U3

-J

.

FOR AERONAUTICS

TECHNICAL NOTE 2744

PRACTICAL CALCULATION OF SECOND-ORDER SUPERSONIC

FLOW PAST NONLIFT13’JG BODIES OF

By Milton D. Van Dyke

REVOLUTION..- .,

Ames AeronauticalMoff ettField,

Labora@r yCalif.

:.

<::-.. . .

. . . .k

‘r

.

.

TECH LIBRARY KAFB, NM

l~lllllulllllllllll’111NATIONAL ADVISORY COMMITIEE FOR AERONAUTICS

C10b5qilL

.

TECHNICAL NOTE 2744

PRACTICAL CALCULATION OF SEG3ND-ORDER SUPERSONIC

FLOW ‘PASTNONLIFTING BODJJ3SOF REVOLUTION

By Milton D. Van Dyke

SUMMARY

Calculation of second-order supersonic flow past bodies of revolu-tion at zero angle of attack is described in detail, and reduced toroutine computation. Use of an approximate tangency condition is shownto Increase the accuracy for bodies with corners. ‘JMles of basicfunctions and standard computing forms are presented. The procedure issummarized so that one can apply it without necessarily understandingthe details of the theory. A ssmple calculation is given, and severalexamplesare ccmrparedwith solutions calculated by bhe method of ——characteristics.

INTRODUCTION

For predicting the pressure distribution over a nonlifting body ofrevolution in supersonic flow,

~linearized theory is often found to be I=

inadequate. In the past, greater accuracy could be achieved only by.-

. resorting to the laborious method of characteristics. Recently, however,< a second-order solution has been found which within its r

cabilit yields greater accuracy than linearized

:heory, while requiringi

rably less labor than the method of characteristics.+—

The present paper aims to give a complete description of the second-order method, and to reduce it.to routine computation.

.Previously

published description of the procedure, which are inadequate in somerespects, are.revised. Shortcuts in the computing scheme are pointedout. Extensive tables of the required basic solutions ere presented,

..—

to be used in conjunction with standard computing forms. Severalexamples illustrate the procedure.

The reader interested only in calculating the second-order solutionfor a definite body, without necessarily understanding the details of.

●

✎

☛

.the theory, can turn directly to the final section Fractical Use of

. .

Method on @ge 26.

NOTATION

a, b, c

,}

functions of t associated withd, e, f source solutions

}

g, h, i, j functions of t associated withk, t, m ture solutions

CP

E

Go

G=

K

M

N

Pn

q

r

R

s(x)

t

pressure

complete

coefficient

..

Hnesr and quadratic

step, corner, and cua- .

elliptic integral of second kind

k= A/(1-t)/(l+t)

function associated with determination of

function associated with determinatio~ ofintervals

with modulus

first interval

subsequent.

complete elliptic integral of fdrst kind with modulus

k= d’(1-t)/(l+t)

free-stream Mach number

7+1 M2—.2 92

nth point on surface of body

resultant velocity

radial coordinate.

local radius of body

source strength distribution function

conical variable()~

x

,

.,-

.

.d

3

. -v

x

P

Y.

f‘

x.

.

\T - (1),. (2)

c1

r!

c

axial velocity corq-onent

radial velocity component

axial coordinate

m

adiabatic exponent of gas

length Of interval between points Pn an”d Pn+l

first-order (linearized)perturbation potentf~

basic first-order solution homogeneous of order m

second-order perturbation ~tential

exact perturbation potential

complementary function for second-order solution

particular integral for second-order solution

Superscripts

first-order value

second-order value

differentiation with respect tm x

,

value at

value at

value at

Subscripts

tip of pointed body

nth point on body, Pn

corner

—

-:

-.

..

.

4

DETAILS OF SECOND-ORDER SOLUTION

.

NACA TN 27’44

The natural way of attempting to improve a first-order (linearized)solution is by iteration. For nonlifting bodies of revolution, thesecond-order iteration equation was solved in principle in 1949 by thediscovery of a partictiar integral expressed in terms of the firat-order solution (referencel). This reduces the second-order problem tothe form of the first-order problem. For supersonic speeds, bothproblems can then be solved by suitable.modificationof the method ofK&m&n and Moore (reference2). The result is the axially symmetriccounterpart of Busemannts second-order solution for plane supersonicflow (reference 3), to which it reduces locally at a corner.

As a preliminary to describing this procedure in detail, the~duction of the second-order problem will be summarized. Furtherdeta~s will be found in references 1 and 4.

Reduction of Second-Order Froblem to.Two First-Order Froblems

At moderate supersonic speeds, the flow past a reasonably slenderbody of revolutio~ is nearly isentropic and therefore nearly irrota-tional. To this approximation, there exists a perturbation potential Qwhose derivatives giye the velocity perturbations (referred to thevelocity U of

Here subscriptsby sketch (a).into the single

the free stream), 60 that

u-=1+%u

v-=Oru

‘1

indicate differentiation, andThe equations of motion for aequation

u

(1)

the notation is explainedpolytropic gas combine

?=-” ‘Sketch (a)

.‘n

I

r

@

where

P2 =M2-1

~ . 7+1 M2——2 P2 I

.

5

(2)

Here all linear terms have been grouped on the left and quadratic andcubic terms on the right. The only cubic term which gives a second-Order contribution is the one involving ~2Qn.

This equation must ‘besolved stiject to the boundary conditionsthat all disturbances vanish ahead of the body, and that the flow is~ent to the surface of the body.

\

lteration procedure.- The equation of motion (2) cannot be solveddirectly because it is nodinem. ~erefore a ~thod of ~ccessiveapproximations is adopted - the so-called Prandtl-Busemann iteratio~procedure.

In the first approxim&tion, the nonlinear right-hand side ofequation (2) is neglected altogether. Hence the first-order perturba-tion potential Q satisfies the familiar wave equation of linearizedsupersonic theory:

(3)

In the second approximation, the right-ha~d side of equation (2)is no longer entirely neglected but is evaluated approximately in termsof the previously determined first-order solution. Hence theorder perturbation putential @ satisfies the nonhomogeneouseqgatlon

second-wave

(4)

Here @ will be taken to be the complete second-order perturbation Ipotential, rather than a correction to the first-order,solution. ~

This procedure cotid be continued to third and higher approxima-tions, subject to the limitation that at some stage”the effects of

6

.entropy variations; which were ignored in assuming potential flow,would exceed the remainder in the iteration procedure. For slenderbodies at moderate Mach numbers, Lighthill has shown (reference ~) that *

this limit is reached only in the sixth approximation. For practicalpurposes, however, only the first two steps appear to be useful.

Particular integral.- Solution of the second-orderproblem isgreatly simplified by the discovery that a particular integral.~ ofthe iteration equation (4) is given in terms of the first-order solutionby

(5a) -

J

This reduces the second-order problem to the form of the first-order problem, because the nonhomogeneous iteration equation (4) isreduced to the homogeneous equation (3) of first-order theory. Thecomplete second-order potential consists of the particular integral plusa complementary function X which is required to re-establish theboundary conditions:

I

\.*u—

.

A

d=++% (6) .

and X is a solutionof the first-order equatfon (3): Thus the remain, ~’ “ing problem for X differs from that for the first-order potential 9 ●

only in that the tangency condition is more complicated. hlethodsforsolving first-order problems are well established, so that in principle ‘the second-order problem is solved. In practice, however, variousdetails require careful consideration, to which the subse~uent discus-.sion is devoted.

Tangency Condition

Because approximations were made in the equation of nmiion, onewould anticipate that a corresponding approximation is permissible Inthe condition of tangent flow at the body. Such an approximation can be

,

made, and it can be shun that the mathematical order of the error is not*.

thereby increased. ‘I&issuggests that it is immaterial whether or notthe approximation is adopted.

.-However, numerical examples show that the *

,

t

NACA TN 2744

approximation has in some cases a large effect upon thethat the choice of tangency condition must be carefully

7

solution, soconsidered.l

Exact and a~proximate tangency conditions.- If the body is definedby r = R(x), the exact tangency condition for the original problem ofequation (2) is

,r,, ,.

-:. __&.., @r= ~f(,+@z ~1 1’. Iat r = R(x) (7)

where the prime indicates differentiation with respect to x. Thecorresponding exact tangency conditions for the first- and second-orderproblems of equations (3) and (k) are

Pand

w = R’ (1+4X) at r = R(x)

#r = R’(l+$$x~ at r = R(x)

Now in equation (8) it is consistent with the approximations of thefirst-order theory to neglect the small q~ntity TX in comparison withunity. Thus the approximate first-order tangency condition becomes

Pr =Rt at r =R(x)

)

(10) y-;{,

Similarly, in equation (9) the term 6X can be replaced by its first- \fkiorder counterpart. Thus the approximate second-order tangency conditionbecomes

@r = R’(l#x) at r = R(x) (ha)

or, septiating the second-order term into particular integral and com-pl&entary function according to equation 76) and collecting knownquantities on the right-hand side,

Xr = R’(l+>)-$r ~ at r = R(x) (lls)).

Smooth bodies.- For bodies without corners,the choice of tangencycondition has no consistent effect upon the error in surface velocity.Greater accuracy in the second-order solution results from using the

. exact tangency condition in some cases, but the approximate condition.

+ lThe magnitude of this effect was brought to the authorrs attention by.. John Huth and E. P. Willisms of the Rand Corporation.

Y

——. . .—

z

8 NACA TN 2744.,’

in others.2 For example, the exact condition leads to ~eater accuracyfor cones, as shown in sketch (b). This superiority, of course, arisesat the tiP of any potited body and persists for sQme distance down-stream. On the other hand, the approximate tangency condition leadsgreater accuracy for the boattail following a long cylinder shown insketch

.875

,900

.9’?5

$

.950

.975

Low

(c), for-which the exact solution h& been-determined by the

I— Exoct $OfUt?Of7

I

I

s\/ “*-\I Second- cvder solution:\

\ .4pprox. tangency

&

J? ‘\ Exact tongency

l? ‘A --

A

‘>, - “-

— Fkst-ordrr solu?lon:

L~ Exoct tangency

Approx. tangencyI I

1’2 3 4

hf

Sketch (b)method of characteristics.of the order of error, thatmore accurate, is confirmed

5

1,12

1.10

1.08

qv

f,06

1.04

1.02

1.00

I I Ifi’rst-order soluilon:

fTxoct tangency, ‘>

— Approx. tonaency~ %> ~

D ‘

-~ Lx

M=3 ~ 4&col. ‘ 7;/,

Y

Second-order sohticw:d

Exact tangency

0 .5 1.0 1.5 2.0 2.5

to

d

“

●

-e

.

x, semicalJbersa

Sketch (c)!Ibus the conclusion, based upon estimatesneither tangency condition is consistentlyempirically for smooth bodies. .

Bodies with corners.- In plane flow, the approximate tangency ‘ “. ‘condition insatiably leads to more accurate first- and second-order u

velocities than the exact condition. The superiority of the approxi-mate tangency condition is most pronounced for expansions, and becomesgreater as the Mach number falls toward unity..

At a corner on a body of revolution the flow is locally hro-

dimensional. Therefore the approximate tangency condition is, at least.

locally, consistently superior to the exact condition for both the

..

m

21n the first-order solution, however, the approximate ~ency condi- .tion eeems invariably to yield greater accuracy. .

2;

.

.

.

..

NACA TN ~~ 9

first- and second-order solutions. This is shown.in sketch (d) forthe velocity just behind thecorner of a conical boattailwhich follows a very longcircular cylinder. (The .exact solution is, of course,given by a plane Prandtl-Meyer expansion.) At moderate *’Mach numbers, tbe superiorityL ,.of the approximate tangencycondition is of considerableTractical importance in thesecond-order solution. Thesuperiority is not confinedto the immediate vicinity ofthe corner, but persists fardownstream. This is illus-trated in sketch (e) by comp-arison with the solution fora conical boattail calculatedby the method of characteris-tics. (For clarity, the first-order solutions are only par-tially shown.)

1 Sketch (d) suggests thatthe large discrepancy associatedwith the choice of tsmgegcy con-dition is in some sense a tran-sonic phenomenon. This isconfirmed by examination of theexpressions for the streamwisevelocity just behind the corner.For expansion through an anglewhose tangent is ~, the second-order solution using the exacttangency condition is

(12a)

f.3

1.2

~

u

r./

I.o

Exact sofufion

I

Second-wder

/

1.35

i. 30

f.25

/.20

qz

1.[5

1./0

1.05

/.00

r? 3 4In

Sketch (d)

First- order so[ution, _—

Approx. tangency

~.fia

\\\ , second-order SOI.

>

<r

Approx. tun’q,

.- --\. .[Ex”cf ‘an’%.

, \. -.

Exact soluf[on(characteristics)

TITo .5 1.0 1.5 2.0

x, semlcalibers

Sketch (e) .

.8.

4

.

10 NAC!Ati 2744 ●

w%ereas the second-order solution using the approximate tangency condi-tion is

(la)

The difference between these two results is clearly of order ~3 andhence of third order in the usual sense, according to which linearized “theory gives the first approximation. However, in the transonic range

(where ~ is of order cl\s for small.disturbances) the main term in 1

the difference is,

(12c) .

which is small only of order ~4/3* Since u/U itself’is of order e2\3in the transonic range, it is seen that the discrepancy has grow to beof second order h the sense of transonic small-disturbance theory.This is simply another example of the fact,,whichplagues all users oftransonic small-disturbance theory, that higher-order effects aregreater-in the transonic range than at other speeds.

Chofce of tangency condition.- It has been seen that although forsmooth bodies neither tangency condition can be preferred, for bodie~with corners the approximate condition is consistently stierior to the “exact condition in both first and second order. consequently, theapproximate tangency condition (equations (10) and (11)) is adopted foruse henceforth.3

The approximate tangency condition has several minor additionaladvantages. As might be expected, the computing procedure is simpli-fied. For example, the second-order velocities on the surface of acone, which could not convenientlybe mitten in e~licit form in ref-erence 1 (where the exact tangency condition was used) are not undulycomplicated if the approximate condition is used. The result is that

3All numerical examples given in references 1 and 4 were calculatedusing the exact tangency condition, and will.therefore not agree pre-cisely with results from the present computing scheme. It should dSObe noted that the solution presented in references 1 and 4 for the3-1/2-caliber-longogive at M . 3.24 is inaccurate near the noselecause linear rather than quadratic source solutions were used forcalculating the complementary function X, which results in appre-ciable error where the body slope is nearly that of the Mach cone.

. .,

.

<-

.

..

—*

A

..-

.

,

.*

.

NACA TN 2744

at the surface of a cone of semivertex angle tin-l e

J-1

u

J1-T2 “4(%HY+

= 1 -e2 “ch-lTfi

-[ Js-@)+’”-’’’2(7~n“=)‘~ *(sech-1T)2+ ~ ‘eeh-lT

where T = ~e.

Another advantage isfirst-order solution(the supersonic counte~art- of the G6thert tie, reference 6). “

that with the approximate tangency condition theexactly satisfies the supersonic siruilaritirule

Hessure Relation

After thecient is given

velocity components are determined, the pressure coeffi- ,by .$lg ●

‘-~~

2

[{ [..

~:7-lMa ~=— - (1+QX)2-7M2 2 “r21}fi-’l“4)

It was shown in reference k that approximating this expression by theleading terms of its series e~ansion cannot generally be justified,and numerical examples show that such expansion leads to unnecessaryloss of accuracy, partictiarly in the second-order solution (refer-ences l“and 4). Therefore the complete pressure relation of equation (14)is used in the present computing scheme.

Basic Solutions of First-Urder Equation

. It has been seen that discovery of a particular integral reducesthe second-order problem to a sequence of two first-order problems.These are best solved by repeated superposition of five basic solutions,which are derived and tabulated below.

Any first-order solution may be regarded as resulting from a con-tinuous distribution of supersonic sources along the axis of the bo~.

12 NACA TN 2744 .

—

(See, for example, reference? or 7.) A source distribution of localstrength S(x) per unit length yields a first-order perturbation poten-

~erefore the first-order problem consists gimply in determining thesource-distributionfunction S(x) which produces the desired shape.However, substituting this expression into the tangency conditionyields an integral equation which cannot be solved exactly.

The K&m&n-Moore procedure for obtaining an approximate numericalsolution involves the assumption that the unknown source function s(x)can be replaced by a broken line, as indicated in sketch (f). Another

x .sketc~ (f)is that the function is approximated by

L 1x

(qui_&eequivalent) viewpointthe sum of a number of linear source distributions hav%g variousstarting points, as shown. The slope of each of these linear elementsis determined in succession by imposing the tangency condition at corre-sponding points along the IMyi (me details of this procedure are

~, clearly described in sauer~s book, reference 7.)

For calculating a first-order solution which forms the first stepof a second-order solution, this broken-line approximation to the sourcestrength is too crude. Although the final second-order velocities aregiven by first derivatives of @, they involve second derivatives of thefirst-order solution q, which enter through the particular integral.(See equations (5a) and (~).) Since differentiation is a roughening

J process, this means that the first-order potential must be one degreesmoother when used as the basis for a second-order solution. This isachieved by approximating the unhewn source-strengthby q@yRticrather than linear elements, as shown in sketch (g). However, as-”

‘(x’& kthe’i’Ofapoti&dbO;ltiere’he

indicated in the sketch,‘thelinear -element is also required for use at

source strength actually rises line~ly.

x x For a Stith body with continuousSketch (g) curvature these two @sic solutions are

sufficient. Others are required, however, if’the body has corners ordiscontinuities in curvature, which require special treatment. A corner

.

,’

I

..

.

..

u

*

.

;.

NACAITN 2744 13

—.

is accounted for in the first-order solu%io~ by adding a source distri-bution of square-root strength, which produces a discontinuity instreamline slope along its foremost Mach cone. As indicated insketch (h), this corner solution must be shifted upstream so that itseffect first reaches the surface justat the corner. In the same way, a r

).curvature discontinuity is accountedfor in the first-order solution byadding a source distribution of3/2-power strength, which produces a ~discontinuity in streamline curvature *X

along its foremost Mach corie. Thiscurvature solution is required also ata corner, because an apparent Z~a-ture discontinuity remains after thecorner solution is added. Sketch (h)

+

Because of the roughening due to differentiation, the particularintegral has stronger discontinuities than the first-order solution.Thus in the case of a discontinuity in body curvature themparticular

&Q@W_~.behaves like a corner solutiori,while in the case of an actualcorner it behaves like the solution at a step in the streamlines(sketch (i)). These spurious discontinuities must be canceled in the

—.

complementary function. For this r

PWose the corner solution is usedagain in the first case. In thesecond case, another basic solutionis required which produces an ~ +xactual step in the streamlines. Asindicated in sketch (i), this stepsolution results from an inverse

.

square-root source distribution.

m “summarize, the first-order solution and complementary functionare’calculated by superposing thejfollowing five basic solutions:,

1. Linear source solution - used at tip of pointed body2. Quadratic source solution - used thereafter for body having

continuous curvature3. Corner solution - used to account.for corner4. Curvature solution - used to account for curvature discontinuity5. Step solution - used to cancel step in ~’ at corner

Homogeneous solutions.- The required solutions are axially symmetricsolutions of the wave equation, homogeneous in the space variables. meorder of homogeneity is integral (1 and 2) in the first two cases, andhalf-integral (1/2, 3/2, -1/2) in the others. Such solutions have beenstudied in debail by Hayes (reference 8). For present purposes cp~m~,

.

14 NACA TN 2744

the solution homogeneous of order m, cm be obtained by taking theIsourck distribution S(x) in equation (15) proportional to ~. It

r

is convenient to choose the source strength as

=:,Pdqx) ● (16) -

where C is a normalization constant, so that solutions of various ‘..

orders are related by

p(m-P) s()

b p @)

s(17)

i

For integral m, the solutions have simplest form if the normaliza- ~“tion constant C! is taken to be unity. Then using various relationsfor the hypergeometric function (see, for example, reference 9) the SOIU- ...tionsare found’to be given by ,.

9~(m)(x,r)=-1,3 ...(1)1) (l.tqm%( m-l-lLU+2

)——;m.$; l-t* (18) “2’2.

Here the corLic&lvariable

,● ✍

☛✎✿

✎

✃✍

✎

✎�

t=~

x (19)

the polar angle-to the tangent of the.

zero cm the axis to unity at the Mach ‘t .is the ratio of the tangent ofMach angle, and so varies fromcone. For intepyal m, the hypergeometric functions which occur inequation (18) can be expressed in terms of products of 4= andsech-% with polynamiqls in t2. The first two required basic solu-tions are obtained by setting m equal to 1 and 2, which gives:

.Linear source solution (m = 1) .

(J)=- -- J’ZP) %=”-+*. -x (sech-lt

If.

. f’‘:.

*

‘i:“

NACA TN 27h&

Quadratic source solution (m = 2)

+[(-+;++-% -@q%-~

~~=--1

x (sech t - J1-t2)

(m’”CPr‘ :x ~– )tsech-lt %7r=–

15

For half-integral ,mjit is convenient to choose the normalization

constant C as ~, so that the solutions have simple values at theMach cone. (The difference in normali~tion for integral and half-integral m- is of no concern, because the connection between them isnever used-.) Transforming the hy_pergeometricfunction into a more .

useful form for this case “gives

The h~ergeometric functions occurring here can be expressed in terms ofproducts of complete elliptic integrals and algebraic functions of t.The remaining three required basic solutions are obtained by setting mequal.to 1/2, 3/2, and -1/~. For convenience, asymptotic values validjust inside the Mach cone (where t = 1) are also given below:

Corner solution (m = 1/2)

Q ‘-fi~~~ (K-E) -0T(

11.-u ——

8 x3/2

3P*-—8 ~3/2

P’ &-lpti=. —— (2-t2 2-t ~—E–—

)

7 .62fu-— —x3/2 r (1-t) m t2 t ~ x3/2 1

(23)

16

Curvature solution (IU= 3/2)

q=. x“l~ ~~~[(3+t) K - 4E] --0gfi

bl-Ft E_K

T

(l-l-t* 2-t ~2— -—%2 ,t” )

-0

-2

-sStep solution (m = - 1/2)

Px ‘1 42 1—. (K-E}

x3/2’ r (1-t)m

pal9T= ——

~3/2 St (1-t) m (+E-K)

‘.

