Vortex theory of an airscrew in consideration of contraction or expansion of slipstream and variation of pitch of Vortex sheets in it

Lab. v Scheepsbouwkrnde

Technische Hogeschoo

Def

Reports of Research Institute for Applied Mechanics

Vol. VII, No. 27, 1959

VORTEX THEORY OF AN AIRSCREW* IN

CONSIDERATION OF CONTRACTION OR EXPANSIOÑ OF

SLIPSTREAM AND VARIATION OF PITCH

OF VORTEX SHEETS IN IT**

By Matsunosuke IwAsAKI

ABSTRACT

The airscrew theory in consideration of the cOntraction òr expansion of its slipstream and the variatioh of the 'pitch of its vortex sheets in the slip-streani is derived. The calculation charts, which are necessary for the application of this theory to practical cálcutation of the performance of an airscrew, are provided. The method of application of this theory to practical case Is d6sdribed precisely. The present theory is applied to the caldulation of a four-bladed windmill, and discussions are made about the results of the calculation. Some informations of the structure of the tangential component

of induced velocity are given. . -.

INTRODUCTION

In the previous paper' thé author described thé method to calculate the

induced velocities by the system of vortices of an airscrew, and showed

calcula-tion diagrams for these velocities. . This general vortex theory, though being able

to give complete consideration to. the. deformations of form and pitch.of the vortex sheets, has not- been yet convenient fOr the. calculation óf propellers and windmills performances, because it is necessary to perform integrations concerning two .varia. bles zuR and r in. order to obtain the values .òf indúced velocities. Up to ùow'some

researches2>.8). of mutual interference of propellers with their. slipstream by

vor-Machinés of propeller-type such as propellers, windmills, fans,. rotors of helicopters, and anemometers of -propeller-type etc. are meaned .by this word after Glauerts use

(Durand, IV, p. 17.1). .

** _{Presented at the Eighth Associated Meetmg of Applied Mechanics m Tokyo Japan Sep}

'tember 6, 1958 (theory and some results of calculation), and at the Meeting of the Ja-pan Society .of. Aeronautical Engineering in Tokyo, April 8, 1959 (fiñal results of cal-culation of 'r etc) ..

1)" iwasaki, M., Diagrams for use iñ Caiculatioh of Induced Velocity by Propéller, Reports of Research Institute for Applied Mechanics, Vol. VI, No. 23, 1958

Betr A. and Helmbold H B Zur Theorie Stark belaster Schrauben propeller Ingerneur

Archiv, HI Band, 1. Heft, 1932. . .

Glauért, H. On the contraction of the slipstream of an airscrew, R & M1 No. 1067, 1926,

4) Theodorsen, T., Theory of propellers, McGRAW-HILL, 1948.

160 M. IWASAKI

tex theory in rather qualitative treatment and the researóh of the determination of the form of slipstreams have been perforrned.4 And, so far as the author concerns,

the theory of the mutual interference between an airscrew and its slipstream in

exact consideration of the effect of the deformation of wakes has not been

report-ed. As stated aboye the number of the papers concerning these problems are not

so many, and few papers are published fôr the last problem of mutilai inter ference. This seems to be due to the abominable trouble mentioned above and the

want of diagrams for the calculation. In addition no precise and quantitative

investi-gation of the tangential component of induced velocity of Vt has been made up to the present. The calculation of the tangential velocity Vt, however, is also an

interesting and significant problem. Now in the present paper the author will show

a new method by which the integration by zuR was changed to akind of swnma-tion, so that the procedure of integration becomes considerably simple and the cal-culation of performances of propellers and windmills becomes possible with tolera-ble labour for practical use. For the calculation of induced velocity by the

method thus simplified, the curves of V1, V and V versus log10 z/r1 with respect to the parameter r/r1 are necessary. The functions ,V etc. which represent the co-ntribution of a cylindrical vortex sheet to induced velocity, were calculated, and the curves are shown in the present report. Since the preparation for the

investiga-tion of an airscrew with its deformed slipstream had been finished, the author calculated the performance of a four-bladed windmill for 2R/ V=2.5, with the assumption of infinitely large number of blades, considering the effect of the ex-pansion of the slipstream and its variation of pitch of the helical vortex filaments in the slipstream in order to prove the utility of the present theory, to know the effect of the deformation of the slipstream on the performance of the windmill, to find the induced velocities in the slipstream, and to investigate how y1 was

constructed in the slipstream.

