national advisory committee for aeronautics
TRANSCRIPT
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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL NOTE 2451
MATHEMATICAL IMPROVEMENT OF METHOD FOR COMPUTING POISSON
INTEGRALS INVOLVED IN DETERMINATION OF VELOCITY
DISTRIBUTION ON AIRFOILS
By I. Flügge-Lotz
Stanford University-
Reproduced From Best Available Copy
Washington
*,~-^ October 1951 >!S rFHBirnON STATEMENT A Approved for Public Release AAAAAAJ/ J A m
Distribution UniimiSd 20000816 107
DTIC QUALITY INSFBCTED 4
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL NOTE 2^51
MATHEMATICAL IMPROVEMENT OF METHOD FOR COMPUTING POISSON
INTEGRALS INVOLVED IN DETERMINATION OF VELOCITY
DISTRIBUTION ON AIRFOILS
By I. Flügge-Lotz
SUMMARY
The Poisson integral involved in the determination of the change in velocity distribution resulting from a change in airfoil profile in parallel incompressible flow is solved.
First,.three well-developed numerical methods of evaluating this integral, all based on the division of the range of integration into small equal intervals, and the difficulties involved in each method, are discussed. Then a new method, based on the use of unequal intervals, is developed, and compared with the other methods by means of several examples. The new method is found to give good results for both the direct and inverse airfoil problems and is easily adaptable to rather complicated problems. It is particularly recommended for all those functions where steep slopes in small portions of the region to be integrated exist.
INTRODUCTION
The ordinary thin airfoil at small angles of attack produces only slight disturbances in the flow of a parallel incompressible fluid. Hence, the influences of camber and thickness upon the velocity distri- bution may be computed independently and their effects superimposed. The effect of camber may be represented by vortices distributed along the chord line of the airfoil section; the effect of the thickness, by sources and sinks also along the same chord line. The velocities pro- duced by these singularity distributions enable one to compute the pressure distribution on the airfoil rather quickly.
Allen (reference 1) has presented this singularity method in a form which has proved to be very practical for common usage. However, in special cases the unavoidable evaluation of the Poisson integral in
2 NACA TN 2^51
the course of the computations has given rise to numerical difficulties. Such integrals are usually computed by the application of finite differ- ences using intervals of equal length. However, changes in airfoil shape, which result in marked changes in the function to be integrated in only small portions of the range of integration, require that extremely small interval sizes be employed in this range, and, consequently over the entire range of integration. This leads to a considerable amount of computational work; hence, it appears reasonable to discuss the possibility of employing intervals of varying lengths for the evaluation of the Poisson integral.
This investigation was prompted by the difficulties arising from the problem of small changes in the shape of symmetrical airfoils at the angle of zero lift. The examples included in the present report are restricted to this case, but the results obtained are in no way specialized and may be applied to all problems wherein the Poisson integral occurs.
This work was done at Stanford University under the sponsorship and with the financial assistance of the National Advisory Committee for Aeronautics.
The author wishes to express her appreciation to Mr. H. Norman Abramson for his intelligent and skillful help in the computational work and for his assistance in writing the final report. The author also wishes to extend her thanks to Mr. R. E. Dannenberg and the computing staff of the 7- by 10-foot wind-tunnel section of the Ames Aeronautical Laboratory, Moffett Field, California, for preparing the extended tables of the functions jn0 and jn0*-
DISCUSSION OF PROBLEM
The basic reference profile may be given by ytr = f(x), and its
velocity distribution may be known from an earlier computation. The problem at hand is that of determining the change in the velocity dis- tribution resulting from a change in the shape of the profile (indicated by the dotted line in the following fig.). The difference of these two shapes is designated as Ay^..
NACA TN 2^51
y 0
Allen (reference 1, p. 7) gives for the change of velocity the equation
Av(x0) 1 X - x„ (1)
where V0 is the velocity of the basic parallel flow. If, by conformal mapping of the outside flow region, the center line of the profile is transformed into a circle by the relation
x = °- (1 - cos 9) 2 (2)
then the profile is transformed into a curve approximating the circle shown below.
NACA TN 2^51
The change in velocity due to a change in form will then be given as
_1_ 2ir
'2it l(ATt)
dx
e - 9, cot do (3)
defining
dx K+9
iM. dx
jr-9
This is the form most often used for computation purposes because the inverse problem (that of computing the change in shape due to a change in velocity distribution) utilizes the analytic form
fel dx 2K.
'2JT Av cot |
9 - 9r
Jxr 10
d9 00
defining
IT) - {¥)
which is strikingly similar. The corresponding formula in the original x,y coordinate system is given by
Lfot) dx
Av
1 / Yo V*o(c - xo) Mo x - xo \£(c~
dx x)
(5)
The evaluation of equation (3) may be accomplished by any one of several different methods; however, all of these methods employ the device of replacing the integral over the range 0 to 2K by a sum of integrals over intervals of equal length A0. The equally distributed points 0n have corresponding values x^ which are not equally
distributed (see following fig.).
NACA TN 2J+51
This arrangement is sometimes favorable, and sometimes not, depending d(Ayt) upon the particular form of
equation (^3).) dx (See discussion following
functions
The use of the angular coordinate 0 has the advantage that the d(Ayt)
dx or Az are periodic functions in 0, and this
periodicity facilitates the organization of the numerical computations. The disadvantage arises from the fact that these functions are usually given as functions of x, and, since the analytic form is not usually known, any transformations made will lead to small errors For example
d(Ayt) A ' lf dx °r y1 iS given at sPecial points which do not correspond
to en = nA0, then the computor must obtain the values of these functions for the values 91} 92, and so forth by interpolation.
DISCUSSION OF SOME OF THE EXISTING NUMERICAL
SOLUTIONS OF POISSON INTEGRAL
The difficulty encountered in the solution of the Poisson integral
arises from the fact that the term cot (° ~ do) or (equa-
tions (1) and (3), e.g.) approaches infinity wnen 6 approaches 0 or
when x approaches xQ. The difficulty is of much less consequence when *(*rt) or Av the function is given analytically than when a numerical
computation is undertaken. As a consequence, any simple integration, performed by replacing the integral with a summation over smaller intervals always requires that the interval in which 0Q or xQ is located be given' special consideration (see following fig.).
NACA TN 2^51
H—I H *m 7 i\-<—i—i- V
xm+l
6m/ m V
H 1 \-
*m+l
A 1 t-
A majority of the solutions currently in use have been developed
to such an extent that, for example, &{90) is Slven by a sum of
products of single values of (A—ii-j and known factors An; that is,
n
f Co) - ,2ir d(Ayt)
fe - er
2n '0 dx cot d0
_1_ 2* US,
<2n-6, d(Ayt) 0* * cot — do
dx 2
!0. x , . *n+l d(Ayt) 0 * 1 \- / \ %J cot — d0
2« / i /x, * dx 2 n
which leads to (see reference 2, e.g.)