I

. .

..

.

(25) ..

e

Here K and E are the complete elliptic integra~$ Qf first ad secon! ‘ ‘. ‘-k’indwith modulus k =A/ (1-t)/(l+t). The second derivatives of the

step solution are not required.

Use of relations amo~ second derivatives.- All-three second deriva-tives of the first-order potential are required in order to carry out

the second-order solution. (See equations (5b).) Considerable laborcan be avoided by calculating directly only one Of them, saY p=.Then Pm and pm can be obtained from the equation of motion and

..

tangency condition. Thus the first-order equation of motion (3) gives.

i~ediately an expression for ~rr:.

.

9

3R ‘.

,%

.

,

\.

.

.

,.. .

‘./

.,

-1

.

.-,

NACA TN 2744 ,, 17

(26)

Differentiating the first-order tangency condition (equation (10)) withrespect to x gives m expression.for C& on the surface of the body:

9* =Rtl - R’q)rr at r = R(x) (27)

The computing forms described later incorporate this simplification.

Tables of basic solutions.- With this simplification, the fivebasic solutions and their required derivatives comprfse 13 distinctfunctions. Each is a powec.of x multiplied by a function of t alone.Thus, associated with the linear and ~ua&atic source solutions me thefollowing six functions of t, which, as indicated, play different roles

in the

e

a(t)

- b(t)

c(t)

d(t)

e(t)

f(t)

two solutions:

\ Ftmctional form

(~++‘9’secK’t-+-eech-’t -m .

sech-~t

t

1I

tRole inquadraticsourcesolution

-q/x=

-~/x

‘r/W

-%x

(v=/@)

---

.(

Role inlinearsourcesolution

---

---

(28)

.

These functions are tabulated in table I for t ranging from 0.100 to0.940 by increments of 0.0Cll.4 Values are given to six simificantfigures or seven decimals, whichever is the-lesser, and ar~ believed tobe correct to within one-half unit in the last place. Linear @terpola-tion results in errors of no more than three units in the last placeexcept near the beginning and end of the table.

4Tables I and II sre modeled aft& unpublished tables for calculatingfirst-order supersonic flow past inclined bodies which were preparedfor the author at the Rand Corporation.

18 NACA TN 2744.

Likewise, associated with the corner, curvature, and step solutions‘are the follo&ng seven functions of t: “

m?Q

!“+ g(t),

Lf h(t)

J!: i(t)

;,1’j(t)

k(t)

z(t)”

m(t)

Role incorner

solution

-“-

*/&

---

-fi %

G ?JP

Role instepsolution

---

---

---,!$

‘t

These functions are tabulated in table II for t rawzim from 0.100 to1.000 by increments of 0.001. The number of figures &-accuracy arethe same as for table I. Linear interpolation resuits in errors of nomore than three units in the last place except for certain of the func-tions near the beginning of the table.

To facilitate interpolation, first forward differences are given .without-their algebraic sign in both tables. It should be noted thatthe differences aretion f(t) in table

The five basicorder potential q

actually negative except in the case of the func-1.

Choice of Intervals

solutions are superimposedand again to calculate the

to calculate the first-complementary function X.

.

NACA TN 2744 19

The procedure, analogous b that of K&&n and Moore, is indicated insketch (j) for a s?moth ~inted body.First, a linear source is added atthe origin of strength sufficient toproduce tangent flow just at the tip.Second, a quadratic source is added ~at the origin of strength (negativefor a convex body), such that togetherwith the linear source it producestangent flow on the body at somedistance 80 from the nose. Third,another quadratic source is addedwith it8 vertex shifted downstream so

Sketch (~)

that its effect begins at the end of the first intervfi, and itsstrength is determined by imposing the tangency condition at somefarther distance 51 along the body. ~ corners or curvature dis-conti.nuities(or stepB in the complementary function) must be accountedfor by adding suitable strengths 6f the appropriate solutions, afterwhich the superposition OY qyadratic sources continues as before.

The proper choice of in~rvals is of crucial importance. Theyshould be taken as large as possible, because the computing laborincreases nearly as the square of the number of intervals. On the otherhand, the inaccuracy associated with using finite intervals rises withthe square of their length, so that too large intervals lead to unaccept-able error. It should be emphasized that the error considered here,which will be termed %unerical error,” is the difference between theapproximate second-order solution for finite intervals and the corre-sponding limiting solution for infinitesimal intervals; it is quitedistinct from the difference between the second-order and exactsolutions.

Fortunately, the tendency for numerical errors in successive inter-vals to accun&te is lsrgely offset by the downstream damping of dis-turbances. Furthermore, successive numerical errors alternate in signin most cases. Consequently, it has been found s~ficient to fo~aterules according to which each interval alone in an otherwise exact solu-tion would cause no more than l-percent numerical error. The entiresecond-order pressure distribution will then be dete@ined ’correctlytowithin roughly 1 percent of the maximum pressure increment.

Silgplification resulting from simihrit y.- The dependence of thefirst-order solution upon Mach number can be accounted for by thesupersonic counterpart of the C&hert @e (reference 6), which is thesimilarity rule for linearized compressible flow. ‘Ibissimilarity ruledoes not hold to’second order. However, cafrying out the usual similar-ity analysis shows that it holds approximately for the particular inte-gral, which is the primary source of numerical error. (The similarityfor the particular integral fails to be exact only b the extent to

20 NACA TN 2744

which p -differs from M, which is important o~y in the transonicrange.) Therefore, any measure of numerical inaccuracy in the second- .order solution may be expectid to follow roughly the ordinary super-sonic similarity rule. It is clear that this approximate result isadequate for estimating lengths of intervals, becawe moderate erro~sin interval length wi~ not appreciably affect the solution. As aconsequence, rules for,choosing intervals which haye been determined.at one Mach number become Universal valid if restatid tith theradius R replaced throughout by pR, the reduced r~ius of the super-sonic similarity rule (or possibly MR, since the approximate similarityrule does not distinguish between ~ andM). Thi6 conclusion, whichgreatly simplifies the formulation of rules, has been cofii~ed bY anumber of numerical calculations.

First interval.for pointed body.- If a pointed body begins tith aconical hose of finite length, the first interval is, of course> t*enequal to the length of the cone. Otherwise, the meridian curve willordinarily begti with finite curvature. For a specified limit ofnumerical error, the maximum permissible length of the first intervalmust be proportional to the initial radius of curvature, which is theprimary length in the problem. The factor of proportionality will, ofcourse, depend won the shape of the body. If the meridian cwve isanalytic, dimensional analysis combined with the supersonic sitil~itYrule indicates that the first interval is given by an expression of theform5 i

(30)

Here ~?, Ro~t, Roirt are the first three derivatives bf R(x) eval-uated at the vertex, and the dots indicate that no appreciable depend-ence upon higher derivatives is to be expected. Tndeed, for slendersmooth bodies even the second.variable Ro’’’/(f2)f2) is no-lly verYsmall compared with the first. Hence it may be assumed that the func-tion Go does not depend GignifLcantlyupon its second variable, sothat the length of the first interval is given by

180=— Go(wO’)MIRo” )

(31)

It is now clear that the body shape need not be analytic throughout thqfirst interval; it.is sufficient that no violent changes in curvatureoccur.

%hat the denominator shouldbe taken a~ MRo rather than @o isSuggested by the result of equation 32).

,

.’

,* ‘“

,-

*~\

● “

NAC!ATN 2744 21

!!lheform of the function Go can be determined by analysis,because the second-order solution at the end of the first interval ofa general.ogive can be calcul-~tedexactly as well.as approximately ifthe interval is very short. Although the result is formidable, itsimplifies greatly in the limiting case when @o’ approaches unity(which corresponds physically to the Mach cone becoming tangent to tHenose). In this case, for a relative numerical error @~@x in stream-wise velocity perturbation, the length of the first interval is

~

J8..= 1

(~-P% ’2)Jm as ,8R0t+l (Y)7+1 MIRo”l

Numerical exmples show that this asymptotic form is, with a revisedconstant of proportionality, a good approximation to the functionthroughout the range of practical application. The relative numericalerror at the end of the first interval will not exceed 1 percent i~

1

It is conceivable that an unusual body shapefor which the curvature would change considerablyso, the above rule would not apply (the variable

,

.-

might be encounteredover,this length. IfRn’’’/(~Ro“2j in

eqktion (30) would not be negligible), 8nd some ~eri&ntation wouldbe requfred ta ascertain how much the interval should be reduced.

—

Internal intervals.- At any point on a smooth body, the length ofthe next interval will be proportional to the local radius, with thefactur of proportionality depending upon the body shape in the vicinity .of the point. If the meridian curve is analytic, dimensional analysistugether with the supersonic similarity rule indicates.thatfor aspecified limit of numerical error the length of the interval from thenth to (n + l)st point is given by

The third variable here corresponds to the second vmiable in equa-tion (~); its form is different because R rather than l/Rtt istaken as the primary reference length. (The second variable here has ‘no counterpart in equation (30) because R is zero at the tip.) Fora smooth slender body, the third variable is ordinarily very small, as

%his rule ordinarily permits greater first intervals than the ruleZio= 0.025/~ times initial radius of curvature which was previouslySuggested in reference 4-.

.

,

22 NACA TN 2744

are all subsequent variables which involve higher derivatives. Thenaccording to the argument used previously, the function G1 dependssignificantly’upon only its first two variables.. This conclusion isreinforced by the empirically determined fact that disconttiuities incurvature must be accounted for separately, but not jumps in third and Ihigher derivatives. Hence the nth interval is giveu by

As before, the assumption that the body is analytic can now be replaced ‘.

by the requirement that no violent changes in curvature occur.

Analytic determination of the function GI seems impractical.It~ detailed form could be determined experimefitallyby calculating a

1

number of solutionsusing intervals of various lengths. ‘However,experience suggests that for the body shapes encountered in practice G1may be taken as a constant. The relative numerical error will appar-ently not exceed 1 percent if internal intervals for bodies withoutcorners are chosen so that .—

‘P12E1‘.(36) .-. .- =

..—

Modification for corner or curvature discontinuityy.- Two pointsmust be chosen at any discontinuity in slope or curvature, one just oneach side, as indicated in sketch (k). A corner so strongly affects

the subsequent flow field that it hasbeen found necessary to reduce the next

r“> ‘nte”d” ‘e”’ativeemorwi” -

apparently not exceed 1 percent if the -interval following a corner i~ taken

Sketch (k) where Rc is the radiusThereafter, intervals can be chosen according to the rulebodies (equation (36)).

(37)

.

at the corner.for smooth

Limitations of rules.- These rules for choosing intervals areintended only as guides and must not be followed blindly. Althoughadequate for most bodies, they may fail for unusual shapes, particu-larly those having rapid changes in curvature._ For example, the rulefor choosing internal.intervals (equation (36)) does not apply to the

.

.

.

.

.

.

b

.

.

\

!

NACA TN 2744 23

comugatA body shows in sketch (L). In this case the variable&Rn2Rn’ t‘ which was taken to be verysmafi in equation (34) is proportion-‘m (fi/k)2,and so become6.arbitrarily” k’?

large as the corrugation wave lengthis reduced. It is clear physicallythat the interval should in this casebe chosen as some fraction of thewave length. Fortunately, the factthat intervals have been taken toolarge usually reveals itself byexcessive scatter in the finalsecond-order results.

Also, the rules have been developed forflows at moderate or high supersonic speeds.become invalid at Mach numbers only slightlythey should involve the transonic similarity

Sketch (2)

the purpose of calculatingThey may accordinglygreater than unity, whereparameter, R’/p.

As in the case of solution by the method of characteristics, theonly infallible rule (which may be invoked in case of doubt) is thatthe intervals are sufficiently small if further reduction causes nodiscernible change in the result.

The rules given above are believed to be somewhat conservative fornormal shapes. In some cases, therefore, e~rience may indicate thatthe length of the intervals can be increased. It seems inadvisable,however, ever to double the prescribed values; not only is the scatterquadrupled, but successive errors then accumulate to such an extentthat the result departs progressively farther from the true solutionwith distance downstream.

Description of Computing Forms

Standard computing forms have.been prepared which largely reducethe second-order solution to routine calculation with a desk machine.Form A is used for bodies having continuous curvature. Form B is aninsert to be pasted into form A to account for a corner & disconti-nuity in curvature. EOvision is made for six points beyond the tipof a pointed body,’which is adequate for most purp;ses. The forms canreadily be extended to handle longer calculations. Copies of theforms suitable for photosensitive reproduction are enclosed.

. 7Thus if one extra point is required, every row on each_sMe o~fom_AI and B which nov extends to column Pa (except rows @ to @, @ ,

0and s of form A) is extended to form an additional column labeled

. PTj and below row 6W of form A is inserted a new group of rows iden-

0; oticalwith rows 6a to 6W on the left and@ ‘o@ ‘“he

right, but labeled 7- - and containing blanks only in colum P7.!

—

24 NACA TN 2744

The desired values of Mach number and 7 =e entered at the top~f form A, together with values of XjR,Rt, and R’f at points alongthe body chosen accokding to the rules formulated above. Then the formcan be given to a computer together with tables I and II. The solutio~for a typical ogive or boattiil can be calculated in from 5 to 10 hours.

As the solution progresses along the body, the results are found asdifferences of increasingly lmge numbers. c~nsequently, it is advisableto carry all computations to six significant figures or seven decimalplaces, whichever is the lesser. It is for this reason that tables Iand 11 mmt be so extensive. It is not, of course, necessarY ~ pre-scribe the problem With such accuracy; it is ‘sufficientto give M, 7,and the body shape to three significant figures.

Details of form A.- The left half of form A is devoted chiefly tothe calculation of the first-order potential q and its required deriva-tives. The particular integral ~ is also found in the last 23 rows ofthe left side. The right half gives a Ywal-lel c~ctiation of the com-

‘Tbesecond-order pressure coefficient ie ~

%:u::f:??~’$? ‘@d the corresponding first-order result,.

Following various preliminary calculations in row8@to@ , eachgroup of from 10 to 13 rows bounded by double lines comprises a sepsratebasic solution. The-first such group (rows q:; ,g)=;gs:ra linear source solution beginning at the origa pointed tip. It my be noted that a stratagem has been introduced ‘in calculating its effect at the tip. There both x and R are zero,so that the value of the conical variable t givenby equation (19)would be indeterminate. This difficulty is surmounted by identifyingvalues at the tip with those at the end of a tangent cone whose lengthis arbitrarily chose~ad unity, as indicated in sketch (m). The requi-site modification of g}ven values in the first column i= indi&ated by _

asterisks in rows @~, ~, and ~

Each of the subsequ~~t six groups ,

(coded @ to @ ) provides a quad-ratic source solution, the firstbeginning at the origin. Each ofthese seven groups is separated intothree subdivisions: first, determina-tion of the conical variable t\ \’

‘ +1 (row @) and interpolation of therequired fuhctions from table I;

Sketch (m) second,calculation of the required

~)strength of.the solution (row -s from the tangency condition; thlr ,calculation of its contributions to

Q)d

-P, -~, ~/p, and -~ (rows -t

to -w, at each of the points P. to I&..

.

iR - NACA TN 27~ 25

.

.

.-,

*

. .

‘ese;=atico’’ribu’i;me8::q:i~~=::%Tcomplete flrst-order results in rowsand (27) pe t the calculation f the remainin two second derivatives,-Qm (row 2 ) and’ c& (row 29 ). Finally, equation (~) for the

o

partic ar ntegral are used to etermineo

$x/M= (row 45 ), Yr/M2(row 49 ), and -*/F (row @ ), the last being requi ed only on eachside o every corner.

On the right half, various quantities required in calculating theComplementary function X are assembled in rows

@ ‘o @* ““follow seven groups of three or few’ rows each which are the second-order counterparts of the adjacent first-order groups, a linear source

Qsolution in rows -

Q Qand quadratic source solutions in rows -to-. For each goup, the second-order tangency condition y elds’

~)a weig ting factor (row ss which multiplies the first-order resultsto give the corresponding contributions to the complementary function.Thus the contributions to -Xx and Xr/~ are found in rows u and@<@. a Adding these together tiththe co@onents dwtothep~tictiar ..i>%e~al gives the complete second-order velocity components -@x

(row @) and @r/p (row @). Then the second-order pressure coef-ficient at each point is determined in row

Q73 from equation (14).

The first-order pressure coefficient, if requ red, is liketise obtainedin row (7J.

Details of form il.-The left half of form B provides a corner solu-

~)tion (rows C- 0)followed by a ‘curvaturesolution (rows K- for thefirst-order potential. Both are inserted at a corner; only the latteris used at a curvature discontinuity. The two groups are similar instructure to those of form A, with the addition that qm/~ is alsocalculated (rows @ ) for later use.

me right half of form B contains the corresponding corner andcurvature solutions for the complementary function. In addition, astep solution is provided.(rcws @ ) which, as discussed previously,is required in the complementary function to neutralize a step in theparticular integral at a corner. This step solution is placed adjacentto the first-order corner solution with which it is associated. Simi-

r-

larly, the corner solution is placed ad3acent to the first-order curva-ture solution, with which it is associated even if the body has nocorner. The curvature solution is not required in the complementaryfunction except at a corner. At a corner the curvature discontinuityis so great that it must be accountid for at least approximately inorder to pres~e numerical accuracy. Its strength cannot be calculatedexactly in terms of previously determined quantities, but fortunatelycurvature and corner solutio~ are so intimately related that it

%Ct maybe noted that the coding is mnemonic to the extent that rows

oand -v are proportional to the first-order velocity perturbationsu and v,

am ‘s @ and @to the second-order values.

26 L NACA TN 2744

suffices to take them in the same ratio in the complementary functionin the first-order solution.

.—

Use for first-order solution alone.- A very accurate first-ordersolution is found in the course of the second-order competition. The

as

present scheme can therefore be simplified i’f only a first-order solu-

tion is desired. Except for rows @to@, only the left halves of

forms A and B are used, and form A car.be terminated tith row ’22 andQformB with row c (because curvature discontinuities needno be

accounted for). Poreover, the folloting rows can be deleted from ,form A:

and the followingfrom form B:

oCe o, and Ct

The restrictions on interval length

and O-w 1s; and o20

cab be considerably relaxed.An analysis similar to that described previously shows that the firstinterval for a pointed ogive can be taken as’

A few numerical examples suggest that sub~equent intervals can be taken /

at least twice as large as for a second-order-solution,so that

[

.2pRD exceq~ just behind a cornerbn =’

~Rn just behind a corner1

( 39)

PRACTICAL USE OF METHOD

The following instructions are intended to permit the reader to - . ..apply the method wi’thoutreference to the preceding detailed discussion.

Applicability

./

~- - _-

The method gives both the first- and second-order velocities andpressures at the external surface of a body of revolution in supersonicflow provided that:

.

.

NACA TN 2744 27

1. The body has a pointed nose, or has a sharp-edged open nosewith purely supersonic external flow at the entrance, or is a boattail

.—

following an infini& cylinder..

2. The body contour is continuous (corners are permitted, but notsteps), and has finite curvature (except at corners).

3.-1/2

The slope of the contour is everywhere less than (M2 - 1) ,

the slope of the free-stream Mach cones.g ~

In order to teke advantage of the tables, the slope must in fact benowhere greater than 94 percent of this value. Furthermore, the solu-.tion can be carried back only to the point at which the radius of theMach cone from the nose has grown to ten times the local radius, as

. ‘ indicated in sketch (n) for an open-nosed body. The solution could becontinued beyond this point only by !extending the tables according to

‘7 ‘-:

“/f\

equations (28) and (29)./

/ i\4 II

,’ 7

Choice of Points

.“ +e-For normal bodies, points on the \

body are chosen according to the fol- \ 1;

10WL~ rules. These rules may fail ifthe curvature changes unusually sketch (n)

rapidly; this wll-be revealed-by excessive scatter in the second-order solution, which indicates that the intervals must be reduced.

.

1. Choose point P. at the vertex of a pointed body.

.2. If a pointed body has a conical.nose of finite length, choose

point PI immediately behind the base of the cone. Otherwise, chooseP1 at a distance behind the vertex no greater than

,

l-~2Ro?260 =

8M]R0’JJ

where R.r ani %’* aretertex.,...,

3. Choose point PInosed body or boattail.

the slope and second derivative at the

immediately behind the start or an open-

‘Although there is no absolute limitation on negative slope, the method “,becomes inaccurate when the magnitude of the maximum negative slope +-exceeds (M2-1)1/2.

.

28 NACA TN H44 u

4. Wherever’the body has continuous c&vature, choose point Pn+l

.

beyond point F’n no farther than

En = 6Rn*

—. “..~ - ,,

where Rn is the radius at Pn.

5. For @ discontinuity in slope or curvature, reduce the precedingintervals if necessary so that a point falls exactly upon the discontinu-ity. Associate this point with the body shape just ahead of the discon-tinuity. Choose the next point at the same abscissa, but associate itwith the body shape just behind the ‘discontinuity. An exception arises,

—

however, if the discontinuity follows a conical tiP or ~inite~Y long 8

cylinder, or is the lip of an open-nosed budy; then (as prescribed byruleB 2 and 3) only a single point is required, ~d iS associated ~ththe body shape just behind the discontinuity.

>-

6. Choose the first interval behind a corner no greater than

c

where Rc is the radius at the corner. A boattail or open-nosed bodyis to be regarded as starting with a corner.if its initial slope isdifferent from zero.

*The previous rules apjly to subsequent intervals. ,

Examples of choice of points.- The choice of points for four typi-cal bodies is indicated in sketch (o).

“e ‘d!E!l “ .

.

(o)

Preparation of Gomputing Form.

Form A is prepared for computation in the following steps:(

1. Enter the desiredthree significant figures.

._

free-stream Mach number M in row @ to

.

J.

.,

..

.

NACA TN 2744

Enter the desired value of the adiabatic exponent

6row 2 to three significant figures (1.40 for air).