This calculation, however, has the same meaning for a propeller etc. as the theory of dead water for the drag of a plane surface placed normal tothe

direc-tion .of wind flow. Namely, in both, cases, the practical states do not, correspond

to these ideal conditions. For the former the helical vortex sheets begin, to roll

up to hlical vortex filaments from the place, just behind the actuator disk. In addition the assumption of infinitely large number of blades, though it. is . very powerful and convenient method for the simplification of vortex theory, is not always an appropriate assumption, especially, in the case of QR/ V is very small. In the latter case of dead water, the region of dead water is not realized in the practical cases, but turbulent flow full of vortices prevails in this region. The theory of dead water, however, has contributed to the problem of drag considerably. Similarly the present theory of propellers and windmills, though it cannot describe and solve the practical states qualitatively, will offer a method to calculate the per-formance of propellers and windmills giving consideration to the variation of slip-streams, and will show some new facts by applying this method. As seen in the note 2) of the section of symbols the helical vortex sheet behind an airscrew of fiñite number

of blades cannot conserve its shape of helical vortex sheet of constant pitch if the distortion of it is taken into consideration, so that the theory in consideration of this distortion of slipstream cannot be established in rigorous meaning in the case of finite number of blades, and only on the assumption of very large number of blades this theory can be constructed in exact form. In this meaning the present

theory will be the one which is made on the assumption most reasonable to the

present problem. In the followirg sections the theory and its application to a

windmill will be described.

S YMBOLS

w

X, y, z, r, O, z,

unit vector of vorticity, components of which are;

radial

w1, : tangential

axial

orthogonal coordinates

cylindrical coordinates of points. No suffix corresponds to pint where

induced velocity is calculated, suffix 1 means place where vortex element exists, and suffix O indicates propeller surface.

i,j,k unit vectors of x, y, z directions, î : radial (x), : tangential

91 _{- (y), k: axial (z)}

R=D/2 radius of airscrew

- ds length of vortex element

=dz,/w

- - - - y velocity vector of induced vel-ocity by total vortex system

r, _{y,, velocity vector of induced}

vel-- ocity due to helical vortex

cy-P

_linder

y, velocity vector of induced

vel-ocity by central vortices

v velocity vector of induced

vel-. ocity by bound vortices

'

'\

_. V velocity of oncoming flow

-r

T'(ro) circulation around blade

ele-ment of radius r0

B number of blades of airscrew r = Bf'/4Tr2RV, nondimensionsi

é' _¡

¡

_{representation of F}

U,. induction coeffifient of radial

direction of circular vortex

z

VORTEX THEORY OF AIRSCREW ₁₆₁

element

U, induction factor of tangential component by cylindrical vortex element

U induction factor of axial component by circular vortex element

162 M. IWASAKI

Thesó th±ee ale the functions of z/ri and nr1. The vortex element is at

z1/ri==O. (see reference 1)).

V1 radial induction factor at (z/ri nr1) by cylindrical vortex sheet of

radius r1, whiôh extends frOm z1/ri=0 to zi/ri=,

V1 tangential indûction factor of vOrtex cylinder,

V2 axial induction factor of vortex cylinder,

These three are also the functions of ztr1 and r/ri. 2=2irn, angular velocity of airscrew

Qr/V=cotê, ratio of tangential velocity to velocity of oncoming flow.

geometrical angle of attack of blade element measured from no-lift line

of blade element

induce angle of attack of blade element effective angle of attaôk of blade element

S2r/V+v1/V

cotq= _l+v2/V , (on cylindrical vortex sheet -it is equal to w/w) c chord length of blade element

s= cB/2irr, solidity of blade element at radius r

¡9 angle of blade element measured between lower flat surface and airscrew surface.

angle of blade elemeñt from no-lilt line

4r/R increase in radial distance of fluid element due to induced velocity from z

axis by deformation of slipstream of cylmder of constant radius

CL=ÍCL' lift coefficient of two dimensional aerofoil section 2irk= dCL/dc, slope of lift curve, ae in radian

C,, drag coefficient of windmill When we treat a propeller, this is replaced

by (eq. 46)

Cd tofque coefficient of airscrew (eq. 47)

C thrust coefficient of àiìscrew (eq. 48)

CQ tOrque coefficiént of airscrew (eq. 49)

C power coefficient of airscrew (eq. 50)

power ceefficient of airscrew (eq. 51)

Note: Oi the dethütion of vortex filaments änd vortex sheets

1h ihe followings some brief explanations will bé given for the words

concer-ning vortex filámëìits or vortex sheets which are used 1h the present paper. Helical vôrtex 'hëet: the vortex sheet of helical shape which springs from

airscrew blades of finite number of blades.