(6)
£(flo)-£> l(^t) dx -in
(7)
NACA TN 2^51
The coefficients A^ depend upon the particular method of dK) numerical integration which is employed. If, for example,
dx is
replaced by a step-curve, that is, assumed constant in every interval (see fig. below), one set of values of Ano would be obtained.
a(4rt) dx
r\ x/c •-
Greater accuracy would be obtained by the assumption that d(Ayt)
dx is
replaced by straight-line segments (see fig. below), in which case a second set of values of Ano would be obtained.
x/c
A further refinement would be that of assuming 1<&L dx
to be composed
of segments of parabolas, and so forth. Since the accuracy of the
resulting values of ^ depends upon both the character of the approxi- "o
mate curve and the size of interval taken, it is apparent that the same degree of accuracy might be achieved from many different combinations
of interval sizes and approximations to the function d(4rt)
dx
dx
NACA TN 2^51
Naiman (reference 2) has used Simpson's rule for computing the Poisson integral, which corresponds to the replacement of the product
^ yW cot — by segments of parabolas. The "critical interval"
- > 00) was carefully treated by using differences V ■ / 2 I d(Ay.)\ of higher order I including the fifth derivative of I. Naiman
divided the period of 2 it in 20 or I4-O intervals and calculated the corresponding sets of values of AnQ. Other workers at the NACA have
extended the calculation of these values of Ano to 80 and l60 intervals
(unpublished information).
Obviously, the time required for computing ^v increases with the
number of intervals taken because of the increased number of multipli- cations to be performed. In addition, greater preparations for the computing proceoa are necessarily involved, particularly since the
values of —>—— needed must usually be obtained by interpolation, dx
This interpolation has to be done rather carefully as it is often not sufficient simply to take the values of the plotted curve of d(Ay ) . This curve should be checked by difference tables if the
dx
values d fel
dx Jn
are 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 Poisson integral which has apparently been overlooked until the present time. This method has the advantage of leading to a computation which does not involve the derivative of Ay-j..
1Naiman has also suggested a second method for computing the Poisson integral (see reference 3). In this second method he uses
d(Ay+) Fourier polynomials to represent the function —*—**- . The computing
dx
procedure is very simple; however, the results depend largely on the d(Ayt)
degree of approximation of —*——<- by such a polynomial. Thus, for Q.X
large families of functions results are good; however, cases are known to the author where results were not satisfactory because regions with steep gradients may not be represented well enough by a Fourier polynomial of moderate order.
NACA TN 21*51
Equation (6) may be written in a different form (reference 1, equation (lj-3)) as follows:
Av = 1 rd(Ayt) sine ^o nj0 ^ cos ^ " cos ^c
d9 (8)
and this may be rewritten as
i2 rw de äv = i £ Vo n c Jo d9 cos 9 - cos eo
(9)
Equation (9) is strikingly similar to an integral ocurring in the theory of the lift distribution of a finite wing in incompressible flow. There, the induced angle a± is given by
ai = d.7 de'
2it Jo d9' cos G' ~ c°s e
where y is the local dimensionless circulation.
(10)
Multhopp (reference k) has given a solution for equation (10). He divides the range of integration into (nu + l) intervals (see fig.
below)
with
n m + 1
7n = 7(&n) (11)
10 NACA TN 2^51
and computes a± at the points 0n. He assumes that 7 may be expanded in the form
y =2L,C
H 8±n ^e
or
m-, m, 7 = —— >~! 7n ^"1 sin \iG„ sin u9 (12) 7TE^nSsiD ^en si
+ x n=l n=l
He then obtains the expression
mi t aiv = Vv "SV7n (13)
The prime on the summation symbol indicates that n = v is to be omitted from the summation because that special term has already been considered in the first term of the right-hand side (i.e., bvv7v). Reference k presents tables for the coefficients bvv and bvn for
mj_ = 7, 15, and 31. Applied to the problem at hand, m-|_ = 31 would
appear to be rather small; therefore a table for mj = 63 has been
computed and is included in the present report (appendix A). As a comparison: For n^ = 63, A0 = 2.8125°; for Naiman's method with
l60 points, A9 = 2.25°.
Utilizing this method of integration which was developed by means of Fourier series, an expression may be obtained for the velocity distribution as follows:
Ü* ,
The great advantage of this method is that of simplicity: (l) The actual computational procedure is very simple and (2) the derivative
vyt' is avoided. The simplicity of computation is reflected in the dx
fact that the time required for computing {=&■ at one value of 0Q vo is approximately half that required by the method of Naiman when the intervals have approximately the same size. It should be noted,
NACA TN 2451 11
however, that the accuracy of the method of Naiman will be greater than that of Multhopp In those cases where the differentiation of Ayt by- Fourier expansion (equation (12)) does not give good results.
A third method of evaluating the Poisson integral became known during the course of the present investigation. In a paper by Timman (reference 5), the integral is studied in the form
(0) = _ J, / ä(t) cot LzJL a* (15)
Timman assumes that ä(\|r) is not given analytically, but only at equidistant points. An interpolation polynomial (reference 6) for ö'fy) is employed, and these polynomials replace the function ö"(i|/-) in a single interval by a function of third order. The polynomial function thus introduced has a continuous first derivative,2 and it is evident that this continuity is essential for the attainment of good results.
Timman has divided the period 2JT into 36 intervals of equal length and established a computing scheme. The function a(ty) is separated into its symmetrical and unsymmetrical parts so that
aW = s + d (16)
Then
18 18 T(^)=z:-k2sk-2ZßkZ dk (i7) k=0 k=0
where the factors o^ and ß^ are given in tabular form. In the
d(Ay+) present particular case —*—^- is antisymmetrical (equation (3)) and
£%■ is symmetrical (equation (h)). Thus the separation indicated by vo equation (l6) does not require any additional work.
Timman's method should give good results provided that a sufficient number of intervals are taken - the division of 36 intervals over a period of 2it (i.e., 18 intervals over the chord of the profile) appears to be insufficient for an accurate representation of the function which
di^tl or ^ dx V '
^ 2The polynomials used in the classical interpolation formulas are less smooth (see reference 5, pp. 7 and 10, figs. 1 and 2).
12 NACA TN 21+51
The time required for computing one point by the method of Timman is approximately the same as for Naiman's method with the same interval
size.
Other methods of evaluating the Poisson integral have been suggested. They will not be discussed here as it is the intention of this section to consider only the most practical of the known methods. The three methods already discussed have their own particular advantages and have been especially developed for rapid and simple
°-(Ayt] AV computation; however, all three of these methods, when v^ °r ^~ s
change rapidly in magnitude, become cumbersome, and require that very small intervals be taken over the entire range of integration because the scheme of equal interval size is utilized.
EVALUATION OF POISSON INTEGFAL BY A METHOD
EMPLOYING UNEQUAL INTERVALS
Development of Method
As the change in airfoil shape, or the change in velocity distri- bution, is given originally as a function of x it appears logical to retain the coordinate x in selecting the size of the different intervals. Hence, the Poisson integral may be studied in the form
^ (18) T(X0) =-i / a(x)- 0 Ä - xo
which corresponds to equation (l). Conforming with its physical
meaning cr(x) = ^^ is assumed to be a function which is finite in dx _
every point of its range of definition.^
Define
Ax„ =
with
:n - Vt-l ~ *n
n = 0, 1, 2, 3,
(19a)
3This restriction will be dropped later; see discussion beginning with the first paragraph after equation (32).