3. In the col~corresponding to each of_the points

29

7 in

ever, in column PO (which is-wed only for a pointed body) indetermi-nate forms are avoided by replacing the abscissa, radius, and secondderivative by unity, the slope, and zero, respectively, as indicated onform A by as-risks. The origin for measuring abscissas must be takenat the tip of a pointed body, but is arbitray for other shapes.11 Theunit of length is arbitrary, but it is usually convenient to measure insemicalibers.

4. If th body is not pointed, strike out column P. and rows Od

‘0 @and & ‘0 @*.0

5. If point Pn lies just behind a corner or curvature disconti-

@nuity, cut out and discard all rows labeled - . Replace these bypasting in form B for a corner, or the portion of form B below thedouble line for a curvature discontinuity, with the first column alinedbelow CO1- Fn of form A. For example, sketch (p) shows schematicallythe modification requiredfor a discontinuity betweenpo~ts Pz and P=, as on Top of

the first body shown in Form A

sketch (o). Note that a downtoboattail or open-nosed body rvw@ ‘is to be regarded as start-ing with a corner unlessthe initial slope.is zero,and with a curvature dis- Fwm B

continuity unless the

initial curvature is zero.

computing

Iv,, ,,,,,

II ~/~ir-m,,,/,,

II! v’“; Iu

Bortom of

Form A

below

row @

The computing instruc-tions on forms A and B are

Sketch (p)

intended to be completely self-explanatory. As noted, all calculations

should be carried to six significant fi~ es or seven decimals, whichever

10Care should be-taken to give R1 and R“ the proper algebraic sign.11

An exception arises in the unlikely case of an open-nosed body or

boattail which starts with zero slope and curvature. In order toavoid indeterminate forms in this case, the originmust not coincidewith the start of the contour.

.-

—

30 NACA TN 2744

ia the lesser (regarding given data as exact to that accuracy). Thetables should be interpolated li~early,“notingthat the first differencesare given without algebraicsign.

Bectiusethe computat ons are rather involved, with only partial

@checks at rows 2 band 6 , it has been found expedient when possible .to have two computers carry out the same solution simultaneously withfrequent conqmrisons.

The quantities of

First-order quantities

oRow 21 :

0Row 22 :

Row o83 :

Second-order quantities

Row o62 :

Row o63 :

Rowo73 :

Typical shapes can be solved in from 5 t: 10 hours.

Results

interest obtained at each point of the body are: ,

%(1)

u(=)l+$X=,*

(#’)

Only three significant figures should be kept in the final results.

Exmples

Before calculating a new case, the reader may wish to check hiscomputing procedure on the first few columns of a known solution. Forthis purpose, numerical values from various intermediate rows of thecomputing form are given below for a 6-caliber-long circuLar-arc ogiveat a Mach number of 3. The significance of these rows is also indicated.

.

.

—.

—

.

+——

a

,—

*

.——

.“

?

.

.

.

.

\

NACA TN 2744 “ 3J-

Dimensions are measured in semicalibers, and the intervals have beenchosen slightly smaller than the limits prescribed by the rules in orderto give simple values of x.

%t31P. ~2

T

2.00 2.8C

● 3X ● %14

13 *1

14 *. 168

15 .168

16 *O

20 .0U8926

21 .0441146

22 .0593969

23 .0364553

45 .wMQ64

49 , .0037346

62 .0567766

63 .9mm

. x:

R:

R’:

R“:

-9:

+x:

~r/$:

-P:

_ ~x/M2:

~r/M2:

- #r/P:

–- 14.:

3.90

.546. ,,

,/

.139 I .U8 .11.2

=4-= -.0141

.0549784

.023~71

.0491439I

.04521j48 .0395979.

.0052442

.0006239

-.0019$91 -. ma893 -.0028234

.0475034 .0439176 .0386489

.9634o4 .968955 .975150

.0606 , .0506 .0403

.0514 ● 0459 .0376

&):

(#’), b73 .0830

83 .0660

Note: The ,asterisksserve as a reminderthat in colwnn P. ‘theactual values

. ?. .

.

of x,R, and R“ fit bethe value of R’, and O,

replaced by 1,respectively.

.

.

32 NACA TN 2744 .

*-

The first- and second-order pressure distributions for the completeogive are

numerical

As a

shown in Bketch (q) in comparison with a solution by the

.08

.06

.04

.02

0

-.@

4

0 Method of characteristics

\ — Second-order so/ution\ ,

——- first-order solufian1

\\

o 1.5 50 4.5 6.0 Z5 9,0 @.5 12.0

x, semicalibers

Sketch (q)

method of characteristics given by Rossow in reference 10.

further example, corresponding results are shown in sketch (r)for a 3-caliber ogive at a Mach number of 1.7.

.4

.3

.2

GP

.1

0

I I I I, t 0 Method of character fstlc$

0—Second-order so[ution

—--First-order solution\

. .

0 I 2 .3 4 5 6

x, semico[lbors

Sketch (r)

Ames Aeronautical LaboratoryNational Advisory Committee

Moffett Field, Calif.,for AeronauticsMay 12, 1952

.

Q

a

.

—-—=

b-

.-

—

●

..

.

NACA TN 2744 33

REFERENCES

v1. Van Dyke, Milton D.: A Study of Second-Order Supersonic Flow.

T’h.D. Thesis, CIT, Jime 1949. (Available as NACATN22CXI, 1951. ) ‘

2. von K&r&n, Theodor, md Moore, Norton B.: Resistance of Slender.Ibdies Moving tith Supersonic Velocities, with Special Referenceto Projectiles: Trans. A.S.M.E., VO1. 54, DO. 23, WC. 15, 1932,pp. 303-31.o.

3. Busemann, A., and Walchner, O.: Profileigenschaften bei ~erschall-geschwindigkeit. Forschung auf dem Gebiete des Ingenieurwesen8, ..-

Bd. 4, Heft 2, March-April 1933, pp. 87-92. (Available as R.T.P.Trans. No. 17%, British Ministry of Aircraft Production.)

*4. Van Dyke, Milton D.: Tirst- and Second-Order Theory of Supersonic

Flow Past Eodies of Revolution. Jour. Aero. Sci.,,vol. 18, no. 3,Mm. 1951, Pp. 161-179.

5.’ Lighthill.,M. J.: The Position of the Shock-Wave in Certain Aero-dynamic Problems. Quart: Jour. Mech. and Appl. Math., vol. I,pt. 3, Sept. 1948, pp. 309-Q8.

‘6. G&hert, B.: Ebene und r~umliche Strhung bei hohen Unterschall-geschwindigkeiten. Lilienthal Gesellschaft f~ Luftfahrtfor-schng, Berlin, E!ericht127, Sept. 194-0,PP. 97-101. (Availableas NACA TM 1105, 1946.)

●

7. Sauer, Robert: Introduction to Theoretical Gas Dynamics.J. W. Edwards, (- Arbor), 1947, pp. 71-81.

8. Hayes, W. I).: Linearized Supersonic Flows with Axial-Symmetry.Quart. Appl. Math., vol. 4, no. 3, Oct. 1946, PP. ’255-Z61.

9. Magnu6, Wilhelm, and Oberhettinger, Fritz: Formulas and Theoremsfor the Special Functions of Mathematical Physics. Chelsea ~

Pub. CO., N.Y., 1949, pp. 7-U.

10. Rossow, Vernon J.: Applicability of the Hypersonic Similarity Rule

to Rressure Distributions Which Include the Effects of Rotationfor Bodies of Revolution at Zero Angle of Attack. NACA TN 2399>1951.

.

.

.

34

—t

.Ux

.103

.lGI

.10:

.l@

.m

.lot

.loj

.lct

.1C5

.I.lc

.Ku

.lJz

.1.13

.-J.I.4

.U

.116

.IJ.7

.U8

.U2

.12Q

.L2.l

.Ez

.123

.124

.W

.I.26

.127

.128

.W

,lW.132,UQ.L33.134

m,135,137138I-39

,140,141142lk3.144

143U6147148149

%.W.153.@

.U5

.196,157

A

m,159

,la

NACA TN 2744

TABLE I.- LINEAR AIUlQUADRATIC SOURCE SOLUTIONS

a(t) b(t)

1.s%’24 m1..9&33k.Sn1.97853 971.96983 ~1.55922 95

1.94970 841.94Cr28 =1.93294 021.s-2169 81,

1.91253 m

1.S0347 ES1.89446 8W1.&3m4 m1.8767’3.87,1.86795 ~,

1.85s$33 ~,1.6X69 851

;:8~~=.

1.01-roz 824

l.me al:

l.fKKw 81(1.79251 m1..78447787

1.7765c ~C1.76960 7e41,.76076me1.75296 rn1.74326 *

1.7376~c1..7m mL.722k7 ~~aL.nk% ~~L.713T% 7s7

L.71X119,aL.692$$ -L.@@ mL.&@@ mmL.67127 ~~

L.@AT mL.65713moL.6WL3 8E9L.6&318 +L.636!25&

1.6(2944em,.62264 675L61!S@3 en:.EZJ318me..60252 ml

..59791 a57’L.*+ .9561.* WE1.57633 -a1.56%Q Mo

1.WW msL.55-CL5eal1.55084-L.5%-457-1.3%34 eln

L.53.215

c(t)

k.&&X swck.77453 4e7c4.72475 4.9e44.67593 •~el4.62802 4701

k.%xn 46144.5* 45P-C4.48977 444E4.44%9 4s.374.40142 4291

4.3%51 42144.3637 41414.27496 400s+.23k27 40CCk.19k27 aeu

:.:45 ~=

4:07829 37084.04C91 =774.00414 3517

3.%797 855s:.9mf& ~~

3:86250 339E

3.82.f@ 8-8

3.7X359 Ozm3.7&n S2S73.7334 Slae;.~69m5 ::

3.6W n0473.6365 mm3.57563 a.q3.54@4 nels3.516%9 2*T4

3*@315 Pmz

3.kw83 ~na3.4n90 .a-3.40437 ~~,3.377’22an

3.59045 2e403.3405 Zw3.29801 a-e3.27232 23353.2M97 2s00

3.2219T .e4ea

Lwlv 24a5

3.1-E94 ,.m1.148% 237’2).I.2519al

3.10178 23U?3.07866 22E23..o55@422503.0333. 2aE53.O~ti 2107

2.9@@ 2170?.96739 nlu2.9k~ *IIT2.9A78 -~e.90386 -e

2.!33320

L(t)

2.9334 laF?.983Q m2.97332 em2.96351 ~P.95@3 eat

Q.94417 ●=2.93464 0442.$%723 Me2.w584 m2.SQ677 as

Z.@-@ elc2.E#28 W3!2.a7925 m2.8703 -72.86144 an

2.85263 av2.@&393 m2.83529 .f572.826@ m2.8M22 w

2.~79 me2.&n43 -2.793M ~2.78k91 ale2.77675 We

2.76.86 G2.76%3 m2.752&5 7812.74475 -e.73690 -

2.7292.2-~.72Uf9 7972.n.g2 ~2.70610 75UZ.69W 750

2.6a .y432.69359 -2,676!23~~2.66895 ~2.66156 *.

2.6%32 7.U2.6qT3 flu?.64030 760?.6329~ -TOS~.6eTe5 me

?.61EM7 eaE?.6U92 m~.@a2 665~.5w7 -2.%u36 e-rr

?.78459 qz2.577&7 em2.97120 0.%2.%56 ~02.33757 855

2.%142 EZUQ.*91 C-47!2.5384464s2..532!31~2.ZM2 ass

2.51927

e(t)

9.94997 9s519.m35 87s79.B279 ewe9.6571c s-9.5-5324 8s0s

9.k7D6 9QE59.3fa81 -79.9214 0-9.2C51O %54*9.U965 WI

9.03574 @J?408.55334 e#058.87235 7s518.7?289 ~1~8.nk74 T8W

8.63796 75478.56249 7418

8.W3U 72M8.k~~ 711R8.Yix?3 70ss

7.93725.a.zws7.87=6 ewo7.81026 -7.-748A an7.69727 0014

7.62703 5ea7.5”57W 98m7.W347 57%7

7.k5iCo 9<7.3$5% Svlu

7.33960 54887.28462 e4177.2X45 SWI.7.1T704 -=7.12441 ,190

7.07251 01177.021.345C4B5.97059 4U7B5.92U4 4m-r5.67227 4W0

5.@67 4-r?sim 47L05.72aw 4-75.69’23545m5.636L9 ~us

;.591.24Uee6.5465$ 440a6.5a2P 4SSI6.4%w ●aw6.4M04 4240

6.373Q 41m6.3R78 dIti6.29344 4m6.2LwX 4CQL6.2Q9w wan

6.169k8

f(t)

I.cird 1(1.W514 1(1.CQ524 I]1.02535 lC1.CC5A5 11

i.w56 111.00767 1(l.mn 11I.@ 11Lw 1s

l.c@ll 111.CGQ2 111.&%33 Ii1.*5 u1.(x656 u

1.CQ6S3 111,036% u1.C@T2 41.m7d ~1.a2716 u

1.oo728 la1.W74Z la1.07753 u1.C0765 M1.WT2 la

l.wpl la1.CCa3 181.Om.6 ~1.m=9 141.00843 18

1.w@56 lal.a)w 141.o@33 1s1.o@6 141.MO 14

L.a%2k 14LC6JM .14l.cDm 141,(x966 14L.C@O n

~.cQ955 14L.OIC@ IEL.0102k 14L.010~ mL.01.@3 an

L.oti ISL.olri13mL,ol@ 10L.01U4 16L.OK@ 15

L.Ollhh lal.orlfl MI-.0U76 Ml.cmglm1.012Q7 10

1.012.23181.012W i-f1.07.256ML.o1272 171.o12@ 18

1.01325

. .

.. .

WQg=-- “-’

..,

NACA TN 274L

TABLE I.– CONTINUED

. .

/,

\

.

.t

.la

.L51

.I.62

.163

.164

.I.@

.la

.167

.169

.169

.lTo

.l-n.

.172

.173

.174

.lW

.176

.in

.178

.179

.m

.181

.182,183.18k

.l@

.185,187.lwM

m.191.192.lm.194

.19

.1*

.In

.w

.19

.m

.201

.m

.a3

.E’Ok

.=5

.x

.Xfl

:Z

.210

.ZL

.=

.213

.21k

.a

.2L6

.21T

.218

.219

.220

a(t)

.735419 207

.532546 Z3C

.929693 =

.- =

.524042 am

.Ym?k am

.51a4al 27s5

.515726 2-C=-m -7.slcQ73 as

.537974 am

.m89k am

.502232 -5

.ksqw Zea?

.4565.$) =S09

.49k351 25!aq

.49177?22Sm

.4@18k =s.

.4&w6 -1.

.484099 2E=!4

.481%1 2sca

.479332 zde.z

.4765% 2478

.474094 ?-

.k~k 2444

.4691&J 24=

.4&1751 2414

.464337 ===

.461939 ==3

.4%5% -s

.4571.8723=

.454834 2ss

.4724P 2324

.450171.aali

.447%0 E8?S5

.44Z63 -

.4432a3 2268

.441O’E =54

.435761 =41

.k36520 =-r

.434293 Rzls

.432C@ ==01

. 4@79 218T

.427’692nrd

.42528 =.l

.423357248

.421z@ Elm

.IJ5AT(-2us

.4169L+92111

.4148X Zo’aa

.WmJ2 20.s5

.41cQi4 2079

.Wg SC-s,

.40s517 209

.404467 ZCss

.40242$ q

.4CQ40L 20L9

.3933% em

.3s6382 1-2

.39k~ 1B61

.3924CB

b(t)

1.532L5 S321.5* al:1-P* m-1.5U82 @I.m -

1.5Ql’79 w1.L$593 5ai1.4&il ml1.48402 S8!1.47817 S.cll

1.47236 m1.46559 E-n1.4@33 S?’2l.hmlz =1.44945 S&

1.443&l MI1.43819 5571.4%2 555l.w-p-f 6=1.M% 54s

1.41&28 5461.41@53 w1.4Q5’a use1.= =51.39447 Sti

L@91k -1.3@!E4 m1.37857 -1.37334 Eal1.36313 n19

L.tiak -~.35779 n=I.352s7 no

L3k757 so-?1.342W son

1.3TM ml.1.33244 4821.3zlk3 4W1.32249 4=L 9755 4eL

I.g%h 4aa1.30775 4ea1.9- 4=l.- 4sl1.=4 472

1.28545 4701.28369 474

1.* 4-n1.2742k 4@B~,=9m 487

1.2&% 4e41.2S024 w1.25%2 4ec.1.ao2 4901.2M44 453

1.24189 46s1.23-/’%453.1.22287 44s1.2XN 4471.22* 4.44

1.21945

c(t) ,

2-@83m2.&@82.8k2612.&2672.FU97

2.n3%2.764-262.74523.2.W6432.7cv83

P.@k?2.673.282.653w2.637732.617w

la?,

lea

lWlea(

lesf

.lal;lraf

l-r-r175[172-!

2.- Ieef2.27485 la2.a5Kn 13Z2.247* 1=52.’376 la?

2.2mz7 lSSC2.2M92 ‘ma2.19371 1S012.1&%3 m2.16767 1-

2.15483 ml

2.Um2 12s1

2.1231 lR4a2.u703 w2.1o465 l=?=

2.03-23 12P2.c&24 m2.W20 Uw2.05526 u2.okJ+k3UT2

2.Om 116s2.- u=2.02956 U421.59314 11=1.S@m2 U

1.97%3

d(t)

2.51927 w2.51.235 w2.- ==42.m44 ax2.4*4 SIC

2.48539 eu2.481.97 me2.4-(56 mm2.46981 b2.46379 me

2.4T@ 5=2.45185 5=2.44S3 ~ee2.44004 s2.43419 S22

2.42837 ~ra2.42259 m-m2.41693 5782.411L0 =2.40541 ma

2.39975 s=2.39412 54$12.*31 5572.33294 5562.37739 5=

2.371&3 C492.%639 546~.3&93 5432.- 5402.3mlo 5s7

~.3k473 S352.339S -2.33405 m2.32877 m2.323% a

2.3M26 =-=2.fi304 SIe2.x787 ne2.30269 5142.~ 5U

2.wE13 w2.2873k SOT2.=.=7 =x2.27723 w2.qz?l 489

2.26722 4882.%224 4052.27’7294s22.25237 4s12.24746 4a8

2.24@ 4i.62.2377’24e-zf2.23X9 4U2.=WI 4792.22328 478

2.21.85047U2.2u75 4=2.m902 4n2.ZM31 4.se2.l$q52 467

2.19495

e(t)

5.97754 37025.9b052 =5s?.SQ393 m15>.85778 35735.8- s=

5.3UJ.O 2a5-r5.s253 28a-75.Z53J26 z-5.22433 =5>.l@y 2837

5.167* ~T5.139ZT a-ma5.u143 -7.09393 m5.C%71 2829

4.&@’8 -4i37359 al,4.84844 24a04.823% ==4.79333 2443

4.77445 24204.75325 -74.7262S 23744.Tc254 =524.6792 ==

4.65572 =-4.63253 =Wk.6@76 _4.53-DO Z2e4.5s464 =X25

L542B -4.52034 a=4.49#9 alaw4.47693 =.~e4.45737 -

4.43403

f(t)

1-OIW 171.01322 17

1.01339 171.01355 171.01373 17

I.olm 17l.oll$q’171.o1424 181.01442 171.01453 m

1.0147+ 1s1.01495 laL.o1513 laL.0153L laL.01549 1=

1.01%7 19L.01Z6 181.01634 1SL.01623 laL.01642 la

L.Ol&IO lBL-01679 201.O1’sg 1sL.o171e 1sL.01737 IS

L,Olm aL.01776 mL.01795 1SL.01815 mL,01835 zo

L,01@5 ~1.0N3”76201.01896 201.01916 SQ1.01937 ao

1.Olm El1.01978 al.al~ al1.- =1.CLQ41 a

1.02062 211.02%3 221.U2105 =1.rE126 221.om8 22

1.021’foa1.02192 =1.02214 a!1.02236 2=!l.- ==

1.02231 221.CL2303a1.02326 a1.02349 =1.02372 2s

l.- =SL.02418 =1.02441 m1.02464 =L.C2J!I&24

L.0ZW2

.

—

36

.

NACA TN 2744 .

TABLE I.– CONTINUED

t a(t) b(t) c(t) d(t) e(t) f(t)

.220 .?.92W 1=70 1.2L9k7 .44e 1.9756Q lale 2.19497 4eB 4.43409 am 1.0271.2m

.22L .3*39 1=9 1.2L733 441 1.96447 1103 2.19030 4- k.4L3m 2* 1.Q2735 x

.22-2 .38848C 104S 1.21062 4oa 1.953h4 10= 2.18%7 4EI 4.35210 E- 1.02559 24

:~~ 3&55 ::: W&j ;:w% ::2 2::%: :% ::TE$ :: ::z~? z

.22> .382&fB LDIE 1.197* 4= 1.- 10SJ3 2.1719 ASS 4.33CA8 mm 1.02632 =

.226 @@& 3 1305 1.19322 4~0L

1.9102k 3.058 2.16737 4s4 4.31030 i?ml 1.0265.6a.22-I 1695 1.* 423 1.8qA6 1048 2.1628L ASI 4.E5W9 leu 1.02683 28.228 .376973 la+ 1.18464 426 1.EWW8 l-l 2.1*D .4s0 $.27044 Ice-r 1.02705 z.229 .37W lura 1.l@3 424 1.87677 Lcal 2.153&2 Ur 4.2wr7 18s1 1.02730 25

.230 .373196 IW 1.17614 4= 1.86346 1029 2.14933 ● de 4.23L26 u= 1.02i’552s

.231 .3’f1332166u 1.17192 421 1.85823 10LS 2.14457 4U 4.21L92 1018 1.CE78C. &E

.232 .369479 lMB 1.16771 41a 1.8MW me 2.lk@$3 44P h.w74 lBCZ 1.02EiyJa

.233 .367636 1sS5 1.16373 .415 1.83802 333 2.13531 441 4.17372 lES-S 1.02830 z

.234 .365@3 132Q 1.15937 415 1.8X03 am 2.131Ec 4s0 k.lw la-rl 1.02856 a

.233 .3639W LELa 1.15522 413 1.818L3 em 2.L27Z2 487 b.136u less 1.CQW1 m

..236 .362167 IED3 1.151C9 411 1.m30 074

.2372.L2X5 4= 4.LL76C 1840 1.CE937 26

.3&1364 1734 1.14698 4Ga 1.798% ae7 2.LW9 4CIS 4.C9920 L3=E 1.02933 =.238 .39s70 J-r= 1LM289 407 1.78a39 Sse 2.LL416 4@2 4..239 .3w@4 1-W5 1.13.332405 -

1811 1.02$y3 m1.7793c B&e 2.~wa4 MO 4.62 1735 1.0298> m

.240 .355JXJ l-run 1.13477 404 1.76978 S44 2.105M ~Ea 4.04489 li-iu 1.03011 m

.2kl .3532ti Ine 1.13073 m 1.76334 =7 2.1OL26 4e-r 4.0270T 17= 1.03037 a

.242 .351490 1740 1.L267L 4ca 1.75097 - 2.@Q9 4= 4.0C941 1- 1.03053 E-?