Helical vortex filament: the vortex filament of helical shape which springs from the airscrew disk of finite number or iflfiuiitely large number of blades.

lt is not necessary for this filament to have constant pitch with respect to the radius nR and the axial distance from the airscrew disk or z/R. The

elements of a helical vortex sheet, are also helical vOrtex filaments. And, if we consider the distortion of a helical vOrtex sheet, these cannot be already consideted as a helical vortex sheet, but as the group of helical vortex

VORTEX THEORY OF AIRSCREW ₁₆₃

filaments, especially far downstream from the airscrew surface, because the pitchs of the distorted helical vortex at various points on it differ very much,

so that the helical vortex sheet cannot maintain the form of helical vortex

sheet and becomes a gathering of helical vortex filaments.

Cylindrical, (helical) vortex sheet : the cylindrical vortex sheet is shown in figures (14) and (16). The pltchs of the vortex filaments over the surface of the vortex cylinders may differ from each other with respect to nR and

z/R.

Cylindrical axial vortex sheet: This is shown in figure 13.1 ®. Its vorticity represents the rotation of the slipstream.

Circular (tangential) vortex sheet This is shown m figure 13 1 ( It is

the group of ring vortex elements, and its vorticity represents the increased axial velocity of the slipstream.

The combined vortex sheet of 4) and 5) have the same effect with the one of .the vortex sheet of 3) on the velocity field of an airscrew. And these three kinds of vortex sheets are used mainly in the airscrew theory of very large number

of blades.

The vortex sheets of 3), 4) and 5) are sometimes called simply the vortex tubes or the cylmdrical vortexes, in the cases when there is no possibility of mis

construing their meaning

§ 1. Fuñdametal formulas

As assumed in the previous report°), we shall suppose that the helical vortex

ifiaments springing from the blades form a vortex tube or cylinder over which infinitely large number of helical vortex filaments are wrapped. Then the total

circulation around this vortex cylinder is

Bdl' (ro)

or

the sum of the

separate helical vortex filaments which go out of the blade elements at the radius

of r0. This assumption is called the assumption of

infinitely large number of. blades, and it is very often

used in propeller theories (see figure 1). Then the circulation around the vortex sheet is Bdl72ir per

unit angle. The sign of minus attached to BdT'/2ir has the function to conform the sign of BdFJ2ir with

the promise of sign in our positive coordinate system, because, as shown in figure 2, the trailing vortex

produced by BdT'/2ir has the positive sign with respect

to the present coordinates system, while BdT'/2ir is

negative. This curved cylindrical helical vortex sheet

with the central vortex BdT'0 and the circular plate

of vortex sheet on the airscrew disk composes the

vor-tex system of airscrews. The induced velocity at

po-in P(r, O, z) by these vortices is given,° po-in

nondimen-5) Iwasaki, L c. ante p. 159)

Fig. 2. Sign of vortex

164 M. IWASAKI

sioiial förm, by

vva+vc±VbLj

dr

d(z1\\2U-!.+CÇ drJd(ì (J)?(U

Ur)±

- tip

-'

tip

V

V

kM)

r rl Wz root O Toot O ztR R\2 r0, zj/R=O ±d10

_/R)(l

_{± {(r/R)2+ (z/R)2}} T U .0 tip + dr f (1) j00 J0

R)ni)

Wo J.

In order to calculate induced velocity by this formula in the case of the slipstream of curved cylindrical vortex shown in figure 3 we must calculate the

values of nr1 and (zz1)/rj [(r, z): point where induced velocity is calculated,

(r1, z1): circular element of helical

vor-11e £-1f çy/iizdiíozl vorta deetgf tex of radius r which is situated at z11. (sta,zt 1" _{Next we read the values of U,-, U1 and} U, from the diagrams of the previous

ciartri9/rn4naz/

VarteJ Lh paper. After that the integration by z1

-

between the limits of zero and infinity

i

r r,, _{is performed. Then the integration by}

L I

- -' z

_r

must be made. These procedures

should be performed for every pOint on

the airscrew disk as well as in the slip-stream, where induced velocity is

re-qu.ired to be calculated. The author - Áe//aI ycr/e1 tried this method, but it toòk more than three months for the calculation of the induced velocities at the points noiR =