NACA TN 2^51 13
and
x™ < x„ < x . m o ^ m+i (19b)
For convenience, there is chosen (see following fig.)
v - xm + xm+l 0 i (19c)
H 1—I 1 1 1- 7? r xm+l
The function a(x) is approximated by straight-line segments (see third sketch in preceding section). Then, for xn < x < x^.-i,
cr(x) = a(xn) + P(Xn+l) ~ g(*n)^x _ ^
Jn+1 Ax. n ~(
x " *n) (20)
from which there is obtained
Ik NACA TN 2451
K)--if>>HV--£ "n+1
*n + a , - a n+1 n
Ax n (x " *n) dx
x - x0 « X - x^ lxn
2 ,Xn+l an +
gn+l - V N —3x^ (x - xo + xo - xn)
dx X - x^
,xn
ifef2^) **n
I *n + an+l - an
AXn (xo " *ri) % x" x°/f
(21)
Also,
,xn+l dx X - X, - Jno
,xn (22)
by definition. The function jn0, in the different regions of x, is
given by different expressions as follows:
Xn+1 " X° loge _ __ for xn+1 > ^ > x0 xn " xo
<U = 1 loSe Xn+1 " xo xo - %
for xn+1 > x0 > xn (23)
l0ge X° "IT1 f°r X° > *n+l > Xn xo " *n
NACA TN 21*51 15
Introducing jno into equation (21), there results
T(xo) = - i JT X!N+I- °n) +X!
2>Jn0 + SK+l
«* + !l^1(Xo
\ / xo " *n ^
**) Jno
*n
Or, defining
(21*)
1 + X° " ^ i Jno Ax- •n Jno (25)
there results, finally,
•(xo) 2 anJno + ]>] (an+1 " M Jno' (26)
Since xn+1 = j^ + Axn, the functions jno and Jno* may be
written as
T — 1 -
Jno logjl + -^ xn " xo
for ^ > x0
Axn = loge(-l + x^ _" j for xn + Axn > x0 > ^
o ^n>
= loge(1 + **n
*n " xo/ for x0 > ^ + A^
(27a)
and this form shows that j and J...* are functions of *IL~ X° Jno only.
Axn
16 MCA TN 21+51
For *n Ax
_>±c ■n
For xo " xn
For very large
1 Ax n
Jno
Jno = 0
"no
Jno = 1
*n Ax
= 6, n
Jno
Jno
->i 1 6 2|2
26 31 + .
For very large negative xn." xo with Axn
xn ~ xo
A*n **,
dno % 26
+ .
Jno ' 21 31
*2 J
(27b)
(27c)
(27d)
These functions are given in figure 1 and in table I.
It is seen that -high absolute values of jno and jno occur x - xQ \
near those values of -~r which characterize the critical interval. Axn
Figure 1 give3 an idea of the characteristic qualities of jno
and jn0* as functions of ^JI—° j however, the representation is not Axn
sufficient for picking out values for a computation. Table I gives the
* _ "n ~ xc values of JnQ and Jno for -1+9.5 < ^ < 1+9.5. This table might
\f x = ^ + Xm+1 (equation (19c)) the critical interval is
given by -0. 5 < xn ~ xc A^
< 0.5-
NACA TN 2451 17
be used for rough computation and for getting acquainted with the method. In general, it is advisable to use those tables which are given in appendix B.
It will prove of benefit to investigate the exactness of that portion of the integral which contains the singularity. Recalling that the function a(x) was replaced by a straight line in every interval (equation (20)), there is obtained:
tr(x) x <lx = - -(CTm+l - CTm) (28)
if x < x < x ... Now, let an expansion of the function cr(x) in
the critical interval around x0 be assumed as follows:
a(x) = a(x0) + a»(x0)(x - x0) + ^"^(x - x0)2 +
-T^i* " xo) + "TH^* " xo) (29)
Then,
dx = :.M«.^mf. 3.' 312 }
lf(am+l " CTm) " ^(ffBi-f2 - Vl)
" \ llsN+l - v )
1_ 36 K+2 " <W)
+ (ffm " Vl)
Comparison of formulas (30) and (28) shows that the error in the critical interval is approximately given by
(30)
18 NACA TN 2^51
nliH^nH-l °m) ' js (Gm+2 " am+l) + {% " ffm-l) (3D
The error of evaluating the whole integral "by finite differences may- be estimated by using two different interval distributions and comparing the results for a given xQ.
However, in addition to that error of the result produced by replacing the Poisson integral by a sum there exists another error. This sum cannot be computed exactly, but has a certain error depending on the accuracy of the given data for an(x) and the tabulated values
of jn0 and j no As the function a(x) =
■3 dx
usually has an
error of en = 1 X 10 J it has proved amply satisfactory to give j no and j to four decimal places, the error being less than
5 x 10 . The error of no
(Xo) 1 it 2><W> +5ZN+I " a^^°
is smaller than its upper limit given by
,fsr«u„i + 2 e \^
Jno 'no + 6, er -, - a j n+1 n )"
(32)
This formula shows that the influence of e^ is stronger than the
influence of e2 as long as \ '|orn| + \ '|ffn+1 - on\ is smaller
than 1 - as it is in our later examples - and the sums / i |,Jno|
and ;? , j 0 are always larger than 1. An increase of subdivisions
makes the sums in the upper limit of the error (32) grow, thus requiring a higher accuracy, especially in a and perhaps also in the values
of Jno and Jno*.
In establishing the solution of equation (18) it was assumed that cr(x) is finite throughout its range of definition. If it is desired to compute the change in shape due to a proposed change of velocity distribution, this restriction must be eliminated, as will be recognized immediately.
NACA TN 2451 19
Equation (5) may be written in the form
ax -*\/Mc xo)Jo Np__Tx_Xo (5)
Omitting the factor \JxQfc - xj) , which doe3 not affect the integration
process, the integral may he reduced to the form of equation (18) by defining
AvAc \/x(c - x)
= (Ti(x) (33)
However, cr-^x) will be infinite at x = 0 and x = c if \§r~\ ^ 0 /Av\ , ^ °'°
and (rr—) f 0; therefore, a special consideration of the neighborhood
of x = 0 and x = c is required. This is done by splitting the integral into the following three parts:
CTnlx) = (Mx) + / CT-,(x) /O x " xo Jo x - xo /€l
x - xo
a (x)_i£L_ (34) x - x0
'C-e2
with e, and e being small compared with c» The integral
2 / \ dx x - x0
el
may be treated as was explained formerly for
°a(x)- to
(0 x - x^
20 NACA TN 2^51
(see equation (l8)) because a±(x) is finite for e1 < x < c - e2>
For the first and third integrals, however, a new integration formula must be developed. By introducing
|i = c - x and CT-LU) = a1 M-(x) = a-ffo)
there is obtained
,c-62 Hence, the method used for the first integral will also apply to the
third. In most cases (—j will be zero and there will be no need Vo/c
for a special evaluation in the neighborhood of x = c.