.243 .349744 1737 1.L2271 am 1.74167 6= 2.w274 4ss3 3.99@3 1= 1.03C50 a

.24h .3W7 1T28 1.LL873 Z= 1.732k> DIU 2.0895L 4n2 3.97449 17ES 1.03J..LTm

.245 .346279 lm 1.L1477 .s.3s 1.72329 WEI 2.oa4z5 420 3.w24 1= 1.03L43 27

.246 .344X1 1710 l.LLC& 834 1.71h2L 301

.247 .3k2851 1701 1.lMEB ml2.c@xB 41= 3.940L2 18s8 1.03170 =

1.7om 335 2.0735Q 417 3.92314 1- 1.03193 =.2V2 .341.L701832 1.l@97 300 ~.m -.249 .3394% 108S I.G%WI MS

2.07173 415 3.936.29 lam 1.03225 =71.63737 - 2 .C&@ 414 3.W.97 1059 1.03252 Ze

.273 .337777 1074 1.0%.19 sea 1.6j@5 W4

.2512.c434k 41s 3.87.@ l-e 1.032&I ZT

.336101 lees 1.09L33 SK9 1.66982 ,wa 2.0593L 410 3.8 72 LeGS 1.03307 ~

.252 .334436 -57 1.C@74e - 1.66L14 w 2.C5521 410 3.d 19 1- 1.03335 =0

.253 .33-2779 1W8 1.09365 Q 1.6522 b 2.0>~ 408 3.82358 lam 1.03363 a

.@ .33L131 lao 1.07983 - 1.64397 3+0 2.04703 m 3.w&3 1- 1.03391 =

.257 .323491 1631 T.C@ly 078 1.635JW .sa

.* .- 1- 1.07=9 a-r? 1.@7fM =?72.cA297 4- 3.79193 1- 1.03419 =82.03892, 40s 3.77603 in-ra 1.034k7 a

.257 .3-%237 1016 1.Ckw8 875 1.61859 wo 2.03$89 4- 3.7r@’5 1601 1.03476 20

.259 .324622 1= 1.o@173 a7& 1.61o39 ash 2.03087 400 3.74473 l!54u 1.03504 a

.259 .323016 -as 1.06CW 372 1.-15 ala 2.132C87 w 3.725’-2615= 1,.03533 m

.2.53 .321418 lBBO l.m ~ 1,59397 -as 2.0’22% 8s8 3.713&3 15M 1.035fL2 ~,261 .319828 Is- 1.09356 Cu3 1.58X4 me 2.ol@3 am 3.@8&. l.s~,1.03791 ~,262 .3L8246 -74 1.04987 sue 1.57778 .901 2.01494 seE 3.63347 lsp 1.03620 a,263 .316672 leas 1.C4619 m ~.%977 -IU5 2.ol~ 33s 3.65943 14e3 1.03649 m.264 .315107 ma 1.0433 Ca4 L.56U32 no 2.CD706 8ss 3.653P L4rn 1.03678 ~

.265 .313549 1ss0 1.o@39 MS l.~w -re4 2.03313 w 3.632-57 1471 1.03’7M B

.266 .3LLm 1542 1.03526 w 1.54&@ ?7s 1.99323 - 3.62396 l~al 1.03737 m

.267 .3L&37 lEaE 1.03164 w 1.5383o ~ 1.59334 S= 3.&3935 14S LJJ3767 m

.z~ .- 1ss7 1.02804 =e 1.53C57 70B 1.99146 err 3.~48y 1440 1.03797 m,269 .3!27395 I=le 1.02445 357 1.&89 TE2 1.96759 Sw.9 3.5W15 1480 l.o~ ~

,270 .305876 WJ I.@@$ me L.51527 _ 1.93374 * 3.*5 1- 1.03857 S1,271 .30436T IW 1.01732 3s6 L.73770 %2 1.97%C- m 3.571% 14- 1.03EW ~,272 .302861 14.T 1.0L377 SYS 1.~18 r.e 1.97&37 832 3.537,273 $

14ccl1.039.L8 a

.3o136k 14~ 1.01024 w 1.@272 7U l.~~ WO 3.52

274 .W73 14= 1.00672 nso

lee? 1.03949 301.48530 m L.9@47 @._ra3.50996 IQW L.03979 =1

.275 2X2X; ;= L.~322 WE L.47794 TS1 1.5Q67 - 3.4%16 ~~ 1.04010 a,276 .W973L =~ 1.47063 ~ 1.XU39 ma 3.48246 I* 1.04G41 =

.295k53 14E0 .996276 - 1.46336 = 1.97713 *-ra3.46&34 1051 1.04072 =,;! .293993 i4B3 .$9394 ~4s L-4%14 71e 1.95337 - 3.45733 1s4!31.a4L04 m‘=9 ~.292540 14*5 -939345 ‘“’ 1.4W - L.94%4 am 3.4419 1* 1.ob135 u

283 .291C95 .985910 L.kklf% 1.94791 3.42@I 1.04167

‘“w+$--

.

,,.

.,

.,

i.

●

✎ ✎

☛

✎

✎✎

✎

● ✎ ✎

✎

✎

.

.

.

‘.

.

.

.

,.

NilCATli27,44

TABLE 1.– CONTINUED

a(t)

..291@5 14s:

.296% Mm

.288= 1424

.2W.XJ1 1417

.=5384 1411

.283973 L403

.Z32570 1397

.WiiaMaO

.2’7’3783mm

.2784C0 1377

.27024C 1-

.263932 le=ll

.!as~ =24

.266247 181B

.26493 1312

.26361’7 la=

.2&2311 m

.2@o11 1*

.Wm -7

.&&JIQ” 12@

.anm 12-?5

.?55573 1.270

.Zl@5 -

.253342 -

.Z5X@ U51

.a* 1.=46,2495.!%=8.2Mi%9 L234.24Tl15 -.zk~7 -

,244665 1217.24M IU1,24223 -S,241032 1-,235832 1184

,2@@3 I.lm.237W9 II-.2@265 llrf,23= 11P,233916 llea

,a”qw lla2!231599 1156.2W432 IISI,W- 1145=36 1141

- ~y@yJ& llGO.224750 1124223-$361=2224% 111*

222.37211102’2326211C4~9159 10BBam% 1035a@@i 103S

215i375

c(t) d(t) e(t) f(t)

=+cJy-”

37

}

I

.=. -

.

38 NACA TN 2744

TABEE 1.- CONTINUED .

—t a(t) b(t) c(t) d(t) f(t)

1.C6335 411.c6376 411.@417 41l.cdfi 4P1.06W 41

1.06%1 421.c@83 421.ct6q 4=l.mw 4n1.067’1o 4E

1.C!@f2 4s1.C6795 431.W% 4s1.06%1 481.c@24 4a

1.@67 441.07011 4s1.07054 44L.oi’oX 441.07142 45

1.07187 441.07231 461.07276 441.073-21AS1.07365 4e

1.07411 461.074% 46L.Owl 4eL.07547 4eL.07593 4e

1.07639 4eL.07685 47L,0’7’7324.3L.07778 47!.07825 47

L.07872 47L.07919 47l.o~% 40!.C@lb 4SL.@ 40

..O911O 4s

..08159 4s

..08a% 4,

..cX32m 4s

..06W3 4s

..c8352 40.C&O1 49..084w 50..08Xm m..08572 40

.08%9 51..05652 so..087m ~.C@yc El.C@.%1 51

.08952 61..C%Q3 51,.0W4 51.WU’ =.@J57 -

.091m

“--Q?p”

..3k.341.342.34:.34}

.3?

.3ti

.34i

.34.!

.34:

.3X

.351

.3Z

.35:

.374

.355

.3%

.357

.39

.359

.36C

.351

.362

.$3

.$h

.365

.356

.367;%

.370

.371

.372

.373

.374

.375

.376

.377

.378

.379

.3m,381.382,383

.*

.X5,386.387m*9

WJ

g393394

3993%397393%

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. 21X75 103J

.214791 1-

.2137L_Llcm

.21263& Lm!

.w%? ma!

.2LOX2 lm

.23+42 low

.2083W 105(

.207336 lM(

.26@J0 104:

.2052k9 Icm

.224212 L-

.2)3282 Iw,2G2153 loa.23UW .101[

.mm.12 101C

.lm ICG

.l@90 lW,l%G% lC”X>1923’36EIm

,Lww se?..@099 3ea,193113 em3.92133 em,191153 *~~

l$X317$l98s189zm ~lWk5 ma18v87 B57le6323 9?=

185376. 348184428 mm183485 840N&@ 8s8181610 331

l@679 02717975.2 e=178829 816177910 SK17&95 811

17&84 00717 177 e03

?17 P74 80s173375 834172h81 8B1

171593 857170703 Ml@3m eeo

1689kI WEM&q en

167193 em166326 -16X2 MO16W2 sw16374g B=

L62893 =e16m44 w=161_19$ mz160357 em159719 *

159695

.M1’291 ml

.790530 mm

.7957m a74r3

.793035 am

.7p33i mu

.787576 =IT

.WJ@9 m7

.782.L52-a,779454 2830.776764 mm

.n4084 za-m

.771411 2383

.7e874e 2658

.7m3 =a4e

.763447 S8sa

.7t@39 =

.7*X33 m

.7>5459 2812

.752947 m

.793343 2533

.747W7 2887

.7451@ =7’S

.742%1 -l

.74CQ1O Eusa

.737447 ~~

.734892 Rs4e

.732346 m

.725.%)7 awl

.727276 z-

.724754 atiB

.722239 -7

.7L9732 24*B

.7L7233 P4U2

.714741 24s3

.7122% 2.47e

.7C9782 z<ee

.70793 Z+ea

.-P4853 Zz!fo

.702ti =m

.m* Z4Q8

.697516 24B0

.697X16 E4H

.6*3 241e

.@247 z408

.b87839 a~ol

.68+3e mm

.683045 ~

.&c659 m

.678XC 2ST2,67P 2-5

,673543 z-,671M5 Z*I,66M37 -8,Wp Em6641% 23S0

&1826 2s2e,6wj04 BIB!657189 -,654EW AI672579 =Br

6w284

1.08689 4SLS1.M196 4901.0776 4a7L.07219 484L.C%735 #

L.062-j3 4791.05774 477L.0~ 474Low 4721.04351 4aB

L.03M2 4471.03415 4-1.02951 ~L.024@ 4Y9L.0m30 457

L.01573 454L.OLLJ.9 4=1.03666 44,L.m217 44s.9976$-2US I

.%3241 4428

.9%913 4406Lm 4ME.9%025 -I.97w~ 40W

.971326 4a.Ie

.$%7010 -4

.%2716 4W

.m4b3 4-

.954193 4-

.9bw63 4204

.945755 41m

.941%8 ~Im

.937402 414T

.933257 4126

.929132 41CU

.ww~ 4034

.9m9& 4*

.9168% w

.9128% ~.M&

.919912 4C!C4

.94828 asss

.9X1823 eE05

.W@ S04e

.8*E =BZ7

.8%985 mm

.09y77 3303

.891.U% SS7L

.8n317 sasQ

.873465.amu

.@$@ .mlu

.65X17 8797

.EWWQ 8773

.8~241 .3761

.85AWC 074Q

.8w737 S7ZS

.8k7012 a70.s

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.83~4 2974

.83594c m-

.8-4

1.74172

1.73859

1.735491.732371.72327

3M

aunlJ

elc

em

1.72618L 723091.7XQ2;.711s

ma

m+‘amBmBoa

1.710831,707791.704751.701721.65070

at!x

ea

002

.

1.69%~.w71.W661.6%571.69363

1.683701.677731.674761.67Wo1.662.85

Sal

201

6!8s

29s

SW7

R37

2m

b?a5

295

.P3e

1.65w1.64836L.64%T1.64.2571.6369

2912802s029B

Eea

1.636811.633931.63107L.62821

S!ea

1.62535

1.62273L.61965L.61&33L.614cxL.61117

233

230

aeo

27s278

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277

270me

L.%)11 21.4a74.274274273

~.n7’i5L.57EfIl!.?7227..%953 .

,..*

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.

.

.

NACA TN 2744

TABLE 1.– CONTI~D

39

.

.

.,

.-.

. /-’

t

.4CC

.401

:$%.40h

.405

.405

.407

:%

.klo

.411

.412

.k13

.414

.41>

.416

.k17

.418

.419

.426

.4’ZL

.kzz?,423.424

.kz5

.426

.4-,W8.km

.k~

.431,b32.433.h34

A35,436.k~~.435,k39

.440

.441

.442

.443

.4wl

.445

.ku

.447

.448

.449

.k~

.ql

.42

.453

.454

.k55

.k%

.457

.km

.459

.4CC—

—

.

40 NACA TN 2744 ,.

TABLE I.– CONTINUED

t e(t) b(~) c(t) a(t) e(t) r(t)

.J@ .u475a a .Pw 1- .Wa - L.412W -4” 1.93026 == 1.I..26?3=

.& .ukug - .@?l~ 18= .6w370 - 1.40@ ,43 1.92495 mm 1.12@ ea

.462 M& = .520235 1917 .634783 m-m lk-rll Q4s 1.919s5 * 1.W755 as

.463 W .z8f18 aele .632m9 em 1.W 244 1,91k3E - 1.12&l aT

.464 .112219 BeB .p64& lem .6&@A - 1.40224 =4s 1.9W13 Em 1.12e89 87

.46s .Ulm -B .5145m 1901 .626&91 mu 1.39931 E4e 1.XJ3E9 ml 1.W3 ST

.466 .11o966 - .512799 lee7 .6237.72Z791 L39738 az 1.8gW9 - 1.13@2 57

.467 .1.1o343 - .aqa 1-1 .621019 =nu 1.39495 e4e 1.8@A 5.17 1.13099 8s

.469 .10972L a7 .@811 leas .618302 - 1.3%254 242 1.8m31 Em 1.KW7 en

.469 .109107 as .5c@26 w .6155& - 1.3903.2 =41 1.8a315 514 1.E!Z368

.470 .K!&g2 m .-5 lam .612895 =7 1.3~1 241 1.67801 mm 1.13293 -

.471 .N@91 - .m3170 Lan .610209 zem 1.3a730 M 1.6-7289 Em 1.13

.472 P ““.llx&5.5 z :i%a :% .m&59 =4 1.3W50 ao Lefz?l - l.1+% ~.107272

.473.@7533 = 1.W - 1.M-i9 - 1.U 30 -

.474 . 1W62 ml .497574 1=5 .6@214 -s 1.37810 .e40 1.85765 BCW 1.13fi -

.477 .1o5461 ma .495719 1-

.476 .1CW53 Em.~~ - 1.37S’(0Zc!a

:?s R1“% :: ;:%% E.596938 _ 1.3733- e!m 1.

.477 .1c4267 -

.478 .103674 SEO.5%?16 - 1.37W - 1.8W6 4W l.~~ 70

.45W8J1 1036 .7317!39ma 1.36954 me 1.837fl 4W 1.@49 n.479 .103C.W see .4ea349 leal .535’103-i 1.36616 w 1.832&I ● m 1.13919 n

.4& .1cc2496 s-

.481. .1019U s=.4@18 1- .W - L36379 =M 1.82-@ 494 1.I.3w n.#6~ 10m

.4&2 .101328 u.93932 - 1.36141 =97 1.8z?j’cl4W. 1.1- ~

.4?W13 =3 .5s13& - 1.359?4 - 1.~778 4W 1.14133 =:% .;yl’r ~ .48.l@ ml). .Tm =s0 1.35&9 = 1.81.W2 4- 1.1- n

. .4792k7 1609 .~62y =0 1.3*32 F LED7B 4rT l.lm -m

.483 .C595958 rm4 .477442 1s01 .573712 =Y.sO 1.3519C - 1.EW.I.24ea H~9 :

.485 .099@34 mu. .kml 1- .ml182 -0 1.** m 1.79%7 4e4

.487 .G9W534 =74 .473849 ~=1 :X$& Z: 1.34725 =Qs ;:% :: ;:;yg :

.4&3 .0y@3& 6849 .k7205A lma 1.34qw =5

.489 .oq321J ca24 .k’pM.9 l?B1 .%369 z~w 1.34255 Qe-i 1.78* 47a 1.14642 m

J@ .0$67W msa .4@-97 l-n-r .%U@ 24aJ- 1.3kU21 n= 1.77w3 47-? 1.lk715 m.491 .0$61988 E976 .4@lo l-rm ..@&l 24n. 1.33787- 1.77k26 4-X 1.1479374.492 .q3pm3 H .4649* lm .*O ma 1.33553 a% 1.769w 4-m 1.14-%4 -m.493 ,Cpm @5a .46D71 17U2 .593749 &se 1.33320 as 1.76477 4m 1.lk9.494 .C$!45339aml .4614@ l-me .551.297*R 1.33@7 2QB 1.76@4 4m T“l.lxl 75

.J+95 .w39938 == .W%m 1=

.496 .C934361 == .4578$+ 174s:= yg ;.93ZW ~ 1.-Z5534 -

.497 .w8 -mR

.456150 1744 .543%9 =418 1:~~ ~;:TWC6 ~ ;:% ;

.49a .cg23 1 mm .454AC6 l-ins

.49q .c91&s78S=0.*X - 1.WX ml 1.W32 4e4 1.15317 ,*

.452667 1m4 .73ml -5 1.31.927ael 1.736$3 465 1.15393 ~

.W .w~w s-s .4X933 1790 :;;pJ ~T 1.316% “z-. 1.-(

.wl .C.937342USaa .449203 1725 1.31465 Z& 1.72% :: ;:;% z

.502 .@cIXllo Eat@ .W14T8 I-rzI .932023 Zen-r 1.31235 zel 1.72X4 45s 1.156?5 r?

.503 .013*K0. m .J!-4mm 1=6 .X36% =-aEs 1.3-ld 230 1.71= 4CU 1.Wl@ -m

.5* .#391417’- .44&041 171.1 .527297 = 1.30774 220 1.71370 455 1.1* m

.m .0W175 E2a .44233C 1707 .724948 Z1 1.30745 2?e 1.7C915 454 1.1- 7s!

.X6 .-17 =4 .44@3 1708 .52a$S7 Zaal 1.30316 a. 1.@.51 45e 1.ln38 7e

.X7 J$75W3 =81 .4389m = .PW6 = 1.3CG5-la 1.702’C94s 1.l&o.6 m

.X@ .~~~ =ea .k3T~3 1-4 .717s53 -4 1.29359 a 1.6%1% 44* 1.16@ m

.~ .0965344 5145 .433529 1594 .515639 es05 1.@530 228 1.6q09 447 1.16176 7s

.~o .C&olw - .433840 le84 .513334 - 1.29401 = 1.68%2 447 1.1= -W .0s55077 EJm9 .432156 1080w

.5uL038 _ 1.29174 = 1.&215 444 1.16336 so.O%m Born .43cW16 lam .*91 - 1.28+!6 =2-? 1.67771 w 1.16J+16-

713 .C@Ap 50s .428-W 10?1 .7.%472 w 1.28TL9 e27 1.67327 a l.ltl$m =514 .0939e.4950al .427130 @a7 .- - 1.2E49 - 1.66885 u~ 1.16579 w

515 .c83W18 uo- .4254-63lEU.P516 .0’32 10 49s0 .423531 lem

&

.y194c - 1.28265 - 1.64445 - I.m SD.

.499637 Ea44 1.2@J339= l.ti 4as 1.16’p2 =.4Z2J.43le= .497443 - 1.27813 - 1.65%9 4s-r 1.169424en

% :%986: k? .4.2f&8glma .495207 22ze 1J?7597 = 1.6z31 4m l.l@q ea519 .Cf@@o 4E19 .418840 184s .4W979 =, uz7361 ==5 1.W 4= 1.- -

520 .Calcml .k17195 .4$Q76C 1.27I36 1.&263 1.17q3

4

m–

.

.

.

.

.

.

,,

‘---

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R.

.

NACA TN 27hk

TABLE 1.- CONTINTED

41

.

.

.

.

.

.

..

t a(t)

.Mlaxll 4ss

.m~lm 487

.c&a2y3 4eE

.0793W m

.W@@ 4m

.c7’85739 47r/

:%%% .?

.0T71.4JL3 472C

.q%w 47D1

. qalg m

.Wmk 4eEE

:%ER z

.W43429 4s5

.07*3k 45T4

.q3426b 49E4

.W?%@ 45s2

.0725174 4*

.072#&2 44s0

.07’161724473.

