d~

.004 ~

AU7

0

1 : ‘ ‘ %

-c o

=.1

I , ;: , - ‘: - ~ = “ ‘z

40: A(3=9s—

-.004 +: 1 80:Ae =4.

, *, I # I I , , , , t 4 160:A0 =2.2s

-.008

I

-.plz I

-.016, I

T

-0 .04 .08 - .12 .16~ .205

(.) Comparison ~th analytical results.,..

Figure 5.- Results obt@ed by Naimants meth~. 40-, 80-,.:

80iuti~8.

.2+

and lmpoint

.

.

.— —— —.. . —

NACA TN 2451 69

.- k-

.016

E f

vSolution I

IQ Nalman 40’pnt

● Natmqn /60-polm.

/.(XZ \

/ o \

~ .008 ‘ t 1 \

/I

.m

A_v

%0 — — —

m

-.00 4

4: 1 1 t #

-.008

-.012 ‘ r

I( @

o

/T

o .01 .02 .03 ~ .04 ,05 .06c

(b) Figure 5(a) plotted to a larger scale.

~Figim?e5.- Concluded.

——. .—....—. ..— ——.— —— ————— .—.-——.-

70

.

NACA TN 2451

(

.Cxto . I I I I.025

/1 Solutlon{ c? Multhopp 31-pint

.016 . k Multhopp 63-pint “ .020/ 11 0 Nalman 80-pint

.Ola i .015

4)

,Ooa Il\ ) .010

I Au d(we-

1 wK

.004 1d(a~

‘-.005

dx&w 1~

0 - @ I -c Q

. -- .i , 1 , *I , 4 Multhopp31:

‘- ‘A.9= 5.625°

dd

-.004 iii , m ,I a 8 * , , , I u * MulJ@hom 63:

= 2.8125”

>

-.008

- .Olz

‘.016 I

v

o“ .04 .08 .lz . ~ .16 ● 20 .24

c

:a) Comparison with analytical results and results of Naiman*s method,

Figure 6.- Results obtained by Multhopprs methml. 31- and 6s-pointsolutions.

.

.

.

—— —.- -—

n

R//.020

I# MuRhopp 15-pomt# Multhopp 3 l-~lnt

.OIG

.Ola

/ a

.005 - t ~

I

/’.0-

..

AV ~

T

r *

-,004 4

-.008-

-.012 !/

A

#lo a

‘.016

/ T

o ,01 .02

(b) Figure 6(a)

Figure

.03 * .04

r

plottedto larger

6.- Concluded.

.05 .06

scale.

..— . .— ... . ..... . ___ ----—— -----—— -— . ..- .. . . ... . . .+ ...- . ..__

72 NACA TN 2451

.$320Solution . .

@ Nojman 4@point# Multhopp l~intX, Timman 30poin12 .,

.016 “ ,020

1

1

.012

.000 .010

x\

I ~e I 1“d(w)

.004 -

1’ k

d(~% ‘ \z-

W

A_uy o 0

Mul+hopp 19:Aer 11.25”

,

!

-.012

‘.016

o-.04- :08 .12 x - J 6 .20 .2+F

Figure 7.- Comparison of methods of Naiman, Multh~p, and Timunwith abalfiical solution as basis.

.

——. .. ————---——— -—————

.MsSolutlon

.020

A Unequal Intwvals

HI , 0 Naimom f33pint b9=4.5e)

.020 ● P/alman 160pInt (~= 2.25°). .016

b 1

rI

.013I 1

41

.012

.010 x?

u ‘\

,008\\

1~” y ,.

A_V

Y

,ti 5~

,8, CIX. .m

1\

@\

dx ~, \< <

b o.

%7 \\ 7 ~ /

0 % .? a u

-Cd -,004

-.o1o- ,

‘o ,0+ .0s‘.009

,12 ,16 ~ ,20 .24 ,m ,3’Z ,36c

-.. —.. .

Figure 8.- Solution of inverse problem. Comp&i.son of Naimenf a 80- and

160-petit solutions with that obtained by the method of unequal

intervals.

-1-Y

‘(a) MctionE AY~(4 .

Flgue 9.- Examples I and II.

.,

xlo-~

I

1

‘1-**.s.-0-0

1’

I

.10 ,20

II dm*

,08

\ \

@J. .7 Example D -J 0dx

.10

.06 -

\

\*IC

I

o0 —,z— ,4 .6 ,8 1.0

.04 -x

j ‘\ !

1G .Oa

\

.02

y ‘-\

0~ -

0

\.~. — — — — — ~ == —

9— —

- ~

-.OE0 ,04 ,0 B~4

J 2 .16 ~ ,20 ,24 #2e ,36

(b) * as functim of x/c.