=

1.0, 0.975 and 0.925 on the airscrew disk, and at the points in the slipstream which

''"

have the corresponding radiuses to these

Fig. 3. Divisiön of cu rved cylindrical values of ro/R and have the values of

helical vortex sheet z1/R and have the values Of z1/R

0.0333, 0.07-5, 0.15, 0.20, 0.30, 0.55, 1.4

and 6.0. Therefore the author reduced the labour by the method which will be

mentioned in the following pages. By this method, fôr only one month of

Aug-ust 1958, the author could calculate the induced velocities at r0JR=0.85, 0.75, 0.65,

0.55 and 0.45 on the disk and at the corresponding points in the slipstream. This simplification is performed by the method which will be stated in the followings.

In equation

(1),

if we assume that in the

i th section of Fig. 3 the radius of slipstream ru and the pitch angle of are constant, then R/r1g. d(z11/

R)=d(zii/rii) in each of these sections. Then equation (1) can be written as the

following form: tip LII)

[

d

Vn cotii). + (

Ç d ( (v11 CP.xI v)± V _j 1=1 r11 - ru rool root

(1 dro (/R)@ z/R

(R 2

{(rJR)2± (z/R)°P'°)

r) UrZI!R0 d()J

. +

JtIPd.(R vcotii).k1,

root (2)

where cot u = (co5/w), dro : strength of vortex filament springing from

centre, and V,, V and V are

121111/ru

Vr,1 and zi = - Ur;t and a t d(z1/r11) (3)

JZjfljit

respectively. The suffix i indiôates the quantities corresponding to i-th division of

the cylindrical vortex. Vr,t and :1 are the funtions of Zirj/TitZ/Tit, zitrjJriz/ri and

r/rij. The value of r1 represents the constant radius of the vortex cylinder between Zi/Tit and zijij/rj. By the consideratiOn of the following section, we can

cal-culate Vr,t and zi for any combination of z/r11zl11/r11, z/TiZuu1/rj1 from the curves

of arno which are not the functions of Z111/riZ and zoii1/r11. but the function of

z/ri (distance of point under consideration from base of vortex cylinder of radius r1) and nr1. In the next section, therefore, the method of calculation of V7, Id from the curves of V IlId will be described.

§ 2. Calculations of V,., V and V from values of Vr, Vt and V2

We shall explain the method of calculation in three cases. j) z/rit > zitti/nt>' Z111/nli

As shown in figure 4. 1 the two vortex cylinders, which have their bases

at z111/rij and ziiLt/nit, have the radius of r11, and posess the same strength and

inverse signs, are superposed to the positive direction of the z axis. Then, as

shown in the figure, only the portion of the cylindrical vortex from Z111/7o1 to

ZUU/nit has contribution to the point P(r/r11, z/r11), while remaining parts cancel

each other. By formula this is written by equations (2) and (3) as follows:

dTJ"rtaidzt d J

fr ada

_)d(_--_)

2jJ lt

(

(r

z

Zi\j,(Zi

I U,C nId Z ,

I

I"t.-\rll riz r1 /

r11

= dr J

(_ur,tandz(L, _:_-)d(

₎

- dr. j C_.mfu1d2(L,_hi_

_{)(4'_)}

(;;)'t<

/r,z<

')V

P Vt l'ut. r, r, LAi

(X,rir,)

X ID4'

/1;

Dividing both sides of equation (4) by dr, we have

e/eme»!: of c t,d,;01i ,

t,

()(fl%,,.-u%)

Fig. 4. 1. Calculation of Vn, V and from

Vr. Vt and V

(

-

(r

z ZJ[ \ f r z

z1

-=di Vr,t an

z -,

_r1

-

_/

i- Vr, and z t'_r1, r11 (4)

-

ru r1j where

Z/ri=z/rizij

or and

(r

ZltonJlj'\

f:

(r

z Zllorii Z1\ Z1

awl z

-r1, ri1 I

r,t uni z - -

\r r11 r1,

1d.

r11 (5) j rj1,

/

r11 166 M. IWASAKI

(;). 4;> 71z5/ > I%

Z)Vr

T, ?

VORTEX THEOR Y OF AIRSCREW

,T)

iL)

¡g, ¡lu

4 4)

p. 4k'

The co,,tiiMfioug of 15e two

amelr1ed partt lo lAie poiat

1' 'asce1 eo clAicp beeu4iIIe

u ¡s Mc odd fw,ctoii tf (%j

iaJ%.)