.CTU701- 44=

.Wm -

.0702&2 4410

.CE@&2 4sea

.C@uA 4-

.0s3s6ss 4=s,@.M31M 4=.s.@@@ -.Ww 4=35

,c%72351 426s.am -sJXX3W3 4&n0J&@33 4aC9.@@24 U1

05s3.233 an,C64R.52 41?U.,C&2$!l.1 e,0638779 411a,0534.% ~

.C63C573 a?=,X2@ 40%

z=C&22,X18@7 4-

,tilk* m99

C61.0393 597’sc&@U. mslD!XY2g ~

0594533’ E305

0595’78 3sa7

05wm -

%%%=C5-7Y4C ma

CmJW a713sc5555;g ~

O%@+ mm

05%32 a-ins

W%@

b(t) c(t) Ci(t) e(t) f(t)

1.17073 w1.17157 s1.172kl &1.17325 51.17klo m

1.17k* m1.175Z0 m1.176.% Ml.lmz m1.178* 67

1.17s!27ai-1.lEW2 871.lW m1.181E7 m1.F2f275&a

I.m@i m1.I.EW3 m1.I.0%2 sol1.H3632 601.XTE2 m

1.lE312 01lJ.8%13 =1.* =1.1* 911.19177 =

1.lWO =1.1* =1.19455 -1.@9 =1.1*3 =

1.19737 %1.1%3 us1.lN 6-s1.Z?XG2 m1.201M3 m

k.2cE14 961.20310 ‘as1.204@3 971.- ee1.*3 es

L.WK?L m1.2@Xl 69L.2C@g 691.- 100L.21C@ 101

-

.●

k NACA TN 2744

TKBLE 1.– CONTINUED

t a(t) b(t) c(t) d(t) e(t) r(t)

.m .0552629 a-rm .3249as 1~~ .571MI lve 1.1406C au 1.40451 - 1 .=757 Id

. ?s1 .0%923 -8 .324592 l~a . 3@lo3 Irn 1.139+9 ’12 l.- =4 1.22%5 1+

.* .@,5233 mm. .323183 Ioes .367932 ITC.8 1.13637 23.1 1.3972J u 1.22573 108’

.783 .@l* cum . 32178a lsez .36%166 17s8 1.13426 al 1.39* - 1.2W31 10e

.%4 .0537911 %35 .320396 Ime .W+w 1733 1.1%3.5 all 1.3$%9 ml 1.2315Q 110’

.585 .053M16 =1.o .319009 18= . 3626% 1747 l.lw~ 211 1.%3 n~ 1.23WJ 1-

.596 .05306% mo .317623 1081 .%0937 lT~l 1 .U793 X1O l.m ass 1.23409 1=

.m ma@ * .a~~ lQ~ .3596 1- 1.=3 =“ ;.g9g ~ :.:2: ;::’

.W .0523477 me . 314%5 1874 . 35’W2 Ln?e 1.12373 =1

.~ .o~l~ -* ,313491 .,.79 .375703 I= 1.uzL62 Z. l:37m+ am 1.23742 II=’

.%Q .0 S636L W- .m 1Qe7 .353931 1.n7 l.~w .20B I.* SEE

.5gl1.233* lla

.Omw S514 .307% 1s83 .3722& L71O 1.11743 S1O 1.*93 35=’ 1.2* 118,.W .0P9316 s= .309391 lam .35@5% 1705 1.~533 *1O 1.=% e= 1.2~79 118.593 ::&cl; :: W@g ;= .3J@949 l~a 1.11323 =B 1.35785 ~~ ;Alml ::’.nk .347150 B 1.lJ.114209 1.3%32 a= .

.?95 .0498973 844e .33593 law .3WE”9 1=7 1.lC’$WJACJ9 1.3* am

.5961.2442J 116

.0495429 242s .303973 1C43 .343771 LOel 1.1cK96 209 1.34729 0s0 1.24535 1la

.597 .&92CCQ WIB .3026% 1~2 .342W0 187s 1.10487 m 1.34379 m 1.24651 115

.% .04W597 me. .3J12M 183e“ .W15 le-99 1.lWr8 - 1.340& -’9 1.24767 118

.~ .0WJ192 ?a.e.C.2999k7 1sS5 .338746 168s 1.lWO =9 1.33631 ~g 1.2b@3 117

.6W .0481812 mm .2386U iml .337083 185s 1.09W 206 1.3333~a4U 1.2= u-r

.691 .@7w5u C&a .Z97281 1a20 .337425 1851 1.09653.=0 1.3Z987 we 1.251.3.7IIS

.6)2 .04-rm4 mm .295953 laas .333774 1- 1.C9445 - 1.32641 846d

l.- 110

;g; .:43! ~, .?34628 1-1 .332.I.%1-1 1.092?rI=6 1.322%5 =4 1.253 11s.2-33337lala .3W law 1.WW 238 1,31951 *a 1.25473 no

.@ .0467165 =m .291989 1=4 .3283% 1=9 1.C.W21 *T 1.31623 =3 1.25592 lal

.@6 .o~l~ - .W@7 lS1l .32-fi24-~ 1.@514-=7 1.31.2&3%2 1.25713 lm

.&)-f .okW20 -B .2@36k laoe .Rml Lele 1.OMT 26s 1.30$23 %1 1.2583 MI

. ma .0455372 m= .2eW% law .323983 Ielu 1.08155Q07 1.30582 .%1 2

.@ .045214C -5 .236752 1=11.259> 122

.322377. 1-37 l.o~ 20-7 I.3wkl QEB 1.26376M

.610 .l)44&~ - .@5i51 1297 .320764 1.s01 l.o~ 207 1.255W we 1.26199 1=

.61.I .0447p5 ales .2@A174 >Za .3.1916315GU l.omp - 1.2%3 ens 1.26* l~s

.612 .0442542 m.aa .2%2@A3 lzel .gm 15S0 1.07’3-pm7

.6131.292’25=7 1.26445 194

.0439374 W .28L569 LEa7 .315978 lW 1.o@5 2CU 1.28383 =7.614 .043!$s22mlm .2&Q& LQS4 .314394 m 1.*Z 1.26551 ~ Ra :

.615 .0433086 eo .27T@3 lam .3E815 Im+ 1.C”5 32-37

.616 .042~ 81OB .2777W Ler? .3n2kl J-!m9 l.cJ%#s ::?% z :::%1 :::

.617 .0426%1 BOBS .276441 1=74 .3C5673 lE-aa 1.063ko = 1.2n4’f - 1.27v71 1=

.618 .0423777 w .277~67 1=71 .@ll.l 1=7 1.06137-

.6~9 .04206w oo~a1.27m.3 S3a

.2T@36 lBW1.273.* 1=7

.3CX554 155-= 1.07929 a 1.26831 E92 1.2R5 128

.6ZQ .0417641 *Z .272629 I* .m 1=7 1.05723 E’m 1.2649 a.?a 1.27453-

.6C!1 .0414599 8027 .271365 12GC .303457 1%1 l.mla Z!C-a 1.26217 U50 1.275& m

.622 .041157z Solz .270105 -7 .301914 lrs.e 1.05312- 1.25957 w 1.277-IJ.Im

.623 .04@& mm .268%’8 12% .3m3 lSYW 1..E

CGJOJ ~ 1.2%57 s29 1.’2784112C.624 .0407564 mea .26759k MI .29m 15=0 1.252N a2a 1.2m 131

.627 ,0402* ,Sea .266343 ~h-r .297322 1= 1.04@~ a, 1.249xI - 1.28KL2 lea

.626 .0359618 mm .26* 124. .2558CE 151. 1.0449 m, 1.24572 m 1.2@3k la=

.627 .0= 2Jm .263852 IS41 .~428’f lme ,1.o@87 ~ 1.24245 =6 1.28366 IOS

.@ .0393732 - .2626= -e ,W778 1W3 1.040% a 1.23919 ~ 1.23459 lM

.&9 .039@l.2 2805 .261373 w .2912-j3148s 1.03878m, 1,235+ - 1.28633 IW

.630 .0387907 ZWl ,2@139 laal .289v4 14- 1.03673-

.6311,23265Qa4 1.28767185

.038=7216zen .25393 1228 .2&@l 14ss 1.0346 Jdc4l.oA-

1.22945m 1.2F$WJ lea.632 .0382141 ZEal ,2Ti6LM =4 .2%791 14*.633

1.22@2 m.0379Z0 es46 .25645 1=

1.29038me.2@307 1479 1.03061- l.me= 1.29174lae

.63A .0376437 *-L .qy34 1218 .28%8 MTQ 1.02677204 1.219T7= 1.29310la

.637 .03736@ au. .2540~6 121s .2&355 Lee 1.02 3- ‘1.21655- 1.29448mt.636 .0370783 Zw2 .eml 1.2,12.2838% 14M 1.CQ 9204 1.21335a20 1.WXJ6 ~

.637 .0367986 2767 .251589 1s08 .279422 1450 1.W!245 SIX 1.21017m 1.29T2518s

.638 .0367199 v .25a381 Mm

.639 .0362427 z-me .2k91m IZ02 :XR := ~:=8~ i:~%:: i::% :::

.640 .0359659 :. .24797’3 .277062 1.01635–” 1.2C@9 1.30145

.

.

,

.

.

.

b-.

*,

—.*.

.

.

NACA TN 2744

~AB~ 1.- COIi’TmD

43

.

.

.

●

.

44

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.722

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.725

.726

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.’730

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.737

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,7P-w‘f%,753$79

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a(t)

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.02@279 1-

.a2q376 LOOI.020~5 1-.a203605 leas.a201m8 lm

.Oly%ml 3.045

.tJ15$3036 IX

.ol@CQ 1=?2

.o194@3 ml

.019569 lmi

.Olwa 17W

.01@779 irrf

.016-72CP 176?

.0185435 Lm

.0183679 1744

.018193> x=

. OIEQO1 1-

.0178478 la

. m76765 1701.

.o175c67 10s0

.o173375 167-s

.o1’116!X I*

.017CXR7 1-

. o16@@ Ze42

.o165721 s

. 01.65c85 I-

.OI.63439 mle

.0161%3 mom

.016cQ38 LSW

.01- 1584

.olmc@ 1E74

.013486 ma,0153923 155U.01*3-70 l-a.ol@2-7 1sss

.olk9294 1522

.0147’T72IEIS.

.0146260 1YQ2

.014479 14aa

.0143266 14@l

.0141783 147a

.0140313 14E2

.ol@2@ 1451

.o1374m 1442

.0135959 14ss

.0134~6 I-

.0133J.0414=,01.316$z214cr2,o13ce93 19s3.o12~ Iaa$

,Ole@ la-m.01Z6141 z=,0K4777 18s4,o123423 -5,0U078 I.W

.o12c@4

TABLE I.– CONTINUED

b(t)

.I.81W5 lolf

.18G426 101C

.1794U 1014

.3.78397101C

.177387 1007

.1763% mu~

.175377 lml

.174374 222

.1733-75e=

.172379 m

.171385 920

.170395 9s6

.169407 eeB

.’16W22 eel

.167441 em

.I&A61 97U

::% ‘X.1635-41e.qs.162572 *

.1616q _

.16064s w

.lW 857

.1%728 ew

.157’774Sa

.15@3 we

z.15 74 Me.1 928 m3.1>3935 S4Q.UW5 n

.l~q B8S

.Hl172 -

H#g. :

.14746 WI

.146739 810

:;FG :::.14375-2Wlo

.1428’32em

.141974 m

.141M9 202

.140167 ws,139268 827

.138371 =

.137477 aQa

.136535 em

:X@ z

.133928 e.sI,13X47 S-7a.* -,131@f s-mJ30421 ea~

.B5Z =>128694 w,12@20 eaaW?@@ S,9lle@9 a57

,12z42

c(t)

.1* 1174

:3% ;:2

.193135 11=

.191973 ll~s

.1K4 LI=

.l@.6& 11,0

. 1&3510 1140

.187364 Im.

.I.8@.21 llee

.18Y383 LLW

.183949 1180

.m2819 1-

.l&l@2 1-

.l~o 1110

. 17*Z 1114

.178337 Llll

.L77Z26 Llcr

.1T6u9 uce

.175u16 1OS-J

.173917 1a85

.172822 10.l

.171731 loen

.17c#t3 1054

.169ffij’m

.lJ$@l’Wlo-m

.167444 1-

.166331 1082

.1.57263I.ms

.3.64198lW

.163m7 I-

.l@c80 locn

.161u27 LO.SO

.L~ 1.c4a

.158931 1W2

.1* lC=S

.l@~ 1C=4:;.99 :0

.y3757 10=

.lmv 1020

.151’713lola

::@ ;::.1M675 1CL05

.14T6T0 1002

.14.6669s=

.ME7, S-al

.14 70 2ss

.143* 287

.m e=,lkrn3 s-.140732 075,139_@ em.M783 e-m

.137813 ‘aeU,136%7 W33

::= ::>133970 Eme

)133o18

(1(t) e(t)

NACA TN 2744 ●

●

t

.

.

.

.

.

..

.

.—

.

,

NACA TN 2744

TABLE I.– CONTINUED

.

,

.

..

t I a(t) t b(t) ! c(t) I d(t) I e(t) I r(t).76).761.762.763.764

.765

.766

.767

.763

.769

.775

.7%

.m

.-m

:~3.7-L%

.793

.791

.792

:;3

.o12074h lea.?

.oll.941eisle

.OU.8KX2 lSM

.0H67* 1.2w

.Ollmx 3-s07

.Ollkll a

.o13.@32 lzrm

.o111653 M

.oti3 ml

.ol@152 3.241

.OI.079U -

.01.C66791224

?.O1O5A 5 J.214.O1OM 1 -.olozK136m

,c@33Ca 1090.W39269 lmo.c@a179 loez.02.Woq 1074.cCe.scQ3 1-

.Co3&w Iw

.m835@2 lCUS

.cdS285k Iosu

.cKW815 lml

.W@7’% 1-

.12z242 Ma .133018

.L2k* Ea2 .132070

.123537 -a .133=7

.l.E2@3 -e .13m83

.12L343 m ,H4T

.+21m3 .541 .128311

.Iimlm S3s .127@J

.119321. am .126452

.U.w - .L255z@

.m765A H .Lal@

.l1682k 8.= .12*

.lls E24 .E?r16

.115172 e== .m. uk%u azo .-.U3532 =s .12CXl%

. W2714 Us .119155

.U1.@ all .I.M2w

.Luca9 Boa .117365

:~T; ~ .-75.Uz@

.m8@ ml .U47G5

.1O-EK9 m .u3826

.1o7070 m .-

.lo627i mp .I12076

.D5481 = .IJ12Q7

.104t@l m .I.JD340

.1o3933 m .Iwkn

.I.@U8 -ma .lck%17

.lcc2335 -ma .lo-nti

.lolm -r77 .lo6@

.1KJ778 m .lc&@

.m3 ‘?-m .Ilti211

.OX* -mm .mk369

.@34610 WS .103528

.C976%1 m .1CQ6%

.= --- .T@m5 -.71=l.9 -T .7@72 =12

Mo .7m 2009 .770W =elLK= . 707a3 -1 . 763A9 =-w .70%= - .7@5Jo -

.810 .m654mA.& .mJ4x&

& :CDw&.

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.83-8 :CIW467

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.&c I.Cm6al

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S2s

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,

.

46 NACA TN 2744

TABLE 1.– CONTINUED.

8(t) b(t) c(t) d(t) e(t) f(t)t

.@c

.&l

.822

.823

.8A

.825

.&26

.827

.828

.@

. 83J3

.89

.832

.833

.834

.835

.836

.837

.838

.839

:&l.M2.eA3.844

.847

.U6

.847

.a+a

.849

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.m,852,873.854

.855

.8$

.857

.8X,8B

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.551,%2,%3.8s4

,L5@.&%R&,85.5,869

,870,671m,873,874

,675,876.877878m9

e.%

.C@&L 017

.cqzi~k 81C

.Coymi as

.@54382 ‘la,

.0353%7 =7

.00yxklo 7SC

.-0 77=

.m~k~ -me

.co@18Q m

.w49724 m

.@@973 744

.- 7s6

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.OcW@ -m3

.@&!kl T15

.0W5326 70.9

.0cW618 7-

.CK!&3917 w

.0043223 em

.0CA2535 a

.0041855 em

.C@Qlee a07

.@WJ15 ..0J3Q39a75 .s53.00392@ ma

,C0385.% 6~S.0037917 62s.@1372@+ -.IX1365% e-zo

,m36@3 plc.

.0035425 em

. C@4819 Sm

.ca3421.9 593

.0033626 B37

.CQ33039 B&c

. CX332455 674

.cu31&35 =e~

.003uJ8 ml

.oo3q57 mn

.cm302W ,4e

.-53 54s!

.OC@lll H

.CW8575 go

.cKr28cJt5 ,2s

.coq~ 51a

.CcE@A 511

. CK126493 w

.Omxl %

.Wb

,0 CQ4995 487

.cK32@!.9 4U,m4026 475,CKe3551 470,CK@%l 46s,CIE2618 46s

,Ixe21.% 452,0a21-ra 440,CfmZ62 441,KImW1. 4=,0CP0387 42s

tCQl$gp

.078&73 eem

.q79=@ ee41

.q72767 eele

.07659x mea

.O-E&61 am

:.= :Z.qy34n 6705

.un~ 073s

.~% e-r=

. 071an 8707

.07UM eeel

.0704787 ees5

.C@@o eeae

.0591931 @aos

Jg!4g .992

.C@rj-fo 6524

.c@246 am

.06%747 M-m

gE& :::

.@53301k esae

.c&!6W 6s41

.CA20305 0310

.C%13939 eae

.@qToo -3

.c&o1437 eas7

.@$&KK1 8211

.WW39 e-

.@28@ else

.w@47 elm

.W7W5 8105

.*WO C-279

.0558331 eosa

:?%2 z:.VJ@52 mm.0s34279 5m7

.0528332 a

.07Z412 m

.051651.8E-7

.q51@J 5041

.O@@lo ,.914

.d+9@6 573-9

.04932f39 s7el

.&374J+7 W.94

.0481713 5707

.047&X6 s

,0A70W 56%,0M46n 5s27,&5@4 mm.$J3# :;57

,cA42z4 mle.CJ$36M9 549s,@31313 Mes,cJ+2+J@ see,@2c4@ %11

,tilk~

.c&q95 7435

.@3133m T4e4

.‘W30%36 74aE

.07mti 7401

.079KXY3 -r2aB

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:0%” ::o-@@ -rE4a.0754473 721=

.q4T261 71m

.C@wxi% 7148

.073’29~ 711S

.L+72X13 70e8

.0718727 TOSS

.071J.672 ~,

.qdldL7 aeea

.c69-76% ernl,@%93 SE=91.C6@@ em.

.0676263 6-

.@56@+ em,

.CE&i157 -

.c@351 ewe

.C”W957S .5744

.06Jt2831=no

.M36u.8 w=

.W36 -

.C&z?’j’%.ma

.0616164 ems

.- 8569

.c@31316 UE=B

.059GM U4m

.@@92 e4ee

.0%3526 Ma,

.omo91 .s4U

.omo69T e878

.0%4314 0%3

.V37971 Usl=

.0551659 m

.0545379 m

.m39129 azzo

.Wm aloe

.qi26T.21elm

.0520953 as-r

.051W36 EOm

.0X340 ecu5

:0=,: z?.049CU7 ym

,ma 5S42

.0478322 5911

.dt72k11 5a.91

.04655% ,.%9

,CJ+62691 sale

.CJ@l&.2 57B8

,0449274 5787

,@4.3317 ,m,&37m 5m5

,04m&5 5ee4

, c&26231

.6ZW

:%d.64+526.6$2486

.Go3k3

.63E196

.636#W

.633895

.6ZQ739

. &29581.

.627419

.62B;~w.c

a=21!25Zlaen140214@

2147RI,4Q215221*PI5s

1.74714 40s

1 .7D93 44!

1.75596 44e

1.76044 451

1.76495 45:

,

1.76l.n G :::1.77872 4831.78340 4ml.-mll478

1.792671.7$-7631.EK12531.EK)T431.81237

481

4m

400404

4 DB

21.3?

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2171

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1.81’j’361.&39l.&WI.@Xl1.83779

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.607En.4

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1.8k3c@1.@k3311.85365

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1.%$$81.675531. W.lkl.m

5a7

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1.&9321. 9c4161. 91CK171. 91#1.y2Krl

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L, 92816 817

eaa

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L.9W35L.W32Z

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1,5%63L.S&J61.97274L.97ML.*3

.XtE1, 56C.W.X@@. X@@.553973

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1.$9’294L.’W@?.a@9?.0u184

MB

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Tcm71a

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2.cz28182,03549?.c42Ea2.05036?.om

m7QS

74 s

7s7

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2.c65~2.073352.@21z.@916~.@97=

770

780

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81623ES

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NACA TN 27kl+

.

47

mm I.- co~cmmn

.

.

.

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J385.W.W7.ELs3.W

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.CQlm m .Chkyx m

.~l$w 41-? .CAc$61y ,-

.CQ19117 413 .o&.&@ .2,

.@m3-@ 400 .0- 5-

.Cm&@ 401 .0393627 I&m!

.Colm Srn .03333.52 8247

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.mli-lu 3s5 .0377837 51=

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. cn163k7 374 .0367%6 5137

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%23 .aX@23 m%6 .IXW&l 187927 .0X%24 Lea,* .m~ 1,099 .@m52kl 1E9

931 .0XIX156 ml931 .OX.4875 17799 .CK@@3 1?’s,933 .OXAm Iw93b .IXK!435 165

935 .CKXM91 la9% .K@c@9 J.57937 .~3372 L-99 .am371a lua939 .0W3%3 14a

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48 NACA TN 2744

TJWLE II.– STEP, COFWER, AND CURVATURE SOLUTIONS

t—.lcn.101.102.103.104

.105

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135136137138139

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TABLE II.- COIKCINOED●

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.l&

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.175

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12973.208

12no1270

5.501M =1

5.46947 al5.43616 alss5.40423 =6%y.T@59 Slls

g .34172 SC@l5.M =95.2@24 ~lo

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llm

lm

3.25J5Pj -4

3.2391 -3.- 1=3.L55J19 1B3E3.17561 1=!