The integral
f1 ffl(x)-£L_ = f'1 Wo -*L- (36) Jo x " xo JO \jx(c - x) x - x0
will have an important influence on the result of equation (3*0 only if x0 is near to ei- First the general formula will be given and
then a simplification will be discussed for x0 » €■]_.
The integral (36) will be solved assuming that
fo = a0 + ai(|) + a2(ff for 0 < x < e, (37)
Only the final formula of this procedure is given here; the details of the solution will be found in appendix C.
NACA TN 2l*-51 21
-W 61 Av/V dx
'O \/x(c - x) x - x0 c ' °
*W 2 + 2 /?
a„ + a. ■*/ Äo
al + a2 */xr 2 * 0 (38)
with MQ given in figure 2, and
a~ = _ /Avi V o/o
* 1 al = ai + " ao
with
al = 26l -(Ü + *®. - (C
a2 + 2 al + B a°
with
2e, ta - 2©61 ♦® 26,
(39)
The coefficients &Q, a-,, and ap may be determined first, as
they do not depend upon the particular value of xQ, and then Fj_(x0)
may be computed. The term depending on a, and a will exert an
influence only for small values of xQ/c. After a brief training the
computor should be able to decide rather accurately when the formula
*iM-i*-k->f>HS) (Uo)
is sufficient and when the more exact expression (equation (38)) is required (also see fig. 2).
22 NACA TN 21*51
Organization of Computational Procedure for Unequal Intervals;
Transition from One Size of Interval to Another
A thorough understanding of the method is best achieved by following through a rather simple example; in addition various short cuts to the method will be demonstrated.
Assume a function a(x) of the type shown in the following figure.
a(x)
It appears reasonable to take raxher small intervals for small values of x because of the form of the curve tf(x); therefore, the following arrangement of interval sizes is arbitrarily selected:
Äx = 0.002 for 0 < x < 0.030
Äx = 0.006 for 0.030 < x < O.O96
Compute T(0.009) with the help of equation (26). Note that the critical interval extends from 0.008 to 0.010. Table II(a) gives the values of x/c, an, an+1 - on, Jn£), and Jno* for the range
with Äx = 0.002. At x = 0.030 the interval changes to Ax = 0.006 and the same functions are given for the range with this size of interval in table II(b). Naturally the range above the broken line in table II(b) is not utilized in the computation since this portion has been considered in table II(a).
NACA TN 2^51 23
X - XQ Note that —- progresses in table II (b) in the same manner
as in table II(a); this is due to .the special choice of Ax. If
Ax = 0.006 were used starting with x = 0 the critical interval for x0 = 0.009 would extend from 0.006 to 0.012. Hence, for Äx = 0.006,
jno = 0 and jno =1 is to be found at x/c = 0.006.
For rapid computation it is best to have jno and jno as x« — x
functions of —- on a paper strip and to place this strip adjacent ZAX
X — X to the columns headed by an and crn+]_ - cr . If n. ■—2. progresses
as indicated in table I, the correct location of jn0 = 0 and jno* = 1
at the beginning of the critical interval fixes the placement of the strip.
In the example just treated, the transition from one size of interval to another is very easy because xQ lies at the midpoint of an interval of the size 0.006 as well as of the size 0.002, if starting with x = 0.
If Ax had been chosen 0.004, such a desirable arrangement would not have resulted because xQ = 0.009 would not be located at the mid- point of an interval of this size (starting with such intervals at x = 0).
As a second example compute the value of T at xQ = 0.015.
^n ~ xo ^n ~ xo Again, —— and —— will progress as in table I. The values
Ax ^
Jno = 0 and Jno* = 1 will be placed opposite x/c = 0.014 for the
region with Ax = 0.002 and opposite x/c = 0.012 for the region
with Ax = 0.006. As long as Ax = 3Ax, Ax = 3Äx, and so forth and if xQ is chosen so as to be at the midpoint of the largest size of interval, the computation may be accomplished by shifting the strip with jn0 and jno* corresponding to table I.
But suppose that the interval sizes are so arranged and it is desired to compute a point where xQ does not lie at the midpoint of
the largest size of interval; for example, xQ = 0.013. The value
x0 = 0.013 lies at the midpoint of an interval with Äx = 0.002; hence,
2^ NACA TN 2^51
for the range 0 < x < 0.030, jno and Jno* may be taken directly
from table I. However, at x/c = 0.030, intervals of the size Äx = 0.006 commence and there is obtained
Xn - xo 0.030 - 0.013_ = 2<833 Äx 0.006
The value of — progresses by 1, that is, 2.833, 3.833, Ax
U.833, .... Thus the functions jno and Jno* are needed for
values of ^ ~ X° which are not given in table I. One might think of Ax
taking them out of an enlarged diagram (see fig. l); however, it is much more convenient to take them out of an extended table, which is conveniently arranged for "advancing by 1." Such tables are given in appendix B.
The example presented by the figure at the beginning of this section suggested starting at x = 0 with the smallest intervals. However, other examples may suggest another distribution of intervals. The smallest size of intervals may lie at any part of 0 < x < c. There are no restrictions in the arrangement of intervals. (See, e.g., discussion following equation (^3).)
Accuracy of Method, Examined by Means of
an Analytical Example
The accuracy of the result depends directly upon the size of the interval taken and the reliability of the data comprising the func- tion ff(x). Because the function ff(x) will be replaced by a broken line, a glance at the curve will quickly suggest an arrangement of intervals. In addition, the error in the critical interval may be used as a first test of the choice of intervals.
As a test of the quality of this new method, involving unequal dCAyt) intervals, a function cr(x) = — has been treated which allows
dx the analytical computation of T(X) = ^ .
"o
NACA TN 2^51 25
The function cr(x) is given analytically as
0 ^ x < 2A ^ "^ = Bx(2A - x) dx
dl Ot) _ 2^< x< c rVJ^IL = _D(Cl - x)(x - 2A) x dx v '
Cl<x<c ifel.o 1 dx
The following arbitrary values have been selected:
c± = 0.35
c = 1.0
rCl d(Ayt) D = Some multiple of B so that / —- - dx = 0
Jo dx
d(Ay+') / x / x The functions Ay+ and ——**- are given in figures 3(a) and 3(b),
z dx respectively.
26 NACA TN 21+51
The analytical computation of Av
for figure 3(b) is given in
figures U(a) and U(b). The arrangement of the unequal division for the
numerical computation of ==%■ is indicated in figure k(a). "o
5 It was desirable to obtain the value of ^ at xQ = 0; hence, vo the first interval has been placed so that -0.001 < x0 < 0.001 and the
function dx
0 for -0.001 < x < 0. Since the function Lfot) dx
is replaced in every interval by a straight line, the error might be expected to be large. However, g = 0 at xQ = 0 will aid in
.preventing the error from being too large.