(;;;)

_{:< "%ir <}

_'%t

r.

(&/'

L- !HYr'%I

(%jjJa%J)

(Ut-%)

e

Fig. 4. 2. Calculations of Vn, and Vz from Vr,, Vt

and Vz (continued)

(r

z Vr,t a,cl i V,1nud z

(r

z

z1

\r1j, ru r1 ) Vr,t and ru J

Each term of the rigiit side of this equation is written, by equation (5), as the 6) Hayson used the same way of thinking as expressed by this formula to calculate the

axial induced velocity v by the rotor of a helicopter situated at the height of the

radius of the rotor R above the ground, with slipstream of constant radius of R. cf Hayson, H. H. Induced Flow Near a Helicopter Rotor (A Review of present

knowled-ge), Aircraft Engineering, February, 1959, p. 43.

o o

($,I';V)

(P:%,l)

n

167

168 M. IWASAKI following form:

(r Z

Zj Vr,t ntid zi =

-j Urt. and z ru ) r11

(r Z

Vrt and z_{\Tjj Tu}

-

(7) These are the contributions of the vortex cylinder, which has its base surface at zili/rit or Ziii/ru and extends to positive infinity, to the point P(r/r11, z/r11), (see figure 4.1 (i) and equation (5)). These are, therefore, the functions of nr11 and relative coordinate Z/ r11 = zJ r11 - z111 1111/ru. Accordingly, if we have the curves of

Vr.t and 2(r/ri, z/ri), we can flïid the values of Vmii (r/r11. z/r11 z111/r11) and Vrt (rJr11, z/rltzinl/ru) from these curves by using r/r11 instead of Tiri and z/r11zlII/rlI and z/ri1ziiu/ru instead of z/ri. For simplicity we shall use the

following symbols:

Thus by these equation, if we know the values of z/r11ziit/rj, z/rjjziiii/r11, and

r/r11, we can calculate the values of Vr,t and from the diagrams of Vr, V1 and

V,.

ii) z11t/n11 < z/n11 C

A vortex cylinder of negative strength, extending from negative infinity to

positive infinity is superposed on a vortex cylinder of positive strength (see figure 4.1 (ii) (A)and (B)). Then the value of V1fl z I or the contribution of the

vortex system to the point P is given by

dr y1 and

z

J=dr.Vt

and

±d.

f(u udZ(

IZI

Z1))d(Z1)

₍₁₀₎

where (co ± co) indicates that V1 tid are refered to the infinitely long

vor-tex cylinder. Dividing both sides by dr, we have, according to the consideration

described in the following brackets [ J,

ail z (C) V&

21(co±o)±

_{í(_u1 and} ( I

Z 1

'\' d('

Z

r11' r11 T1j Ti1 1) \ r11)

in i(D±)± V1

z(rIrii,lzIrii-ziri1Iri1I).

(Il)

[Because U1 and is even funtions of z/r11z1/ r11, the vortex elements of figures * _{The author used z instead of Z in this description and in figures 6.l.-6.7. In our}

calcula-tion the z in these figures have, therefore, the meaning of relative coordinate Z.

Vr,t and z1 Vr, latid j j Vr, and (nr11, z/r11z111/r11)

Vrt anti zfl= Vr.t nd Jfl Vr,t nud 2(r/n11, z/rlLzI1Ti/rlj). (8)

Then equation (6) becomes

Vn Vnl - Vrri

V1 = V11 - V111 for z/rniI>zlrII/rii>ziil/rlI (9)

4.1 (ii) (B) and (C), which have the same distance from each base, induce velocities of the same magnitude and sign at the points P in both of the

configu-rations of (B) and (C). V ad for these cases are the sum of the contributions of these vortex elements to the point P. Therefore the value of - [V1 and z) J

for the cases of (A) or (B), is the same with the induction factor forthe case of

(C), which iS written here by - V1 und z(C), without any modification.] If we write

ir

Zita

V1 ande i (.co±)= V1 and 1(), and V1 and - --

V1 'und ziEl! (12)

then we can write equation (11) as follows:

- V1 and a (U) V1 and z1uI' V alud (co). <13)

If the vortex cylinder of (C) is superposed on,. the vortex cylinder exten ding from z111/rii to ± : then we obtain

V1 and ei = VI and eI + V1 aad ziEl! - V1 and (co), (14) which corresponds to the case of (ii). On the other hand Ur is an odd function of z/rllzl/rlL, therefore

-

JU1r(

d()