3.l@23 191;3.lklc6 la3.1223n m3.1o334 mm3.WH7 16s1

3.W540 la163.cW21 17963.03c@ 1-3.o1241 1-2.59478 L745

2.97733 lm2.* lnc2.94295 m2.W3 ~2.W 1-

2.8- I=S2.6-7625 m2.8591 Ian2.843% Uma2.8$Z792 Im

2.@2J.c J-5852.7*9 lEEL2.7eu34 L5852.7@J9 1-2.7W3T =7

2.73530 14W2.72037 1*TS2.7c558 ~te~2.~3 14522.676k1 14aEI

2.WXQ 142U2.64776 MIa2.63363 Iuo2.61$3 ISSS2.&575 1s79

2.5g2cKl lW2.57a37 lml2.55486 1=02.55146 Is=-?2.5y31.9Iae

2.52503

lles

1178

1.ln9

11.ecl

u.%

114e

.4U*

.4103s4AC9272.b?CtM7M@@%f/7

.-

.w!813

.Mrq41

.401.675

.4aM15

.3B@

.3939.3

.39T472

.3%436

.3*

1112

1105Ices

4.7EY@24.7&2464.73935k.713914.6930L

4.&@64.64354.6Lw84.59$944.57414

4.T~4.y?3klk.yq’%4A&$5b.46397

k.wKF@4.421404.40343k.fl~4.35X9

k.333714.=b2$&yl4.278754.25913

4.23970$.*5k.2o138k.laas4.16376

k.14521

lG-m

lo%+

lc@a

lCE.4

.LC.4s

1-1

1CL2U

lCUO

kQ4

10L91012lccrr

la

m

690

aB5

S7a674

em

ti7

-

-2240

.394382

.393%4

.39332

.39134

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@9349.369399.W737k.%5395.395421

.W453

.3a349

.3=53JEwr9.*1

J7m9.378751.3ria@.37fwl.375*

-

i7al.9

1W8

953

8s8

106C

1-

1s07

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1855

aea

Uca

.ala

.

.

5-0 NACA TN 2744

TABIJIII.– CONTINUED.

t g(t) ● h(t) i(t) J(t)[

k(t) l(t) m(t)

*.220.221

.*389

.s@370;>76364.57@72.5T-W2

291a

2500

a4 e2E4B0

24=

1.427591.423451.419341.415241.411_15

414 2.52503411 2.5JJ99

410 2.4w409 2.W234cm 2.4~~

409 2.46091.4c@ 2.44841402 2.k3601m 2.k23T2am 2.4L153

Em 2.39W384 2.38745SBS 2.37556f3Bl 2.36377am 2.35207

1305 1.62574 188 4.14211.2e3 1.623@ Lee 4.-3Uaa 1.62178 IQ-I k.mWm 1.61@ Im 4.C$uj5Y.* 1.61785 IW 4.o-@T

lam 1.61591 104 k.@+41240 1.61397 193 4.03~612&s 1.61.2C4 las. 4.CQ55UU 1,61oII ISI 4.CCr26812U3 1.6c@20 I= 3.9757

1196 1.60530 180 3.56MI.118s 1.60440 lee 3.55179IJ79 1.602~ .1!98 3.93P1170 1.60363 187 3.9ME.!311.m 1.59976 m 3.9cml

1151 l.~~ “185 3.&35$-(1142 l.$m l= 3.8S35112-3 1.y$Ty2c leg 3.85 PI.Mm 1.5937 1= 3.83L111s L.*B4 1s3 3.&236

= i:?fx = ::%lW 1.* l-m 3.776C4lQal 1.*3O Z-fg 3.764%6107s 1.*Z 17s 3.745&l

1CS3 ~ . y~~ 1-1-1 3.73@510= 1.~&5 In 3.711047 T1.9728 1= 3.77x 61-0 1.5~42 1= 3.6%7810= 1.77167 L= 3.67233

1.538 .3mwJ1.9R2 .374136.1805 .3732281788,m P3723Q

.371425

:E :~giil-m-r .363~173.1 .3678751U6U . 366*

mea . 3a261=7 .365259les@ .3643#lem .3635371=4 .362682

lell .36832l-see .3-

&; ::*

el 4

Boa ::2%1.971471.W51.95313

.222

.223

.Z24

8.%6s0

es4

.225

.226.%+$24.mm.%4027.5’f15S@.Y@c

1.407C91.403041.3$?321.39mJ1.3$uol

1.944101.93Z4L9262~1.91747.l.Sc.E@

l.mlo1.8w2l.m 02

t;.87s;.

2454244S

a4ee

b?418

a4 cm

.227

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87?

ens-m.s=4.229

.230

.231

.232

.23

.23 i

.55375

.5%383-

.55=XJ

.549s-30

.5W12T2

1.387031.383071.379131.37520

e58

Sso

848

83.5

1.37E9 e59

.237

.236

.237

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.S42591.

.740268

.73TZ%

.53s%

.53336$

.5310S8

.528&0

.F6S4

.>2k319

~.367ko1.36321.35%6l.3mJ-1.3X99

1.34a71.344371.340%1.336821.333C6

1.3292Al

HLS1.31&u1.31452

e4e 1.8s7s51.E!A9741.841w1.833321.&53

am

as

awr

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842

.93a

m

e-m

,,

f,7s5

2’2702288

S.5D 2.2838337B 2.27-2773m 2.26179SW 2.2X59.B74 2.24010

2.%! 2.22938n71 2.2@f5am 2.2c&9S@ 2.197-pm 2.18732

1s44 .357642.wal .35@J6151s .355$95’19CM .3537714s4 .394363

1481 .37355314es .3wrk~y: .3519~5

.353J-4714.94 .3X32

B2e.9=3ala

@14

alo

I.alm1.-lalw1.79@1.78543

7ee

781

776

782

.S22C84

. Z*O

.Z7646

.mA44

.53.32y.

.2k3

.246

.2Jq

.2h8

.249

2224 l.’mtil

;:Z%J1:748s?

-ma2214 -?60

7s8 744705

781

-ma7ms

4).

.

aim

.UQ59 2172

. Xf@q elm

.5@73? elsl

.@+@l Q142

.X2442 2131 11.31@j ms 2.177W 1024 I.* 1-=’ 3.677$9 14= .34@l m 1.74M0 -la1.3oj’2f 3.9.92.16576 101 1.5&19 174 3.64376 1411 .348774 7~ 1.73434 zu1.30357 =3 2.lW lca 1.*45 1= 3.6295 ‘ao .3479W - 1.72713 -fl~1.2H so 2.14651 lC= 1.5’6k73 17=’ 3.6u%5 1- .347211 ~ l.-p~ *1,2$634 aeo 2.136w a= 1.%3~ 17~ 3.6m-T6 IS7E .346434 ~ l.- ,-

.253

.2%

.253 .5003u a= I. .29274 B58 2.u677

.256 .4@189 SLlZ 1.28916 w 2 .Ii670

.257 .k%o~ azos 1.2@59 Q5 2.10S91

.Cm .4935q5 20s2 1.28XL4 w 2.05fLL9

.279 .491&33 am 1.2785i3 am 2.G3754

.260 .48@30 ,0.3 1.274!2’Tam 2.o~$15

.261 .487727 2GS4 1.271.45 3s1 2.G584Y

.262 .WK63 Zoci 1.267%5 tie 2.Oml2X AW& 2C45 1.264b7 we 2.oW3

20% 1.26099 we 2.oho32

.265 .479528 == 1.25~3 .48 2.031ch9

.266 .477W 2018 1.25J@ .ti 2.02191

.267 .475484 2CQB 1.2n64 Ma 2.01279,269 .473476 Is-m 1.24722 W1 2.0037~.269 .471477 1900 1.24@ MO 1.$@76

,qo .469487 1* 1.24041 me 1.$.8*,271 .4675a5 197s! 1.23702 mS 1.q6*,272 .465533 1,84 1.233~_ W-S 1.9W8,273 .463559 lmB 1.230.28 03Y 1.~944.274 .461614 1s47 1.22693 aw 1.9%YT6

,275 .45$657 1e3s 1.22359 eas 1.94214,276 .45~9 l-am 1.2202+ ml 1.933%,277 .45* 1s21 1.21657 8s1 1.$2~

@ .453879 le12 1.2136A am 1.91653,279 .451957 1s- 1.21035 020 1.s’c$24

,2eQ .4W3 l,qq 1.85993

ee7

Cl-m

972

em

030

ml

844

S38

SalBQ4

1.561311.55*1.557911.5%221.5549+

1.552851.mg.1.j41=zi31.547e8I.*3

1.74459:.+&

1:539701.73839

1.536481.534871.533281.53169--1.93010

1.28521.x26951.s25381.72382’1.52227

1.52072

l-n

189

Ies

leB

m

3.*EM 1=73.77431 13%s

3.m75 1=63.547”29 law3.53393 -

3.72Cfi9 1814

3.* IGca3.4949 k,3.48154 l==!3.468S9 m

.3&.%2

.344893

.3k43-27

.343365

.342607

.3U852

.34110c

.3403%

.33*

.338%>

.3383-27

.3373%

.33&xl

.335932

.335W

.334486

.333767

.333@

.332339

.331630

.33W3

.33CE20

.329520

.32=23

.3281.29

.327438

m x.70585lea 1.6yw-ma 1.69153ma 1.awm 1.6@23

i=.e ~.67M674a 1.66k’fk746 1.65937742 1,.65145- 1.6U89

-ms 1.63837731 1.6319- 1.62%7-== lxlglo~ i.612v

m l.mkgrle L.&K@-n= 1.59407700 I.*B707 1. 5KL83

+= 1.57577703 ~.5@6a~ 1.56379s= 1.55787eel 1.5%95

1.54615

m 0

ee7

‘sea

err

678!

Ca-r

8ea

em

e5aIm

917

012em-B

8@E

Me

am

874

me

e

859

asl

844

e3s

en4

lM 3.45594 ls!ea3.44328 IA?-3.43072 12463.41826 =z37

3.40589 1==8

3.39363. we3.38142 lao3.36$32 IaI3.35731 11-3.34539 1184

N= ~=11.50

3.31014 11=3.2995.5 1149

3.28707 1141

3.2Tm ,

.347

04.91+lez181

4378ss

aes161

181

3.58

159

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ala

a14

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e.caMa

1s7

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NACA TN 2744TABLE 11.- CONT~

.

/

1

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.

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4

—

52

TABLE II.– CONTINUED

NACA TN 2744

t. g(t) h(t) i(t) j(t) k(t) l(t) m(t)

.340 .34929?

14’81 1.02U6 Zn 1.48387 s7e 1.43714 1s0 2.’fl444 ma .- InQ 1.25337 m%.341 .34781 147s 1. O’.?~> !270 1.47@9 sfe 1.43* 1= 2.w$16 tie .29.3UW =J 1.*3 -2.342 .346339 1489 1.02305 270 1.47~33 $72 1.43463 125 2.65914 m8 .239639 5ss 1.25Q51 asa.343 .34Jt970 14E4 1.02035 - 1.45651 970 1.43338 1= 2.69155 m .285UX3 597 1.24652 ae7

.344 .343406 .14BB 1.01’76 - 1.M391 5= L43=4 =5 2.6%02 Tbe .=$$3 =5 1.24275 o=

,345 .3419W 145B 1.01498 ‘k7 1.45525 503 1.43ca9 128 2.67653 74s .’.%W28 - 1.2* -

.M .340497 1446 1.o1231 m 1.44562. 501 1.4X I- 2.ti~ 740 .a974~ Ssl L.23507 m

.347 .339349 1442 l.- - 1.44401 557 1.*2 - 2.&l@ w .- - 1.23U6 am

.348 .337&7 14US LW699 =8

.349 .336172 L4W 1.02434 mB1.43&A s=1,43289 3? 1.42s9

1.42720 m 2.1= 2.Z’%2 z ;= z ;:X z

.3P .334742 14E5 1.0U69 m. 1.4q37 we 1.42475 J= P.63w2 - .2853& - 1.2W13 -

.351. .333317 1418 .9%W m- 1.4218-5 %e 1.423s3 3.21 2.63248 719 .28W!59 = 1.216215 am

.352 .331.W3 ~414

.353 .33048Jt 140e :ZH z: ::!1% :: RR :: UR :: :~,~ ~~ i:= :

.354 .w76 14w3 .99U91 MOT 1.405s9 ma 1,41W IRI 2.6u02 707 .2833w sle 1.- ae3

.355 .327673 Leas .XW84 =01 1.4c@.l S= 1.M.869 l~o 2.&395 7M .282--(84614 l.’ml~ m

.3% .326275 ~se~ .935993 25= 1.* s= 1.41749 118 2.5%+ - .=70 mu 1.1

.357 .32@33 1387 .s%3393 a~a=r 1.M954 SW 1.4W0 =0 2.% 639 .28~7

.3% .3234% 1ss1 .%@fJ3 e,w 21.@-24 S= 1.417.0 ,,8 2.582W @R, .&12z :: :::’4 z,359 .3-5 la77 .9-FW4 25M 1.3~ W 1.41392 lle 2.57@6 me .Z30738 ❑ur l.- -

.m .3~7w la~ .ml Esea 1.37373 521 1.41273 1,8 2.%9G3 ow .a9w31 Scu 1.18371 mz

.361 .319367 law .973ch352555 1.3@12 518 1.41155 11s 2.@234 a,g .275725 w 1.1.M19 BSo

.362 .31.&Dl 1=1 .~o~ ,58. 1.36333 53.7 1,41037 117 2.55555 an’ .27%22 BOZ 1.1’7659 C-47,363 .31.@@ 1355 .%7974 e545 1.3%16 510 1.40920 X8 2.54978 872 .!2-7q2Q K-3 I.lw we.364 .315285 1851 .%54’3 25s9 1.35303 SD 1.4cfD2 118 2.542% me .2’-@z2o 4ea 1.I.6976 e44

.365 .313934 1Q*,5 .962893 26* 1.34791 50.91.46% 117 2.53537 m .27~22 4W 1.W32 ~2

.H .3@”99’ l~o .95Q3X - 1.34283 m7 1.40%-9 118 ~.%@ eel .2* 4- 1.162W 041

.367 .311249 1s05 .557833 ame 1.33776 .ws 1.40453 lle 2.522U 832 .276731 42s 1.15949 =

.368 .309914 10Q1 .955314 =12 1.33q3 se 1.40337 lU! 2.51553 ~ .@2y3 482 l.1.fiu as7

.369 .3ca5i33 lazs .95XJT2 * 1.327P 49s 1.40222 Us 2.** esl .275746 4E3 1.lp7k m

.370 .307’@ lasfcl.

.%2 -s. 1.31Tf6 CM 1.39W 114 2AW1 643 ,2T4T69 d.n ,.,4A3, mla4aa 1.32273 49-?1.4olq lL.Y2.5924e M-? .275?57 4a2 1.14* =

.371 .305938 lHB

.37Q .304623 1s11 .*@l 24@@ 1.3r282 4BI 1.39378 11, 2.@@ e40 .274282 434 1.lk2T6 saa

.373 .30331-2 1’=s .%28~ 2400 1.~91 4BQ 1.3763 M 2.k83ti 837 .273793 4- 1.13946 -

.374 .302c07 1~01 .54!03322474 1.3oyl 4= l,395y 114 2.47631 a .27331.4 4a 1.13619 u

.375 :W7007 12-:ER ::: Hi: ‘E :::3::

llB 2.47o48 am .272833 += 1.13292 -,376 12s0 U.3 2.4W.8 528 .~23S3 4- 1.12@ w.377 .298~ 1287 .93m5 S454 l.- ● - 1.39310 1- 2 A5792 ‘3=’4,27W5 4~ 1.126W =1,378 .2@33 1R51 .93cb4812449 I.28369 470 1.=j91g 112 2.45163 =- .271393 4?s 1,1.23~ ale.379 .295552 1270 49zeu32 2-R 1.’27850 4-?5l.- 11X 2.44%9 m-r .27094 474 l.- 817

,~ .294276 1- .w% s4ca l.qkl~ 4-m 1.38gk llR 2.4w tio .2’@4p 471 1.U689 ala,@ .29# lm’? .9231* 24,1 1,26942 471 1.33%2 111 2.’3319 -lo .26~9 470 1.11.373 814.%2 .m1737 =S .%WZ3 =24 1.~7J- 4Be 1.38t383 .m.74 1- .9182-9324..9 1.@X2 4.s7,.38% :: :::22 H :2% :; ;:% ::* .289237 12s3 .9191 2412 1.25535 4m 1.3a530 111 Q.41456 mo .263773 4, l,lcL436 -

,385 .287* 1249 ,913469 24ca I.m-(1 4W 1.W19. 11O 2 .408%! ‘5s7 .2@.08 4e4 l.lf)l.q me,3% .2867~ 1548 .911063 “a400 1.246@ 4eo 1.38309 108 2.40301 52$ .267@A 483 l,q3319 m

,309 2@32,387 .28 P 1240 .-3 =94 1.24149 +$.91.382cE2 110 2.39706 ?s1 .2671Bl 461 1.05513 a

.936269 A3SS 1.23691 4s8 1.38J39J389 :282938 = .XHM1 - 1.23233 454 1.37931 :: ::3% z :EEE? ::: i:rg 3

,~ :~~ :2 .931499 2977 1.22@l 4m. 1.37872 103 2.3~2 ,5.92,26*3 4s6 1.G3525 -391 .8591z2 2371 1.2-2330 45Y 1.377@t 103 2.37363 ~ .3P .2793~ 1=-17 .896-ELIE3’S5 1.21Wl U8 1.37656 Im 2.3678I me

.~~ ;: ::% ~

393 .278M4 121= .%386 2~- 1.21432 U5 1.37548 2@a 2.31W5 s-m. 264439 4= l.~ ma394 .E-RW-2 law? .89m71 ~= 1.205%7 U4 1.37440 107 2.3%32 m. 263987 430 1.0-(41> 234

G393 .27 5 I=os .83%74 we 1.20~3 441 1.37333 107 2.3~ *. 263537 US 1.rp.21 29Z3% .27 llBS .W73% g-s 1.2Q102 440 1,37226 lo-r2.34-W w .263%9 44s l.me al35-7 .273283 11s6 .~w 2SS7 1.l@2 M7 1.37119 10s 2.33930 .= .26z@0 U8 1.@,,@ 2s9398 .27X83 118C .EL32GT72WU 1.19225 4m 1.37013 10s 2.3336!3359 .270aw 1169 .W0316 2=5 1.1.87-a9403 1.36937 im 2.32810 3: *$ G ,‘:%2 ::

4C0 .269712 .mwl 1.u3356 1.36&31 2.32254 26137 1.0567sI

.

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.

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.

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. . <-u -–

NACA TN 2744 53

.‘IM13LE II.– CONDNDEDf

.— -.

mI h(t) i(t)

1-18356 4=1.17924 4s01.17494 4EE

1.17CK6 4s

1.16340 424

.,J$m.401.4Q2.403.404

.%,4MM-(.4CA8

.%

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.414

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.2651m3

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.26M3’2

.2&J381

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z.362761.361pl.-1.355551.35%1

1.35-E+3L 35551.355531.354511.37349

1.372471.351461.372491.349441.348A3

1.347431.346&31.345431.344431.34344

1.*451.341461.340W1-339491.3@51

1.33753I.335561.335591.334611.333-54

1.332@1.331p1.33U751.*1.@@i

1.=1.3X931.3’2yY3l.-1,32*

I1.3231.51.E22.l1.321271.32034I1.31940

~l.ga~1.2754I1.316&ilam1.gk-n

1.31%51.=31.*1.-O1.3102U

1.30329

104 2.w16, =; .259113 4351C4 2.28976 w .’@%@ 4sslm 2.28AW * .2@24Y 6=104 2.z7@ Q .2Z623 4s0Us 2.27374 e .257383 SZB

Lue 2.26845 - .256354 42a12s 2.26319 524 .255- 4-27lCQ 2.2Y95 ~lCG? z.2mk SIB ::% ::Iaz 2.2+757 51s .255291 42S

101 2-24239 ST.= .25W328 4W101 2.23726 512 .21+Ao7 4=10L z.z~k - .2535@/ ● m101 2.22-K6 m .253~ 417K-3 2.- % .25352 410

lCU 2.2L696 m .2~36 *15lca 2.2u9k 43s .*321 41s100 2.2C696 4s7 .WW 41.2S9 2-2ol~ 494 .2514* =0es 2.197C5 4s.2 ,251Cf6 410

SB 2.1W3 4~ .2x676 40Sm 2.13T23 487 .25rzx8 407es 2.1%236 4=s .2k5M1 4Mm 2.1~51 4a3 .2k9k5> 4c4se 2.172@ 4m .2k~ ●CS

S-7 2.167M 4= .2W 4=69 2.16310 bm .248246 401e-? 2.1534 474 .2k78k5 auu87 2.15365 47= .2k7M6 Us9w 2.1M38 4ea .24@T *67

1.o&26JLox

1.037@1.03k3k1.03E0

1.CQW71.026161.023461.um781.Olalo

1.01*1-OK-791.01016l.cC@1.c0493

1.W3233.~b?.59-D-K?.@612.-3

.9?%6

.96-rc02

.-

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.97$%9

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z-m

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275

274

273

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u10 .&3762 2240Llln .FAl~ *5

L.Ill .839287 usUm .83705=3~l,ca .834833 -

lW .832614 QZIY10ss .830359 2s0s10W .EQ@o Zm3sL0B7 .825985 21’aslass &37M a-

Ire-s.m~l ==1076 .&9401 -es1071 .81-7216z7eLo@ .815037 =17sices .812tK.2217L

lGY1 .&lc@l am103 .mz> Elm109.3.&@65 21mLWs .&!@@ 21511M5 .@2059 =47

lM 1 .79931.1214LLoaa .7~o Zlsa10s4 .795532 al=10W .7935U3 EUa1- .79J.372 21Q4

10= .789248 allam .7b-l130=!lls1015 .787215 mm101Z .7EK@ -ElmICas ,7-1 =01

Lms .R87cn m1001 .T7w mu807 ,-fi4512amw .772427 -ml .770342 =-070

9U7 .76%?63 Z074SE3 .7651Ep mUao .7@.19 =m-e .7- mlm .75%93 2057

Wm .757936 =m .75* ms

E. :%?: s

.7WLL6

.417

.418252

.419

.420

.4.21

.422

.423

.k2k

;~

.@--(

.428A*

,Ji30.i31..ky.433.43k

+9

m

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.2k03k9

.23x73

.2?&2i12‘

.237134

.