A more exact solution would be obtained by putting
g = 0 Ax n — < x < 0
Ax 2
x 0 < x < Ax
dx i&Xl
For the interval - — <x <^£ equation (26) would yield
'Ax/2
- Ax/2
dx 1 jt
(gl - 0 ) Jno* + ° X Jno = - -gi
and no error is introduced. For xQ ^ 0 there is a very small error
which may be avoided by respecting the change of size of the interval
near x = 0.
NACA TN 2^51 27
Also given in figure 4(a) are points of the ^ curve determined by *o
the method of unequal intervals. Figure k(b) presents the same information plotted to a larger scale.
For comparative purposes the same problem has been treated by the three methods of computation discussed earlier, namely, those of Naiman, Multhopp, and Timman. Figures 5(a) and 5(b) show the results obtained by the method of Naiman; obviously, the 40-point solution does not use
a sufficiently accurate representation of the \ ^) curve, while the dx
80- and 160-point solutions are quite good, with the exception of the
maximum and minimum points of the &£ curve. In order to obtain a »o
value at approximately £ = O.O36 a solution involving 320 points would
be required. In this respect the method of unequal intervals is more adaptable to special conditions without involving much new work than is the method of Naiman.
The results obtained by Multhopp's method are given in figures 6(a) and 6(b). The 31-point solution (in Multhopp's somewhat odd manner of designation) corresponds to A9 = 5-625°; the 63-point solution, to A0 = 2.8125°. The computation is very simple and the results of the method with 63 points are comparable with that of Naiman with 80 points, with the exception of those near the region 0< x< 0.01 (this is
shown most clearly in fig. 6(b)). The very steep peak of V^W cLx
at x/c = 0.02 requires rather high harmonics for the representation of Ay^; consequently, good accuracy in the region near the origin may not be expected. This is substantiated by the fact that for the 63-point method the highest effective harmonic would have three waves in the region 0 < x < 0.04; obviously a sufficient degree of accuracy in the differentiation process cannot be obtained.
As mentioned earlier, Timman's method might be expected to give good results if the size of interval is properly chosen. Inasmuch as
only a table for A0 = ^- = 10° was available, the result of the
computation for 0£ cannot be expected to be good, as is evidenced bv
observing figure 7. The result obtained is comparable with that of Multhopp's 15-point and Naiman's U0-point solutions.
An excellent method of examining the accuracy of these methods still further is simply that of solving the inverse problem. From the
Av d(Ayj-\ curves of — just discussed, values for v ^ have been computed
28 NACA TN 2^51
and are presented in figure 8. The method of unequal intervals gives good results, indicating that the arrangement of intervals chosen was as good for the inverse problem as for the direct problem. It is apparent that Naiman's method requires even smaller divisions than l60 points in order to avoid inaccuracies near the point x/c = 0.04.
The reader may wonder that the inverse problem is not given by Multhopp's method. It must be recalled that Multhopp's method of solving the direct problem does not involve the differentiation of Ayt;
that is, it is particularly fit for this problem and presents, on the other hand, no analogy for the inverse problem:
dx 2* J0 V0 """ 2 ~ rt J0 V0 coa 0 - cos 90
Because more extended tables for Timman's method are not available, and the results obtained from the 36-point method for which tables exist are very poor, no further examples of the application of this method will be given.
COMPARISONS OF METHODS OF NAIMAN AND MULTHOPP WITH METHOD EMPLOYING
UNEQUAL INTERVALS BASED ON ACTUAL EXAMPLES OF CHANGES IN
AIRFOIL SHAPE
The method of unequal intervals has shown good qualities when applied to a problem where the function cr(x) is known analytically. However, as mentioned earlier, this function is not usually known in analytic form. This section, therefore, will compare the three principle methods, those of Naiman, Multhopp, and unequal intervals, on the basis of actual design problems, solving the direct problem
for — and using these results to solve the inverse problem (excluding *o
Multhopp for the inverse problem).
Figure 9(a) shows the Ayt relations for examples I and II and
a(Ayt) n „ R^t)! figure 9(b), the V^ ' relations. Note that the slope of [_ ^ J
for example II is more than twice that of example I near x/c = 0.
NACA TN 2^51 29
The direct problem for example I by Naiman's method is given in figure 10. The l60-point solution does not show any appreciable
deviation from the 80-point solutions at the region of [§jr-) ', \ o/max
however, near the origin, at (Tr-) > the influence of the smaller-sized V o/min
intervals (8O:A0 = U.5°; l6O:A0 = 2.2^°) is quite pronounced.
The solution by Multhopp's method is given in figure 11; 31 points around the half circle are not sufficient for a solution comparable with Naiman's 80-point solution, and even a solution based on 63 points does not offer much improvement. The results are poor, as might be expected, in the region very near the origin (see preceding section).
Figure 12 presents the results obtained by the method of unequal intervals, compared with results obtained by Naiman's 80- and l60-point solutions. The method of unequal intervals gives results corresponding to those established by Naiman's l60-point solution. The subdivision used is shown in the figure.
As before, the inverse problem was solved, and is given in fig-
ure 13. In each case the computed curve of ^- was the one used in
obtaining the values for the —^—=^- curve. Both methods give good dx
results, thus proving that the chosen number of divisions was sufficient in Naiman's method and in the method employing unequal intervals.
<l(Ay+) , The value of —^—^- computed at x/c = 0.171 is of some interest,
dx This point was computed by the method of unequal intervals in two different ways: First, the arrangement of intervals shown in figure 13 was utilized to compute the lower point. Then a new arrangement of intervals (Ax = 0.018 for 0 < x < O.36) was set up and the same point computed. The idea was to determine 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 ®L} errors of
''o only small magnitude would be introduced; this is fairly well sub- stantiated by the results shown in the figure because the error thus introduced is approximately that of the deviation of Naiman's 160-point solution.
Recall that this method does not involve the differentiation of Ayt.
30 NACA TN 21*51
Now, turning our attention to example II, which, it will be
recalled, has a slope of —^—^i. of approximately twice that of dx
example I, the results given in figures Ik to 17 are obtained.
For the direct problem Naiman's method of l60 points and the method employing unequal intervals give results which are in good agreement. For the inverse problem (fig. 17) it is apparent that the
method using unequal intervals is superior (see the deviation at
yyV given by the method of Naiman). Multhopp's method gives to -W / a rather good result (fig. 15), which may be attained when the Fourier representation of the Ay curve is adequate.
The two examples thus far presented are favorable for Naiman's method because the steep slopes of cr(x) occur near x = 0 where the points Naiman uses are close together. However, going to still steeper slopes near x = 0 would require a rapidly increasing number of points. The new method offers another possibility here. Assume that in that critical region xk < x < xk+1 (xk may be 0) <x(x) may be represented
by cr(x) = ^ anxn. Then the integral
l0 * - *o
may be split into three integrals
(hi)' /xk+1
x - xo
The first and third of these integrals may be solved in the usual manner using the functi
be solved analytically.
manner using the functions jno and Jno*. The second integral will
NACA TN 2451 31
This simple form, due to the use of the coordinate x in the Poisson integral, allows a rapid integration, because the integral
can be solved by recurrence as follows:
n n kn,o = "^ — + x0kn_x 0 for n £ 1 (k2)
with
n
^° = Jno(xXk " X; 1 (*3) vxk+l " xk/
Thus even very steep slopes cause no difficulties.