Figura 9.- Concluded,

I

I

.04

.02

,A_*

~o

-,02,

-,04

-,00

I

. .

r

I I

#-- Solution

+ +Nalman Z()-pointA-u ● Nalman 40-point

/yo NalmQn 80-pint

e Q Nairnan 160-point

\

\

. v

84

a) +-

\

,04

d(&& ‘

Clx,Oz

●dJ@

I

\\ dx

~

,1- ~ ~

—. + ~T,1

-am

0 .04 .OB .12 .16 ,20 x .24 .28 .32 .36 ~E

P&

Figure 10. - Direct problem for exemple I by ITaimau’E method.

.04

~ f,

k>

‘!:

\

\ ‘9olut)onAu—

j !!wK’pp :::;:1’:;

,02 I/ k d Multhopp 63- folntd \

I f

\

o \

{- ~ - ~t .

+ , I Nalmqn 80II I I I 1;

I

I Multhopp 63

-.02 + :‘ dx

Multhopp31

\

.04

-,041Cl@)

1

#dx

$ s02>

‘.06 I

)

d ~-

‘.08‘- b“ ~b- ‘“ ‘- -..

~-~--o,\3-- 1

T

0 ,04 ,08 ,12 .16 .20 ~ .2+ .28 .32c

.36

.-

FiP 11.- Direct problem for exemple I by Multhoppfs method.

!3

,04 d -% AU

Tr. Solution

o Naimo.n 80-pOlnt

SOz L A Unequal lntarvoh

I ● NJa[man 160-point

\

\

A_V\

v, o + ‘* I

+ d(aw)77

-.02

\ ,

.04AX IO.002 o (% < 0.010

I AX cO.QQ6 OJOIS { x< 9,*

\A% =0.olg o.090<xtnJ60

-.04 ,m

1$ da)

dx .

-.06 “<

/ ‘/ ~

!!~

-.08 $

, 1 ID

f.04 .08 .12 & .16 .20 .24 .28 .32

c.36

Figure 12. - Results obtained for exemple I by mthcd of unequal Intervals

compared with results obtained by I’Jaiman’B 80- end. 160-@nt solutions.

I

1

.

Jo

.06

-w

.04

,02

d(A%)‘z-’

C3

-.o~lo

a

4 ● Ncdmran 160-pint

\

oAX .00006 0.04EKX4 0,16R

AX =ao18 0.162fXL J@

A

/

I

0

tiit:

I .04 .08 g .12G

Figure 13. - Solution of inverse problem. Results obtained for example I

by method of unequal intervals compared’ with results obtained by

Haiman’s 80- and lti-point Solutions.

I

I

1

.08

.04

-.04

-.00

-.12.

‘.16.

e So)utlon

9 ~ Naimom 20-point

/ c

+ AU o Naiman 40-point

.0 Naimcm 80-point

! . N~imcm 160-pint

Y

1 + ~ _T u ~

, \

~ *

\

d~

dx\

n

\ # <- ‘- . .- -

=s=

.04 .08 ,12 .16 ~ .20 .24 .28 .32 .3—c

Figure 14.- Direct problem for example II by Nahumts tnethod.

E’s’

I

1

1

I

I

1

I

I

/

II{

1

0 , .0+

SolutlOn ?O Mman 80-pol nt

~ Multhopp 3J -point!2

~v

i\ A

\- ~

J’

-. 08 ‘

—. w

* .

dx

o

— -~ ‘

-

.00 .12 ,* .16 .eo .2!4 .28 .32 .3s

Figure 15. -(3J

Direct problem for- exemple II by Multhopp!s method. #

I

I

.0sSolutlon

nA Unaqual lntarv~ls

.04

0 - ~ ~ — — — —~

*

,.

\-,04 - .0s

J /I d-

4. $ dx-,08 ,04

_\

6?-.12

‘.16 4

0 ,04 .00, ,12 x .16 .20 .24 .28 .32 .36

z

Figure 16. - Reeults obtained for example II by method of unequal intervals

compared with reeults obtained by Nalman’s 80- and 160-pcLnt solutions.

L

..Zo

Solutlon

h A Unequal Inwr-va Is\ o Nalman ZIO-paw

.16 m ● Nalman f60-po/nt

e

,12

\

.~8 k

.04

d~

dx

o -

,

~ -.

I I

1 I I I I I I I-.*O

I I I.04 .00 .12 .16 x .20 .24 .Za .3Z .3b

E

Figure 17. - Sdution of ixrverae problem. Results obtained for example II

by method of unequal intervals compared with resultm obtained by

Ndman’ B 80- and 160-point solutions. 8’

84NACATN 2451

.ZQ

.10

\

60

, I t 4 * * , 1 L!naqual Intervals1 9 , 1 G 1 Q

I-.10 * t I Naiman 80”T

11 * I1 I i Naman 1601 , ,

.20

4Soiutjon

.10 A un~qUQ! lnt~rv~l~)1 ●.Nalman 160-pojnt/

A_uV.~

fu #

v-.1o-

.02 .04- , .06 g - .00 .10 J2 ●1+

z

figure 18. - Results obtained for example III %y methcd of unequal inter-vah compared with results obtained by I?aimn*s 80- -and 160-point-solutions . .

NACA-LaUZM -10-25-51- 1~

.— .