. .236ql.23X1Q.233957.2325C4?.2Q8fi

.230%9

.223-r6&

.228730

.=Z96

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am

2453

2444

24=

E421

2.1W20 4=2.13952 40s2.13+87 <882.13024 -12.12%3 4SS

.246$73

.246254

.2k5e&3

.2k54&

.24wz4

-96@3-952393-m~.957&h5.955230

.k35

.436

.kfl

.4y3

.439

w

m‘as

66

m

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65

84

S5

94

.

ael

.#o

.441

.~~

.443

.4M

~.=554C.22463$.2.?3$58.222533.Z?lm

.220563

.Z?lm

.~m

.a75&3

.=6566

:2@.=24X5.23.262>.211649

.2W3

.244293

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.2k351.6

.24Y.3C

.952353:=

-*5W.9k3y35

2.12105 457

2.U648 454

2.LU94 45s

2.10741 45a

2.1CE91 44s

am 235:

@-9B

3ea

Sz-3 2324

ea3

.445

.446

.447

.448

.449

2.0@3 447

2.09396 +44

2.09952 U2

2. @i10 440

2.ca’J70 4ss

.2k27kk

.2423&

.Alm

.2im5J35

.24=4

.24%35AC&

.2m03

.239327

.2W3

.233*

.23&03

.237838

.23746e

.237C93

2.07631 AM2.07195 4s4

2.05761 493

2.@28 4S0

2. C5595 42s

2.* 4s7

2.- -2.c!4617 a2.04194 4si

2.03773 <l*

2.O@I

.450

.431

.452

.453

.454

m

9a

93

92

Ba

07s

3?7

a-m

37e

a74

L.220S76.qti.2c@4Q.=7777..2wS18

.2053-S2

..

.453.-

—.

●

&

NACA TN 2744 .

.TABLE II.– CONTINUED (

t I dt) I h(t) 1 i(t) I J(t) I k(t) I l(t) . I m(t)

.460

.461

.k62

.463

.464

.9*179

.9W2

.947437

.944@7

.940746

.237099

.236731

.236355/ .23mi -23%35

E1S5

2145

elm

Zla’r

JI1lS

2109

Zlcm

2!0s0

Zc-s?.

m-m

-

*5

arn e

‘2cD3e

2030

2020

201 e

rim

1-

19M

W7*1S71

1we

8s27

.467

.466

.46-/

.469

.W

2.0B3s 40D2.cc.W6 4m2.cd+’fo 4052.0C055 4031.99M2 4-

.937419

.934104

.930&12

.927312

.92k234

32B0

.470 .1%4W w .727% 1.= .9x958

.471 .19= 916 .725596 1=7 .917714

.472 .194632 913 .72- 1= . 9M+7~

.473 .L93739 90* .72M27 197W .9L1242

.474 .19283Q ,0-? .7Q.549 1,,, .$cW24

=54 1.30332

3s!42 1.23943=& ;W&

SW 1. *’m

9195 I.mwl31s8 1.29503~172 1.2941681e1 1.293258148 1.29242

.31ss l.~1~al= I. 29%92.lle 1..2598231- ~.p~~mm 1. 2f@lo

~e$ 1.28724-1 1.X639ml 1. 2855J!am 1.23468?.340 1.28333

8’X?8 1.28299mle l.%ak8038 1.%1302s97 1.2%3452S97 1.Z@l

- 1.27878Em 1.277942957 1.277102Me 1.276272836 1.27W

mm 1.27ti12e1e L.z73782S07 1.272?6Zasa 1.27U2=7 1.2n31.

2877 1.270493e.97 1.26937aasa 1.2@3%amq 1.~4=-s I.26723..

89 1.99260m 1.-Be 1.98k62as ~.w87 1.9767I.

m’ 1.9727887 1.W

8’? 1.964y

07 1 .*1W07 1.93723

= 1.95339= 1.94955= 1.94573= 1.94193= 1.93815

.s3 1.9343985 1.930$,3ea 1.9z@9B5 1.92317w 1.91946

e5 1.9157704 1 .912Me.% 1.*S4 1.5XM7983 1. 5QU6

84 1.8975s04 1.e9394a 1.89336Ea 1.8%79m 1. ~32j

m 1. 87W@a 1.87616B3 1. 87z+5Lm 1.8691Lm 1.%$5

m 1.8$2X3.91 1.8w2m 1.85*81 1.871@81 1.81@3

400

Bee

386

.233*7O

.233113

. 23Z79T

.232402

.-’7

,

.477

.476,477.k78.479

.191923

.191023

.lpuc

.189223

.lea323

.187b39

.l%yi2

.18x%7

.lam

.1839X

.183033

.182.161

.181293

.UW427

.17954

.176705

.177848

.17699

.1761d

.1~

.174452

.17%10

.17~2

.17195

.171104

-903Scb

ee7

ee4

eao

.717673

. ns702

.73-3735

.711772

. 7C9813

. 7078%

.70596

. -lo~~

.70m15

.70W73

.@8139

.6%-

. 69@77

.692352

.690kfl

.69@i.b

.695503

.684689

. &@783

.&?@?O

.67@&3

. 6T7c@

.675192

.673YM

.6T1419

1W71 .934817Lm7 .901622lem .898439le,e .895ti71s,, .892@.“= .-571=7 .8953-91=4 .-3Lao .8797771s= .876k73

la= .8733791s29 .8702961e28 .e672251s21 .%41641917 .@61114

1s14 .8*741s11 .8zQ46line. .852027lWS .akw1916.20 .84@22

lam .8430351.93 .84m591= .837c92,@E, .8341.35.1- .831M5

.231694

.23L342301

300

3ea

..23@$Z.23CIS41.230232

!WQ

:+2.483.494

8878B3

ml

878

al 3

Beu

37a

877

1854

.485

.4%

.487

.488

.W

,490,491,492,493,494

875

874

372

371

s-se

. 2aWL7 1s38

1U30.227875,227533. m93. 22s953

lezz

1B14.Pas

898 1WV

887

sea

MS

ee3

leese57

esa

89 L

848

e44

.@in

.22@to

.2255Q4

.225n69

lasl1ea3

lamlEST

.224835

.224w2

.224170

.223839

. 2237X

.49>k%b~,498,499

*1

e?!8

.3B7

am

!3s4

1861eA2

83.5

6a6

.902

eao

.lam

184e

1@S7

.22379

.222851

.222523

. 2221*

. z21870

1@2a

lap

1800lam

1?84

1m.?

1Tm

1778

17M

175s

350

we

347

‘m .1663.69 812 .66m83 lML .813721% .165357 eLO .65$322 1ss7 .810844

.16k%7 - .655467 Is,. .m7977z .ti37kl u .65.46u 1850 .@l.19539 J62337 BOL .652762 1-7 . &s22’p

.221545

;=

. ‘223254

340

344

344S41

341

310 .162M ~B .650915” ME . -@p511 .161338 -mS .6&Q72 IMO .79s503512 . Mo543 7= .64722 1-7 .793783513 .1597% 7s0 .64?395 lEa~ .79C973514 .l~$61 767 .@+3552 1=0 .78&72

23 .155174 7= .641732 1s6 .7853ec916 -1773* 78X .639906 1s24 . 782p7

517 .156@3 77s .6@XJ2 Ieao .779.523Z8 .lm?l me .636252 lele . n~oy519 .15y5g n, .6344M 1014 .774303

2e2e

am

2810

2801

2792

1.2%421.%%11.26W01.264CC1.26319

.219333

.2L9514

.219295

.2Mp77

.23..%59

.=8343

.21ElM

.217713

.2173$$,=7@6

17@a

1748

1?38

1-?s2

1725

1718

1711

1705

Lasalaw

1.262391.261s1.260791.259j91.259m

815 .&m633815 .79233?314 .798224ala .7%Y9ala .@wl

.79S29

27a9

2 Ti4

Zves

z7ne

e74i*

●

✍✍✍

✎✍ ✞

✎

>20I .154282 I .632632 I .77VE6 1.25840 I 1.8u70 .216774

.-&

.TABLE II.– CONTINUED

55

.

.

.

.

1

.

‘1

t g(t) h(t)

.597035 1747

.yw% 1744

.593* 1741

.*3 170s

.- l-m,

Z9392

i(t)

.7579% > a-

.755262 20.9!

.752577 m-n

.’W5”XQ -e

.Tk7232 h

.74%72 z91

.718433 2570

.715%3 -1

.7133ce am

.Tlqbg -

.7C&?a3 Esaa

.7-5 -

.703135J-

.7CKA13 mm

.- -

.@~l 24w

.69X52 Zu’aa

.f@5w 2<.@f3n6 =-r-r.m39 me.r58317024@E

.6%709 2464

.678254 EL7,67@7 N40.673367 Z4M,67c93k e4a5

,65.Qw =.65kLo7 =%.65173L _.649362 -,tA7CCm =56

,621#

j(t)

M&240 71

1.25761 $-I1.2%E2 7C

1.2k592 751.24z6 m1.2ti39 m1.2k363 TC

1.2k287 -K1.24.2U m~2k137 m1.24059 m1.23* -m

L235K0 761.z@33 m1.23753 n1.23683 74

1.23639 m

L2353k 741.* ‘E1.23385 741.233U ,41.23237 74

1.=63 m1.23@o 74L.23016 m1.22343.74

L.2zKg 7s

1.=796 -mL.w23 -mL.Z2@ mL.22m -?sLam -m

1.22433 m.L.22@ 7s!1.- nL.-6 71L.Z21k~ m.

1.2.27373T2L.22c01 ~1L.’2I93O71L.218w 71

L.21T@2 71

k(t)

1.81170 9a?

1.&8%-k aes

1.805L9 m1.80@ -1.79574 m

1.79552 am1.79233 as1.7ED14 ala1.7@@ ale1.7E990 au

1.7-I* a141.T76z u1.77339 u1.77@7 maL.76n7 ZUB

1.76W -1.761cc so?’l.m~ ma1.754w 30s1.75U33 m9

1.748-M UCu1.7kyf WI1.74276 ml.wrfz ===L7367e =Ss

L73@ =S71.73@3 2s3L727M -1.,72k93 =1.7z?Cm -

I.nw 2011.71616 -01.7u26 ~1.71037 EaB1.701%9 2s-?

1.-@52 -1.7m76 aW1.6@91 2.s41.69Q7 -I,6XW4 w

1.6W42 2SI1.6&r6L m1.634Et2m1.&L2Q3 Sr?O1.67925 n

1.67648 =61.67372 ==1.67w7 =41.C&Q3 mL@~ m

1.65278 -zL6&w =’01.6x37 =1.65468 -1.652c0 =7

1.64933 a971.64G5 2s51.64401 z=1.64137 HL63373 ~

1.63610

.776567 I=

.7749k5 lmls

.iT3330 Lao

.7-n720 1-

.7k526a U?On

.743755 1540

.742255 14e4

.740761 ~~

.735=E 14e3

.m 147a

.-@yl 14m

.734838 14ea

.733376 14@2

.73190S 14s7

.73@~ 14s1

.7- 1447

.727s53 lUL

.726LL2 14M

.72676 14s1

.j’2j2k5 14=6

.72L819 1421

.720398 L4L5

.73.@3 1411

.717572 14M

.716166 1401

.71k765 l-~

.713370 ml

.7L1979 M

.71C793 la=

.TQ3@

.

TABLE 11,– CONI!II?OED

.

NACA TN 2744 .

.

...

.

.

-—-. –—... _.<a.‘.-4M&&”

8R-

.

.

.

.

.

.

TABLE II.– CONT12iT_IED

t

x.641

;%

.645

:%.648.64s

:$3.652

:%

.655-w.657.6%.659

.653

:2.663.664

.665

.m

.657

.&a

.@

.-670

.67

.6P

.6

.6x

.677

.676

.611

.67e

.675

.ml

.&m

.6&

.W

.694

.6

.6%

.@

:%

.*

.69~

.%

.%3

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.@

.&

:%.69s

.m

Tg(t) h(t)

.0792363 4840 . %35639 14SZ

. 07-a74234 . k*lk~ 1.4BI

.omzx3’2 48a .432654 u=

.Omtil 4 .431169 14 E=!

. O’7T2TL8 4 .4296@ 14a!

.07678% 4 .428197 14.w

. 07630E 4EQ . &6Tt5 1471

. 07ZM9 ~ec . ti5z236 14n

::=? :~ :Z% ::

I.0743830 474 .-.0739392 4723 -W&n.0734354 4710 . kl@&I.0729644 4mL .416401. 0724%$3 4878 . W935

.-m 4.554 ,k13k72

.07E-526 4emY . kuoll

.0TLC991 4e18 . 4UZ

.07%375 452a .*

.O-(elm 4673 .407640

14n

u%14e7

14=

148C

14al

L45s14!37

1453

145C

.@TG9 45s1 . Wwr 1451

.-38 4=1 .kJ4736 144s

.- 452, .kl@T 1447

.c@3n2 43m . 4016M 144?

.067’9%7 4487 .4C0395 144s

.c.67@%3 44- . 35%952 1441

.067’0U!2 4450 . ~ 14.s5

.0C65m? 4432 . m-p 14a1

.O’=wfo 4414 .394637 14=

.C@a16 4@96 .393202 14=

.0s52420 4a7a .391767M=

.C@@dQ 4’959 .390335 14E9

.0643$534== . W* 1427

.I%393414- . 387k

.c%35018•~~ .3& ~:

.C@m2 4=.7 . 3eA630 Z421

. @6k25 4270 .383209 1410

.@@U5 4== .3817Y3 141s

.0617933 4239 .3 En372 la=

.0613659 4ale .378%6 1418

.o&9472 4132 . 3~43 1416

.-3 4= .376131 1410

.@1072 41W .374721 1402

.Oms@ 414s .37393 140e

.m%=7@ 412@ . 3~~ 140c

.0%%534 4- .3iw@ L4aa

.W@E@ -3 . 369~ 140.1

.-9 4078 .W@9 =we

.04763534092 .365301 lam

.057=94 4042 . 3f3w4 3.825

.05=252 4@E4 . 363%9 I-

.(?%428 4CQ7 .36c2u6 I.-

.056w21 Wo .360724 z=

.0556231am .359337 m

.W=58 mm .35-D47 1-

.Cmml 33n .349= lam

j(t)

Llnm -1.1735 =1.lH 88L171eg s=1.1-(M27 03

1.17064 C=1.17CHY2=l.l@o 621.16378e=1.1cX116 -

L. ~6754 =1.16@2 a1.16631 ml.,lm 611.16ya 61

1.16w7 s?1.16385 a1.1632’4 UII. 16263 EL1.16202M

1.1614281l.l!ka a1.16020 m1.1* 801.1- a

1.17839 wl.l~m ~1.15.719 ~l,l~m ~l.1~ w

1.1* 601.154&l ~1.l~420 se1.15361 sz1.153ce m

l.lpk? 501.15

7“1.13U1.1= %1.1* E-9

1.1494-95s1.14ea9 521.14330m1.14772 531.14714 52

1.1= 581.14597 55I.14539 m1..14481 91.14423 g

1.14365 m1.143X3 581.1* 671.14193 581.1413 57

1.14078 57l.lti 571.13* ,71.135q gl.m 57

1.13793 ~

I.beu 139

1.43315 1S5l.w@o ~sd1.k2@6 M1.42432 l%

1.422y3 1=1.42CM 13s1.4153 1911.416Z2.lel1.41471lel

l.blz!a 1901.43.093 1631.*1 lea1.- 1=I.@k lm

1.4.0336 MT1.4o149 m-r1.3962 I=1.3977’6 12s1.39591 1=

1.39406 1351.39z.21 l=1.m38 le41.38!354 w1.3e.571 1e2

dt )

.molk SE=3

.58Q34 -

.5&@6 074

.WJCe2 an

.583uo -

.5&21kl 3sa

.W75 ma

.?=2= Eao

.m SS7

.5*97 S55

.Smko e

:%% %

.574493Me

.573559-1

.5m09 -

.5.71671x

.570736m

.56WJ3 -

.5@37~ 320

.s7946 ez4

●

.

TABLE 11.– CONTINUED

NACA TN 2744

t—.7ct.701.?CQ.703.704

.m

.706

.707

.-m

.W9

.no

.-w

.73.2

.7U

.714

.-m

.716

.73-7

.-M

.73-9

,720.721.722.723.724

.‘l&

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.-m

.728

.729

.730

.731

.’732

.733

.734

.735

.736

.737

.738

.79

,m,741.742,743,744

,747m,747>7487k9

.7557%777‘/%759

760—

g(t) b(t)

.C52ml aas4 .3b966b Km

.q24927 B3S7 .348285 1.974

.qJ21G$0 e@20 .346911 la-m

.0717270 seas .34* K.7C4?.05M7 37= .3 l~o ~~

.B0962J Srlo .342803 Lena

.W59n em .341k37 1=

.-% eme .~72 w

.04$AE!422azm .y!qcl.oleal

.0494702 9703 .337349 1QS2

.OJ+m -T .335993 les-r

.048731.2Sa-ro .334633 L8r-9

.cA836& ~s .3332i7 18=4

.0479989 eae.7 .3315-231sss

.0476331 w .330571 10s0

.o&72730 mm .-l 1s49

.G4691.2S m .327872 1~7

.0465737 3s.-?2.326527 lwE

.0461967 sum .33180 1*

.0J@54q em .3’2= ?-841

.0454870’C.m .322497 lM1

.0431347 ame .3m5k Ma

.0J+478398401 .31981.6 1m7

.0444348 sm .318479 1-

.044G373 a% .317144 1C88

.C437414 M* .31- l=

.0433971 .S427 .314479 lam

.0430544 C411 .my-kg 1-

.0427133 8=5 .311821 I-52-l

.0423738 aoeo .310494 1f325

.cJt203g8- .3cg169 18s!4

.@+lm we .3mi

lC.E1.04136J+7 ma2 .30652 1821.ti1031~ a-e .305203 181’9.- Q=l .303W5 1817

.W0369E saw .3- I.e.ls

.04C0414 -0 .30L253 1S14

.0397’3.4432= .~39 131-a

.0393@l 32~e .298527 1s10

.om53 - .25T31,7 1008

.0387430 ~07 .296cD9 1807

.0384223 Sea .34701 low

.0381031 a17e .293397 1-

.03v895 B1eo .=1 lam

.0374695 31- .- lm,

.0371549 elm

.03#419 E114

.0365305 31OO

.036ZZ05 30e4

. Omlz+l m

.0356052 ma

.0352998 mm

.0349960 GO=

.0346936 em

.0343928 2338

.0340935 R27’2

.0337956 es68

.033493 C!248

.03Y@5 E-

.032$m12 =919

.0326193 QZ.X

.283cQ7

.28uT6

.28C426

.27919

.277851

.2765=66

.275282

.27WX3

.272-720

.zzwo

.270163

i(t)

.3e62c4 1e8E

.3$4s16 law

.WY leec

.3fK@.2 iwn

.379475 1.97C

.377@2 1868

.376133 M.YS

.37446% llz.a

.372826 ‘lese

. yp.M8 lass

.369b93 Iwn

.367~2 DW7

.366iD 1644

.361551 Iwo

.362w lew

.3&274 l’asa

.359641 ma

.35”Em2 1’926

.35&@.5 10==

.3%764 le.le

.353145 lme

.351529 ‘161E

.349317 lee-a

.3483G9 ma,

.346703 lml

.3491ce ?569

.3435a3 1s95

.3kl@ Iml

.340317 -

.338729 IEeB

.337144 1=

.335562 1678

.33398A 1575

.332- 1571

.33c@8 l~e

.yg2’70 Is’=

.3277C5 1~-

.326143 Z-

.324X5 I=U

.323CQ9 1s=

.321477 Iws

.319$29 l-e

.318383 l-a

.316W lose

.317302 15=

j(t) k(t)

1.15793 ST1.13736 m

1.375$’5-1.374G51~

I.13679 m 1.37228 L-mI..13623 57 1.37C93 1701.13566 m 1.3@72 L=

1.13 10 se 1.36693 1*?1.I.3* 54 1.36518 ITS

1.13398 57 1.36342 1761.13341 59 1.3616T In1.13285 5s 1.35992 175

1.I.323C se 1.35817 1741.X3174 5.9 1.35643 1741.13u8 se 1.35A69 Im1.13C62 55 1.3=6 lTS1.13m7 54 1.35123 1=

l.~1 55 1.34571 in1.12@6 B5 1.347EKl I.-m1.12841 88 1.34608 1P1.u785 ss 1.344523 1701.I-2730 5s 1.34269 170

1.126m 53 1.34C98 I.en1.W6N s 1.33W m1.M365 54 1.33’76a 1E91.EXJ. us 1.33591 I.E-r1.1245.5 33 1.33424 ME

1.12401 * 1.33256 1871.123A7 55 1.33059 lm

l.- 54 1.3-2923 lea1.I.2238 s 1.32757 le51.U2184 - 1.32591 10S

1.1213Q 5= 1.32426 X=l.lzm 54 1.32261 1s41.12W.l = 1.3X’$J7 1~1.U967 ~ 1.3195+ 1=1.U914 ~ 1.31770 1=

l.llwo * 1.31607 1=1.WW Be 1.31445 lea1.11753 - 1.31283 ml1.u699 BO 1.31J.22 lel1.IJ.646 = 1.3C@l lB1

l.llm 50 1.3090C Leo1.11539 5s 1.306J+0 leol.llw -= 1.* 1621.1.1433 m 1.3G3’U 15s

1.UW = 1.30162 1ss

1222 .M65 less 1.11..327 sme .n=33 Iem I. W?-(h sa1- .3m-703 1627 l.11.za s

12s5 .303176 IW l.lH@ m1&a9 .307673 1s20 1.w6 s

leel .3M133 1s8 1.IN64 53leza . 3CJ+615 1014 1.11o11 s

laes .30310r 1s11 1.lc959 Sa1237 .301y90 lwe 1.1c96 sL2e3 .3Gw@ 160D 1.XWA w

=- .*577 1=2 1.l@Q2 ‘=1202 .297075 1433 1.1O-(P 5s

lS- .295576 14- 1.10S98 5s

- .- 14= 1X%46 m

la-f-f .292587 L4~ 1.1C594 m

u-la .291037 14s7 1.lo5A3 Sa

1.30C03 MaM?C2Jg 157

1.296% 157

1.WQ 157

1,29374 1-

1.&218 15s

1.’29M2 1601.2@f36 1551.28~1 16s1.28596 154

1.2844E 1-1.28289 Ian1.28135 m1.* 1=1.27829 1=

1.27677 I%

l(t)

.1-11402ec4

.171198 3ca

.17C935 am

.170752 m

.17059 202

. 170W 202

. 1701a5 201

::% z.169584 zoo

.169384 zq

.16gleAIs3

.16s995 lea

.168787 198

. 16s589 1s0

.163391

.168193

.167996

.167W3

.1676d

,1674JX.167a3.16703-8.166’323.166523

.1625435

.K-5242

.I&oJig

.165856

.165654

.l@W(’2

.M5281

.165093

.Ml@w

.1647C9

.164520

.164330

.1641kl

.163953

.16376J+

.163576

.163389

.1632V2

.163015

.162629

. 1626J13

.162497

.162272

.16X87

.~6@33

.16177.8

.161535

.161351

.16m58

. 16@A

.-

.161%22

.ltiko

. 16+X2X

.16m78

. VW@

les

188

1s4

128

izs1%

133

1s318s

1Q2

192

lel1s1

13Clel

1.53

13Q

133

L3n.lEa

l@a

107

187

187

lea

188

mu

lea

1=1s.4

133laz

l@-3

ml

lm

lal

m(t)

.567946 8,4

.567022 w

.*1CO au

.%zm *1I3

.*265 ,14

.~22 E9a

.Y~24 me

.557@8 =

.55135 =0

.5zJ2h5 eae

.554357

.353k72

. 55&’99

.551709

.mo%l

.54=5

.54@2

.7482L2

.547344

.W78

.q~ls

.544754

.>43895

.9*

.%2187

.541336

.740487

.539641

.533797

.537955

. 53w6

.536278

:;;?;. 933W

.53*53

.7W

:;3%.5@55

.@a48

.52&34

.yr@z2

. yX412

.5-

.5247s9

.5239%

..523194

.5=3%

.Sr?lm

.52c&4

. 521XM

.mza

.718433

.517647

.516%3

665

83s

ml

877

878

S7Z3

mo388

-

ea3

ml

&ame

3ss

851

346548

844M2

em

Eao5.F!B

as

ezn

Ela

ala

a17

U4

Ma

81.O

S3B

305

7’s872773s

‘me‘ml

-ma

m

7B4

732

--=5%=

.