As already mentioned, examples I and II correspond well to the qualities demanded by Naiman's method insofar as the rather steep slopes occur in those portions where the points 9n are close together. If those steep slopes should occur in other portions of the chord, how- ever, a very great number of points in the Naiman method would be needed in order to represent c(x) adequately, and to get reliable results. In such a case the method using unequal intervals shows its advantage by allowing a free subdivision of the chord.
A third example will serve to illustrate this. Figure 18 shows a
function cx(x) = -i—£1 . The essential values of the function lie in dx
a part of the chord where even Naiman's method with l60 points is not sufficient to represent the function accurately. This is forcibly
shown by the two curves of ^-. If the function a(x) is modified vo
(dotted line) so as to eliminate the high peak, then the —- curve by ^o
unequal intervals can be made to agree with the original ^ by Naiman's ^o
l60-point solution, thus definitely proving that, in this example, Naiman's method with l60 points is insufficient.
Table III indicates the computation for the point x = O.065 by unequal intervals.
32 NACA TN 21+51
CONCLUDING DISCUSSION
The new method of evaluating the Poisson integral developed herein is to be recommended for all those functions ff(x), where steep slopes in small portions of the region to be integrated exist. In these portions a very small size of interval may be chosen without requiring that this same size of interval be used throughout the region of integration. In this manner, the work required for computation may be maintained at a reasonable level even for the most complicated problems.
The analytical treatment of special parts of the integral is possible (evaluating the remainder by the new method; see preceding section). In those problems where a transition to very small intervals in part of the integration range would require the determination of a great many values of an, this idea might be used to advantage.
It should be noted that the smoothness of the function a(x) and its accurate representation by single points is essential for good results. If, for example, single points crn are simply taken from a curve for xn very close to one another it may be compulsory to check
these values by a table of differences.
Stanford University Stanford, Calif., December 6, 1950
NACA TN 2^51 33
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NACA TN 2^51 35
APPENDIX B
VALUES OF j no AND jn0* AS FUNCTIONS OF Ax
The values of the functions JnQ and jno* are presented as indi- cated in the following table. The values are tabulated in a form selected to minimize the necessity for interpolation except for the region contain- ing the singularities of the functions j and 1 * Jno For ease in computation, tables B-I to B-VIII, inclusive, are arranged so that the vertical increment of
xn—x -£- is unity. Table B-JK gives additional values
for the region containing the singularities of the functions J 'no
Jno and
TABLE NO. FANGE OF Xn X° INCREMENT OF Xrr*xo Ax Ax-
B-I -I89 to -90 1.0 B-II -89.5 to -40.0 • 5 B-III -39-9 to -20.0 .1 B-IV -I9.99 to 0 .01 B-V 0 to 19.99 .01 B-VT 20.0 to 39.9 .1 B-VII 40.0 to 89.5 .5 B-VIII 90 to I89 1.0
B-IX -1.000 to 0.000 0.001
36 NACA TW 2^51
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APPENDIX C
DETAILS OF SOLUTION OF INTEGRAL (36)
Fn = v0 dx = 1 / Av _1 L J£SL
10 Vx(c - x) x - x0 - JO V0 Vf ^1 - J J - ^ (Cl)
Introduce | = - and find c
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I - ^r (C2)
With the expansion (see equation (37))
#Mi + ii +^ 2 b 8 ) = ao + (al + § ao) S + («2 + § ai + t ao) *'
* * 2 = aQ + a1 | + a2 | + (C3)
Hence,
-' a„ + *t2 1 / 1' aQ + a1 I + ag*|
°Jo VT (ft - to) d|
= 1 c
'el/c
dl
I- d|
+
,el/c
1 Jo >TT(e- £0)
>/!(* - to)
(CU)
NACA TN 2451 55
As the occurring integrals are all of the same type, define
Tl/C ^« La= / Jn \ N (C5) JO ^/T(£ - l0)
These integrals 1^ are easily solved by recurrence.
Ln = ^ L., + X-^~Y (06) 2
with
1-W^ Lo = "—= !oge ;== for xo > «i (C7)
and
xo
^Ht L°= T7r loge fe4 for xo < ei (cQ) V5° 1+ ,fo ±
The function
1 - \- 1 - i/^a MQ = loge La and loge V 1
is given in figure 2 in order to provide a more rapid computation in x g..
the event that — or -± is not very small. 1 xo
% NACA TN 2451
€1 If — « 1,
Mo = •2h,- + H^ + Ä. Ij_±. +
5 Vx0 (C9)
The integrals L0, L1} and L2 are needed; these are given "by
Ll = ^oMo + 2#
3/2 °/M# fci If -i « 1, xo
Lo =
, --3 ,-5 fi + i./ii +i,/ü +
£oWxo syxo 5Wx0
^--w^-^ +
L2 = -2 ^(jß: + .
-/
(CIO)
(Cll)
NACA TN 2451 57
With these expressions the integral F-, is as follows:
F, = J[aoLo + ai*Li + a2*L2
-^ Mo + *!* (\^o" M° + 2^Sj + a2* ^3M0 +
2*o# + f#3) Mr
a° + a *\/t + a *\/L ,+ 1 V ° 2 V °
Vf(^ * ^*) * f ^ (C12)
The coefficients a0, a-,, and ap of the expansion of — are given Av ^o
/Av
2e.
^k
x=0
/Av) + k
'x=0
Av
X=€i ($ x=2e. >
a2 = 2 6, -) -2(r) +(r) o/x=0 v?/x=€l \
vo/x=2e.
Av V,
(C13)
58 NACA TR 2^51
REFERENCES
1. Allen, H. Julian: General Theory of Airfoil Sections Having Arbitrary Shape or Pressure Distribution. NACA Rep. 833, 19^5«
2. Naiman, Irven: Numerical Evaluation of the 6-Integral Occurring in the Theodorsen Arbitrary Airfoil Potential Theory. RACA ARR L.4D27a, 19Uh.
3. Naiman, Irven: Rumerical Evaluation by Harmonic Analysis of the e-Function of the Theodorsen Arbitrary-Airfoil Potential Theory. RACA ARR L5H18, 19^5-
ij.. Multhopp, H.: Die Berechnung der Auftriebs Verteilung von Tragflügeln. Luftfahrtforschung, Bd. 15, Nr. h, April 6, 1938, pp. 153-169-
5. Timman, R.: The Numerical Evaluation of the Poisson Integral. Rapport F.32, Nationaal Luchtvaartlaboratorium, June 16, 1948. (English)
6. Schoenberg, I. J.: Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions. Part A.- On the Problem of Smoothing or Graduation. A First Class of Analytic Approximation Formulae. Quart. Appl. Math., vol. IV, no. 1, April I9U6, pp. 45-99; Part B.- On the Problem of Osculatory Interpolation. A Second Class of Analytic Approximation Formulae. Quart. Appl. Math., vol. IV, no. 2, July 1946, pp. 112-141.
NACA TW 2i+51 59
TABLE I.- VALUES OF jn. AND j no •Jno FOR -I4-9.5 < x -
n Ax < 1+9-5
*n - *o * xn " xo . #
Ax ^no Jno Ax <lno Jno
-1+9-5 -O.O20I+ -0.0102 • 0.5 I.O986 0.1+507 -1+8.5 -.0208 -.0101+ 1.5 .5108 .2338 -Vj\5 -.0213 -.0107 2.5 • 3365 .1588 -1+6.5 -.0217 -.0109 3.5 .2513 .1201+ -1+5.5 -.0222 -.0112 1+.5 .2007 .0970 -1+1+.5 -.0227 -.0111+ 5-5 .1671 .0812 -1+3.5 -.0233 -.0117 6.5 .11+31 .0698 -1+2.5 -.0238 -.0120 7.5 .1252 .0613 -1+1.5 -.021+1+ -.0122 8.5 .1112 .051+6 -1+0.5 -.0250 -.0125 9.5 .1001 .01+92 -39.5 -.0256 -.0129 10.5 .0910 .01+1+8 -38.5 -.0263 -.0132 11.5 .0831+ .01+11 -37.5 -.0270 -.0136 12.5 .0770 .0380 -36.5 -.0278 -.0139 13.5 .0715 .0353 -35.5 -.0286 -.011+3 11+.5 .0667 .0330 -3l+.5 -.0291+ -.011+8 15.5 .0625 .0309 -33.5 -.0303 -.0153 16.5 .0588 .0291 -32.5 -.0313 -.0157 17.5 .0556 .0275 -31.5 -.0323 -.0162 18.5 .0526 .0261 -30.5 -.0333 -.0167 19.5 .0500 .021+8 -29.5 -.031+5 -.0173 20.5 .01+76 .0236 -28.5 -.0357 -.0180 21.5 .01+55 .0226 -27.5 -.0370 -.0186 22.5 .01+35 .0216 -26.5 -.0385 -.0193 23.5 .01+17 .0207 -25.5 -.01+00 -.0201 2I+.5 .01+00 .0199 -2I+.5 -.01+17 -.0210 25.5 .0385 .0191 -23-5 -.01+35 -.0219 26.5 .0370 .0181+ -22.5 -.01+55 -.0229 27.5 .0357 .0178 -21.5 -.01+76 -.021+0 28.5 .031+5 .0172 -20.5 -.0500 -.0252 29.5 .0333 .0166 -19.5 -.0526 -.0266 30.5 .0323 .0160 -18.5 -.0556 -.0280 31.5 .0313 .0155 -17.5 -.0588 -.0297 32.5 .0303 .0151 -16.5 -.0625 -.0316 33.5 .0291+ .011+6 -15'. 5 -.0667 -.0337 31+.5 .0286 .011+2 -1I+.5 -.0715 -.0362 35.5 .0278 .0138 -13.5 -.0770 -.0390 36.5 .0270 .0131+ -12.5 -.O83I+ -.01+23 37.5 .0263 .0131 -11.5 -.0910 -.01+62 38.5 .0256 .0127 -10.5 -.1001 -.0509 39-5 .0250 .0125 -9.5 -.1112 -.0567 1+0.5 .021+1+ .0122 -8.5 -.1252 -.0639 1+1.5 .0238 .0118 -7.5 -.11+31 -.0733 1+2.5 .0233 .0116 -6.5 -.1671 -.0859 1+3.5 .0227 .0113 -5.5 -.2007 -1037 1+1+.5 .0222 .0111 -1+.5 -.2513 -.1309 1+5.5 .0217 .0108 -3.5 -.3365 -.1777 1+6.5 .0213 .0106 -2.5 -.5108 -.2771 1+7.5 .0208 .0103 -1.5 -I.O986 -.61+79 1+8.5 .0201+ .0101 -.5 0 1.0 ^9.5 .0200 .0100
6o NACA TN 2^51
TABLE II.- COMPUTATION BY UNEQUAL INTERVALS, TRANSITION FROM
ONE INTERVAL SIZE TO ANOTHER
(a) Äx = 0.002.
X
c an an+l " an
Xj! - X0
Ax Jno 1 * dno
0 .002 .004 .006 .008 .010 .012 .014 .016 .018 .020 .022 .024 .026 .028 .030
a0 01 a2 a3 oh ff5 o6
oik a15
al - a0 . °2 " al (To - OV)
ah - ff3 0-5 - oh 06 - 0-5 a7 - aß
CT15 " al4
-4.5 -3.5 -2.5 -1.5 -.5
.5 1.5 2.5 3.5 4.5 5.5 6.5, 7.5 8.5 9-5
10.5
-O.2513 -.3365 -.5108
-I.O986 0 I.O986
.5108
.3365
.2513
.2007
.1671
.1431 ,1252 .1112 .1001
-O.I3O9 -.1777 -.2771 -.6479 1.0
.4507
.2338
.1588
.1204
.0970
.0812
.0698
.0613
.0546
.0492
(t>) Ax = 0.006.
X
c an ffn+l " an
Xn - XQ
Ax Jno Jno
0 .006 .012 .018 .024
a0 0-3 06 a9 012
a3 - a0
a6 - ff3 a9 - a6
OJJ2 - 0-9 CT15 - o12
-1.5 -.5
.5 1.5 2.5
-I.O986 0 I.O986
.5108
.3365
-0.6479 1.0
.4507
.2338
.1588
.030
.036
.042
.048
.054
.060
.066
.072
.078
.084
.090
.096
"15 ffl6 ff17 "18
ff25 ff26
ffl6 " CT15 tj17 - a±6 al8 - cr17 ff19 " al8
o26 - CT25 ff27 _ ff26
3.5 4.5 5.5 6.5 7.5 8.5 9-5
10.5 11.5 12,5 13.5 14.5
.2513
.2007
.1671
.1431
.1252
.1112
.1001
.0910
.0834
.0770
.0715
.0667
.1204
.0970
.0812
.0698
.0613
.0546
.0492
.0448
.0411
.0380
.0353
.0330
NACA TN 2^51 61
* o ö
CO H t— ON >- t- ON O t>- H -4" IfNOCOooOLfNOJONOO VOO-4-OOHHHHHOO
OH I
O Ö
,|_3
f
^1
bD •H
cd
lltf
luf
ON ON o * H
I
ONonVT)00(Y-)\£)HHt— ONONOCO OJ mir\ov3 OVÜ4- (M C\l CO W C\l H
O H
LfN IfN IfN IfN • • • • OJ 0O-4- u~\
o H O • o
II
Uf
o o o o • • • ■
H OJ 0O-4-
IfN o o
113
ITN ITN IfN • • • H I
I
CO no O o
13 t?