.

.

.

●

..

.

.

..

NACA TN 2744

TABLE II.– CONT13KJED

.

●

.

.

..

.

,

60

—t

z.8z

:E!.&)

.82:

.82(

.82’

.82(

.@

.83

.83I

.83;

.83:

.831

,83;,83(.83783t83s

,%.

,%,844

&%-/EMeJt9

%852M3w

m855857873859

E&3

%853W+

2MT

%

87087L872873874

873876877878879

W—

g(t)

.o17731L ma

.olm@ A7c.4s

.0173= 20BU

.01.7u67 s-l

.0169146-~

.0167136MM

.0165142 zsm

.0163M3 l-a

.016u92 mM

.0159236 lMa

.0177293 1929

.01s5364 1s17

.0153447 1904

.0151543 18B1

.0149652 Ian

.0147775 1085

.0145910 1-

.0144058 1=8

.0142219 l==

.0140393 lele

,013858) 1801

,0136T79 17ea

.0134991 1775

.0133216lTm,0131454 174*

,Ouyfoy 17137

,0WW68 17S401262U 1711,0124533 1.s90,0122834 lem

o12114e 1074,0119474 I.ealon7813 I=qon6165 .I*M.0~4525 1=4

CELI’293’51011!olI1’z34lnm0109696 15ea0108110 157401c6536 MEI

0104975 154e0103426 15!!-9Olole$x) 152001o2365 Innm% 14W

WV359 14=W%7 1475IX@+392 1452~ 1451K$?1479 14e-s.

CwcxAl 142a-15 1414CCh9T’20114mC085799 lC.90w 1~-r’r

@o& 1-C081666 1858CXW313 184?CKr@9-n lowCW/-@Ll 1s17

0376?24

h(t)

.1%1% llW

.1** lle(

.193774 llm

.192% llW

.1913$7 llW

.lgnal llmr

.1= llW

.187842 llw

.18%59 ala

.l@A@ UK

.184m 117’8

. 183u9 II=

. 18L941 n-n

.u%T65 um

.179m 1174

.178416 nm

.lj724.4U71

,17tm73 1170

,174W3 11-.173734 1107

,172%’7 1M7,171403 11=!170236 11C4

:::% ::

M&48 ImO,165598 Ilue1~29 11m,tim use162u5 us~

l&$@I 1154I* Uw‘W3553 mIsm n~156352 U80

i= H15* Da151762 Llw150518 1143

14g475 U.42146333 1141lVP-92 X140l@’2 llae144914 1lsr

143~ 11=14ZT%1 Lienlkl,y% Zlw1k0372 u=13924-0 llaa

lym.9 llno1365q8 IUS

31849-1 w llST133594 llse

132468 1124131w U.$!41302N 1-1* l-ml1279V w

W’f

NACA TN 2744

TABLE II.– CONTINUED

i(t)

.m6~ lam

.*53 18sn

.@Jm 1*

.’a2fm6 1s10

.’Xm&f 12.17

.m9170 1e15

.19e@5 131a

.U77542 1810

.156232 IW9

.194924 1.9425

. 193G19 lam

.192m6 WOI

.191017 L2E3

.myp6 1=6

.18%20 m

,16-p.26 w?+.l@a* lE-.184545 .lz~,18- 1Q85,181973 12a9

.~m =,179410 127e,l@1321270,176256la74

,1* 1s?72

,174310 128s,1-fy341m1717-fk 1=s,lm MM,169246 12.50

1679% lasB,J66 a U=!J&l Uw,26ka7us!16W5 l*W

y&-222 124s

157y79 1R41

l%W 1a40

155498 1207

1%%1 12s6

173026 1233

151793 1=1l= 12R8

149334 1==?7lk8107 1RE51465%2 1-14- -I144439 lfile

143221 121714- 1S!141407’90lwa139577 LR1O138367 1200

I* mm135952 leas134’f471-133547 ImlIw lles

1W46

J(t)

l.om 4’

::%% ::1.07434 41

1.073854.

1.07339 4:1.07292 4(

1. C@+6 47

1.WB G

1.07172 47

1.mo5 471.W’W3 4t1.07012 4e1.05$66 471.C6919 4C

I.f=n 471.C&26 ~@1.061@3 4fl.w~ 441.066M 4e

1.-2 401.1%* 4e1.06m 461.0S%14 461.CEA58 46

1.@12 4B1.0657 a1.M321 4a1.05m 4s1.@2jo 4a

1.MM+ 401.C61.39 45l.CE’f#+ 451.- 401.G%03 46

1.059% 4s1.05913 451.qw8 451.058q 46l.q-fa 44

L.mw 45

::% ~l.- 45L.m 43

1.mlo 44L.Q5k& 441.05422 45L.05377 u1.05333 44

L.WE& 44L,052@ 4eL.@2D3 wL.c51% 44L.- u

L,- 40L.05CW 44L.d$fil 441.04937 44L.0W93 48

L.*

k(t)

1.19229 lUl.l$@ m1.MX.9 lE-

1.1!939 la,

,1.187’10 la<

l.lm lm1.le472 l=1.18323 IE(1.183-5 Is?1.1EW9 lEI

1.1*0 la-l#I@l M1.17d.1.17

?“1,17 33 LR!

1.17308 M1.171& l=:l.lm 1=1.16932 1=1.16&17 12:

1.3.65821241,16~ la?1.16A35 1E41.16311 m1.16189 m

l.- 1=?1.15942 lQS1.1%326 la1.l%&3 1=1.177T6 I*3

1.15455 la1.15334 la1.15213 la1.1X92 m1.14972 Iac

1.14852 lac1.14732 12C1.14612 11s1+14493 1101.14374 118

1.1425-5 1101.14137 1101.14019 110l.1~1 1171.137s 11*

1.13666 lle1.13550 1171.@33 1171,13316 1161.13m In

l.l@ 1151.12969 1101.E%53 115~.~8 1141.12624 115

1.

7

115

1.123 1141.lz2 1141.12166 llS1,1X53 lM

1.11940

l(t)

.149692 la

.149531 lec

.149371 la

.149211 la

. lky.352Ins

,14E93 lEI

,148734 150.148Yc Inn.148417 15J.148eJ$l15

.148m2 160

.147944 MT

.lknm M-?

.lk763Q Im

.147474 1=

A+7318 Im.M7M2 lW.lk7cc6 Isa.146851 1~.1465* 155

.lM&l 154

.1463@ 15s

.146232 154

.146078 l=

.147925 1=

.14?TT2 Lm

.145618 15-

. 14%66 IW

.145313 l@J!

.145161 1-

im::.1447ck lEI.144555 la

.lW Lm

.144* lsl

,144103 190,14395Jd ::,14- 14s

slkm 14m

,14335.514s

,143’208148

.143059 140

.1423U 148

,142763 147

,142616 14s

,142468 147,142321 147,14=74 14a

,M?328 i4e>141M2 14E,141736 14e

r ““,1415,141 5 145

.14Hm 145

.14n55 145,141010 la,)-f+&& 14s,1- 144

,140577

m(t)

.45401.6az

.453394 @20

.k5~ 61S

.4snjs EM

.471537 015

.45.cgz2m5

:?8 %

.44X3 011

.44%72 m

.447854 a07

.#7m 6CU

.446651 eu

.446#47 aoa

,445444 ’501

.4’44843m

.444243 see

.44* 897

.443d+7 see

.W)2451 w

Ak1857 59s,441264 5s1

:%: ::

.439494 B87

.4397 Seu,438321 -,437734 *.k3n53 S=.436571 w

.435991

.

u

4

..

.

m.

.

.

.

.

.

..

NACA TN 2744

—t

G.6s1:g.834

.

:3.@:E.%.ay

:%

.W

.@?

:$3.*.%.W.W

:E.@.X5.X

:%.55

.glo

.931

.93.2

.9L3

.914

.915

.9L6

.917,93.8,919

.m

.W

.$@

.93,pk

.=

.926

.927,s28VW

,930,931,932*93J

,937,9%,937,933m939

m—

g(t)

.0722493 100%

.W5 Lm44

.Cfrolzm Iceaa

.azq=m 10ss4

.C675944 1CU18

.&sl@a3

.

TABLE II.– CONiIiNUED

L(t)

.ce47m5 1U2S,@&09 U1O,C&@9 l11s4,c61* Ill-m

@=’E9 111,s

j(t)

1.* ~,l.dw 4:

1.@763 ●.L dt719 4:1.04676 41

1.04633 &l.c!+t~ 4:Ld+546 4:L.&m Z(1.CW6C ~:

1.04417 4:l.d+flk 4[

1.0433 4:

1. CA288 4[

1.C.W5 4:

1.c!llm2 42

L. CJLl&J 4:1.CAU7 4:l.dlqk 421.04.oy 4:

1.- 4Z1.03947 <c1.0* a1.03W.2 421.0* 4=

1.03778 421.03737 421.03693 4aI.q%m 42l.o~ 4E

1.03~T 421.031.03-R ::1.03U2 421.034W e

1.03358 411.03317 42I.ow 41

?1.03-23 421.031$2241

l.q?l>l 411.03110 421.qW9 41l.o~ 41l.- 41

1.CE945 411.- 421.CQ&’2 411.@82L 401.w781 41

I.qkl 41l.ce~ &l1.02@% 411.*7 40l.m 41

~.m% 41l.~k~ 40urzk55 41l.ceklk 401.@374 40

L.@334

k(t)

1. KL94C ~:l.naz-fUC

l.~nk IICl.11%1 m1.L14@ lU

l.llm Ill1.11265 m1.ID* 1111.n.cA3 ~~Llwz m

l.m%il Ilcl.lqu 1111.1C6CJ3UC

l.lti~ 10E

1.10381 UC

1.1- 109

1.1OMZ Lc-a1.1CQ53 1=

1.W 1081.@36 108

1.cq’z7 loa

1.q619 10-71.- ma1.@4c!Jt1G7l.- 107

l.@lw 1071.C5#33 lo-r1.05976 Ioa1.-O 100L.@-(@ 1C9

l.cfks~ lCU

;:% ::

1.ch923710E

l.cfo.w lW1.CW28 1C51.0793 Iw1.qf319 1-1.CVTL6 I.%

l.qaz 104I.q@ lm1.q405 103l.qz Im1.q’2m lm

l.qcq 102l.- la=1.6593 l@zL.C.5791101L.066~ 1=

l.ti~ 10Ll.c@l& 101L.C&f% 100L.C&2E!6101L.- 100

L.c@35 Iml.- xcmL.WW5 1001.05785 =L.- w

1.05597

61

.

.

62

TABLE 11.– CONCHJIIED

-.NACA TN 2744

t g(t) h(t) i(t) J(t) k(t) 1(t) m(t)

.940 .COI.8Z7 @ .c616413 1- ‘.06E@l.l,Lcs3a 1.02334 41 1.135~ m .132391 M .4!334&5~~

.gbl.0m78% alo .Mq@ 10544

:%%

L32713 1.0~3 40 l.@@g m .13-2 I=.942 .0317236 m93 .mn~ 105S4

.40 1 505

Y10052 1.02253 40 1.053t39 a-a .IW32 x28 .& 76 Em.9k3 .w16687 s.sa .0fi47a LOE=.9W+

.W’3585 ~0- 1.02213 40 1.05292 es ,13m4 1- .431973 502.cola~ 577 .0574257 10614 .0fi2743 10EZ? 1.02173 41 1.05192 Da .l~7~ leg .401kn ml

.945 .IX115522M7 .q437k3 Ims

.* .~149m m .05>3238 1c424.0571916 Icao. o@.lc15 lQms

1.oa32 401.-2 40

l.@@, ml.oh$@ ● = .13618 U?a

.l~7& 120.~t@ 4s2. ~o wl

.947 .C014393 w .0~2~44 I.WeE. .Omm 10778 l.m~ 40 1.* m ,1349 MM .39$970 4,7

. 9!+8 .CQ13854 w .033Z279 la-m .0739733 1- 1. 023L2 40 L.om 27 .131352 lm .3$9473 4ff

.949 .co133m Eu .om~u IC4CU .0528771 10747 1.01972 53 l.ok~ n .l~5 - .393976 4*

,99 ;oW&7 ~ L?C?; :%: .05H3024 m- 1.01933 40 1.04605 D-r .~07 =7 . 393AM 49s. Bl.952 .WU782 431 .04YM6 lMSS

. W37z93 10=4 1.01893 40 l.oh~ = .1* 1*7 .397995 4*3

.o&95579 1- 1.01853 40 1.04!+13 m .l~33 127 .3974~ 49s

.953 .CKl~l 4ti .ok~~ 1C427 .CM%% X~- 1.01813 m 1.ok316 m ,130726 - ,3*W 4C1

.954 .CO1O91O 4Ea .0469553 1W17 .04nlg 10aa7 1.Ol~ 40 1.04m m ,1- U-1 .395* 490

.935 .0310341 4S0 .04vu6 lwB .0b64529 lmm 1.01734 40 1.04124 - .13a473 10Y .39@18 400

.9% .CWX91 -s .0kw28 1-

.957 .~9433 4= .0438332 1032s,0453877 1CU37 1.01.94 m 1.040@ ea .133347 Ua .- 46s.0443240 1- 1.01655 43 1.03f132 25 .l~ * .3j ~ 467

.9% .CJYw 4=7 .042~42 lCQTS .043261i

rlbeca 1.01615 Q 1.03837 SE .lW - .39 553 4=

.959 .cmx3%8 U7 .0417553 ~~m .04zaX lc=~ 1.01576 ~ 1.03742 ~ .1- - ,3g4C6-(4F-S

.* .OXMlql 4- .0407193 lcKUO .okl1424 1057B 1.01537 40 1.03646 04 .*? Us .393* 4a4

.961 .m7745 - .0%833 l~m .04m@t9 10s53 l.olkg ez 1.03552 m .l-297aJ m .~~fl 464

.952 .m7349 am .03a5483 10=1 .o~zql 1- 1.01473 m 1.03457 M .1~~ US .*14 462

.%3 .o@6954 874 .0376142 10=s

.964 .CKX2@X SeA .o~lo 10-.0379746 ~- 1.01419 w 1.03362 V .129470 - .392132 ●w.O*@ mm4 1.o13eiJ 40 l,032@ m .1Z9344 X .391652 4s0

.565 .c036226 m .0355488 1-

.* .0CU5872 C48 .0343176 Im-.03X704 l~m 1.01340 = 1.03174 = .12j’2zz - ,39u72 4-.0348% lc4- 1.OIW1 as l.oga = .129398 L24 ,393693 47a

.967 .CW5529 ans .033k872 10~4

.* .~>% = .032k578 IO=.0337723 zw~ 1.0M62 so 1.0s5$% a= .128974 = .3sQ215 ●m

.969 .M75 - .0314294 lo-.03272% 1c4~ 1.07.223 = 1.02E93 m .12Mp = .~@ 47’U.036%2 l-- l.ollell se 1.02W = .I.28727 224 .3- 474

.970 .CO04%3 aal .0304018 1a28a .0306364 lW 1.o1145 .98 l,02-@6 u .12%33 1.2a .~7&3 474

.971 .c034262 - .oW3772 10=’EU .-42 lMM I.o11o7 ~ 1.02514 = .1.2Wo - .3&a314 4P

.972 .o~3971 Qm .02834% 1024e .028~~34 10W 1.01M8 - 1.o~ = .1=358 - .387841 4~

.973 .m3691 m .02732@ lam .07r5140 1-7= 1.olo2g as 1,02428 32 .l~y, U=! .387369 471

.974 .Cw13421 UO .026~lo 1- .0264762 ICUM I.m m I.@336 m .~~~~3 1= ._ 470

.977 .om3M.1 =40 .oe5278~ IO=2C .0254% ICL=49 1.CMm = 1.02244 m .lq~=. 3954B 4c-n

.976 .ofXE912 - .0242~1 10~0 .02440y 10* 1.lX913 es 1.my 22 .~~lzl. ~Y.977 .mH13 - .02323 I I,oz02 .0233716 10S20 1.m874 ae I.l)a.& 92 .1277b7 = .~p z

?.978 .0G02U5 21s .02zzt 9 10102 .0223397 lam 1.(X!836 m 1.Olgw ‘al .l~62y 191 .38 25 4=

.979 .0WJ2226 R07 .02U957 lolea .0213092 102S0 1.m7g 38 1.01877 ●l .127734 122 .WEB +~

.980 .0C!Q231-9Iae .0201774 10174 .oa)2fD2 Lm 1.m79 aa 1.o17& 01 .Jmp Ill .~~~ 48%

.981 .CO01821 1e7 .o191a Lola, .01925% laal 1.oo721 00 1.ol@ 91 .127-261 120 .~~fi 4m

.982 .CIW1634 lTZ .0181435 lcom .0182265 1- 1.CX%82 m 1,01634 =1 ,1271kI vl .~~6g 4-

.93 .om14% m .017~9 10147 .0172W19 10- 1.w64k = I..Olw w .Iflom - .382’p5 461

.* .CCOI-289 ~e .016u32 Iama .01617M La, I.m.l .01423 m .~mm. ~k5 ~

.995 .030LI.33 147 ,o15q94 low .ol~l~ 10QO4 1.0yi5g m 1.01332 w ,E67& MO .381.785 45m

.* .-w, z- .01= 10L=O .01W65 lolee 1.ca52g m 1.01242 m. I.%&Q-. 3a1326 +=

.937 .O)mw M .013746 10I.11 .o13117b loln 1.00491 as 1.oll~ m .126* m .Y@&& 457

.~ .~724 Lla .0=35 10XO* .Omool =01 1.03453 = 1.01063 20 .1264m 11* .~~ 4,6

.* .WCW2)8 I.ca .0110533 1- .OllC#O 1014. 1.@415 w 1.m.373 w .~~1 11s .379957 4s,

.9* .00)05U2 95 .0100440 loo= .01W694 lom 1.W377

.991al .00%4 Ea .1261.82 w .3795@ 4,,

.CW0407 =, -3% ~~74 .-%2 Iollm 1.C0339 m 1!m795 m .126262 I.la .37~b~ 4SS.W .0X0321 -m. CG63282 1CKU7 .ca83443 Iolw 1.oo~l.s93 .moo246

37 1.03706 m .125944 II* .378%2 b-ea. oo70215 1- .m3703j9 10W 1.m264 s 1.CC.517 m .1255-25 11* .3781W 4,2

.’j-y4.CIW31a 5. C&ol% LcXna .0Cff3249 lmp 1.CX3226U1 .m528 ~ .125707 ~f@ .377~ ~5~

.995 .CQCKW25 :: cm= :.0 .Wmln Lw 1.COM8 so 1.mMo m .H* llB .377238 4W

.596 .CW@& am .Cdollo

.977 .UG0451- 1.alp a7 1.Cm351 .98 .125k70 U7 .376789 Ua

m. WKXMo.* .CaXm20 @

lW .w3c062 lCON I.cQl13 ~ 1.CQ’X3 ~ .E5353 ~~~ .37634o Ma

.?39 .0mcnl~,m~~8 1c014 .m= 1- 1.CCQ75 87 1.CQ175 m .12p35 117 .375832 44a

8. CQ103O4 100C4 .OO1OW7 10CO7 1.0CD38 m 1.CWf!8 m. -8 Lla .3754% 440

.CEo o 0 0 1 1 .12%W .37m

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m FORM A: Calculation of 2nd-Order Super-

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P. P, P. P, P+ Ps P6

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Keep only 3 alg. figs. In final results

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Calculate only on ●aoh std. of ●v.fy o.arnar (that {’, ,x-Ily tor ,v.ry,

Golumn wnfch has a @ SomoWh Ore above, and tho column pr@OOdlnQ it).----

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