+
oö
K|o
IfN IfN O t-HVD IAOHVOt^04-ODHH OHOJlfNOOONOOO OOOOHHOOOO
I
IfNO O t^M3 IfN IfNVO OJ O O VO OJ H OHOOONOOOOO OOOOOJHOOO
I I O
oo r— OOOMDOlfNOlfAOOOO VTJVOMDt—l>-OOCOONOHOJ OOOOOOOOHHH
62 NACA TN 21+51
4.0
| \ 3.0
\ 2.0
/ \ \ < 4 h»
•'", ■
XJ 0
1.0 P" \ \ *>
1
/ *
"—— —* -.—
/•»• o 1 1
-1.0
"" ■ 1 1
V. **
\l -2.0
—*S \ \ \ \ \ \ \ i \ i i \
-3.0 i -5i?s5
-40 3 -1 ? i _ 5
*»-
0 5 i i.
iiX
Figure 1.- Characteristic qualities of jno and jn0* as functions
of xn ~ xo
Ax
NACA TW 2^51 63
©r 6i
Figure 2.- Function MQ for computation when x^e-^ or eWx,., is not small.
6k NACA TU 2U5I
z?
K 1 < ►
s.
*
X)Ü
- (vJ
'S
O
>> o
fi u u CO
M Ö
•H -P ra <u
-P
U o
<H
<U H
§■ a;
CO o
•H
H
1 m
•H PR
tf cJ
NACA TN 2^51 65
X\u
a o ■H -P O
"2"
66 NACA TN 2^51
(a) Plot for 0 < x < 0.25.
Figure k.- Analytical computation of ^ for figure 3(b) and comparison
with results by computation with unequal intervals.
NACA TN 2^51 67
& Unequal Intarvals
/Analytical
(b) Part of figure 4(a) plotted to larger scale.
Figure k.- Concluded.
68 WACA TW 21+51
.Otfc.
(a) Comparison with analytical results.
Figure 5.- Results obtained by Naiman's method. 40-, 80-, and l60-point solutions.
NACA TW 2^51 69
.02© Solution I
@ Naiman 40-pomt O Naiman 80-point • Nairnqn 160-point
(b) Figure 5(a) plotted to a larger scale.
Figure 5.- Concluded.
70 NACA TN 2^51
.Ctt.O
Solution £ Multhopp 31-point / Multhopp 63-pomt o Naiman 80-pOint
Multhoppil: -i.e= 5.62.5'
t— Mu|-thopp63: T A9 = ZA\tS
-.016
.025
,010
.015
.010
dx
.005
.20 .24
(a) Comparison with analytical results and results of Naiman's method.
Figure 6.- Results obtained by Multhopp's method. 31- and 63-point solutions.
NACA TW 2^51 71
.ozo
.0\G>
.OI2.
.006
.004
V.
.004
-.ooa
-.oia
-.016
Solution ' ¥ <f Multhopp 15-point
4 Multhopp 31-point </Multhopp 63-point o Naiman 80-point>
(b) Figure 6(a) plotted to larger scale.
Figure 6.- Concluded.
72 WACA TN 2^51
.0Z.O Solution ® Naiman 40-point
</* Multhopp 15-point X Tim man 3&poirrc
Figure 7.- Comparison of methods of Nairan, Multhopp, and Timman with analytical solution as basis.
NACA TN 21+51 73
aCto
1h NACA TW 2^51
et I c ) 1
3 < 9 ?
H K *. vO <5 rJ -
<0 o 0
7
J /
/
/ '
X ^
,^
^
/*>
a
/
'
u.
^ \
/
/ .
/ / K
"^^
i c »
"— — -
. Ti ^■^ a X cd
S./
,-p H o N £ ra
H M f*
Xlü g crt •H X,
>fi +5 w 1
ON
^-^ <L cd M
(VJ S • •H
s o (0 o O
I 2>|duj-oxz> JOJ. ^Xv
NACA TN 21+51 75
Z aidujvxg JOJ. ^~
o tvJ
00 o o
0 \
<4 ■Z.J
\
\ U)
*
1
*l< >
^ I 1
n I E d
'/ c 2 91
' 3. c
■0 "
3 C 5 / /
//
N
a F
E M
0 UJ
^.-"
ts-
X|U
o O o M o r
j d/duimd joj. (^v)p
ü
o Ö o
•H -P O
cd
-d
-d <D
d H o Ö o u
<u u d to
■H
76 NACA TN 2^51
xlo
o -p
a
a •H
CO
H &
0)
O In
a <D H rQ O
ft -P U 0 U
•H o I
o
•H
NACA TN 2^51 77
o -p
ft ft o -p H
H ft
O
a Q) H ,Q O U ft
-P O
78 WACA TW 2^51
w w H Ö cd O
fc? CM <u 3 <r> -P H " Ö o
•H CQ
H -P CO Ö 0 -H
on O1 O <M <U ft
Ö 1 P o
vo CH H o
-d -Ö Ö
«- o 3 PJ A • -p i
<D O a co >> w P -
o CVI M 3
co !>5 X P 0) «o -d « M <u o ö
CH -H CO
xlu -0 -P <U ,0 Ö o
N ■H cd to » -p -p
£> H O 0
w w cu •P fn H <o P Ä o CO p • d) -H « I*
i -d • <u
■*
cu M H 3
o & <D Ö *H O Pi u
(in
NACA TN 21+51 19
xi u
CM & ^ 10 g J>> a ^
* 0) T<
(1) ^ Ö O -H
<<H d
oO -p
a; 0 ö
•H CO CO -p
+3 H rQ P O CO
<U ■*- W ^ CVl •P
d & P -p W -H <D > K
tl J <U • u • 0 0 Co to 01 <U ft Ö - H Ö O
,Q O -H O U -p ^ P ft ra H
H O 0) cd to
<o CO > U U -p . <u a> a t> -P -H 0 0 0
•H -H ft
«in H O 0 cö \o
P H Ö a< O (Bti " •H Ö Ö -P P cd P H <H 1
XIO 000 CO CO
T* 1 O w
d> . .c - q en -p ö H 0 ro a a
<U -H h >i CO
ä'Q B •H
* pq
80 NACA TU 2^51
o -p CD
CD
CO
M
a; H &
(D
O
a CD H
■s ft -P O (D
•H
<D
g) •H
$*
11 NACA TN 2^51 81
o o o
00 0
1 |
5i"
\=&o
o •p
ft ft o
X! -P H
S >>
H ft
0
o
a 0) H
ft -P u 0) fH
•H P
m
(^
I
82 NACA TN 21+51
f \
\ p
p iO C S d Q o > °-
\ \
O 0 +f CD
c o a
1
o a 3 c co F T -
- ^ rt rt r u 1 1 2: =5 ^
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2 NACA TW 2i+51 83
[—
ej <0 * ^ a o O X ^ -o
O r
8U NACA TN 2^51
1 1 i 1—1—1—1—1 1 1 1 Unequal Intervals
— 1 Naiman 80
1 Naiman 160
Solution A Unequal Intervals • Naiman 160-point
Figure 18.- Results obtained for example III by method of unequal inter- vals compared with, results obtained by Naiman1s 80- and l60-point solutions.
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