= -

Jurd(

JUrd (i)

+ J Vrd()_

Jud()o

(15)

VORTEX THEORY OF AIRSCREW 169

* _{In figure} 7.2. another method of calculatiòii of V1i for (ii) is shown.

and the expression, which corresponds to equation (13), is

V11111 (16)

Therefore we have

V,.=V,.Vjij

for the case of (ii) (17)

Consequently the following relations are obtained:

V,.1 V11 - V,.111

V V11 ± V111 - V1() for z1111r11 <z/rll<zJJll/r1. (18)

Vz1 = V21 ± V1iti - V1(co) iii) z/r11<zllL/rll<Z11I1/rll

By the same consideration mentioned in i) and ii) V are

V1 = V,.111 Vr1111

V1 and el = V1 and zlLI! - V1 ajud cil!. (19) Consequently we have the following relation:

Vn V1111 - V11111

¡"11111 - J"11! for z/rIj<zÌll/rll<zlw/rll. (20)

=

+

(O,

»,<I

j-%,=: for Vt

[TIJ,bI,

it ,*::O

-%,* I for

. cancel_-L other

Eig. 5. Calculations of V, Vt and ¡z

170 M. IWASAKI

Thus we can see that Vr, iid can bé calculated from

v

_{anii zÌ and H which} are found from the diagrams of the induced velocities by the vortex cylinder

extending from z/rj=O to infinity, if we know the ratio Of r/r1,

_{or the ratio of}

the magnitude of radius of the point P to radius of i-th Vortex cylinder element and z/rijziij/rit and z/rllzIfIL/rlj or the nondimensional relative coordmates of z-axis of point P to the bases of the i-th cylindrical vortex element.

§ 3. Calculations of V;, V and V and diagrams of them

f

As stated above for the calculation of induced velocities by èquation (2) we must know the values of V i'd or according to Section 2 the values of

Vr.t luid ZI and il. For this purpose it is suffucient for us to know the values

Vr.t and inri, z/r1)

-

aid

(nr1, z/rizi/ni)

.d(zi/ni). (21)

VORTEX THEORY OF AIRSCREW 171 parameter of nr1. In these functions, V and V2 are written, by formulas, as follows (see figure 5) O , n/r1<l

vtUt(J,)d()± LirJ(r/ni)

ir

. r/ri<l

rri

For Vr we have zJ'ri J,

Ur(f,*_)d()

f

ud()

because due to the fact that Ur is the odd function of z/r1, the contributions of A and B to the point P cancel each other. In equation (22) the author used

the following relations:

If we perform, for example, the graphical integration of the left sides of

equati-on (24) by using the diagrams of U,, we have the results, which coincidé with 0,ir!2,

aud .ir/(rJrj) by the accuracy of thzee significant figures for each corresponding

val-ue of r./rj. On the other hand, if we consider about these integrals by the manner which will be stated in Section 7.2, we can find that equations (24) and (25) are

correct. By using equations (22),(23) and the diagrams of U, U, and U2 the

graphi-cal integrations of equation (22) were performed. The results of calculation are shown in figures 6.1 6.7 and numerical tables 1.1 1.6. The author could not have

the values of them which were accurate by more than four' significant figures. The

values in the tables, therefore, are correct up to three significant figures for most

of them. The forth significant figures iii these tables are described only by the

reason that the curves written by these values are smoother in the case of four

significant figures than by three significant figures, when we make large diagrams

Of Vr,tand . There are, however, the cases, iñ which the number of significant

figures decreases due to the subtraction which appears in the course of graphical

or numerical integrations. These cases are

n/n1

'

1 for V2

n/ri < 1

for V,.

Consequently in these cases some results of calculation cannot contain more

than two correct significant figures. Therefore for these cases the curves of V,

(figuré 6.3) and V2 (figure 6.6) show some irregurality. But the absolute values

, r/ricZl

Ucd-)]

=ir/2

, rJri=l

(24)

T1 rdalri=O and

rJ(r/ni) , n/ri>1

r/rj<l

ziri=i = ir/2

, r/rj=l

(25) O

,r/rj)-1

L

07

( '2A .1 9 2!J .

--.

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Nrri

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N

z/rj N

1.00 1.005 1.01 1.02 1.04 1.07 1.10 1.15 1.20 1.30 1.40 1.60 1.80 2.00 2.50 3.00 3.50 4.00 5.00 10.0 0 5.908 4.772 3.992 3.173 2.703 2.335 1.909 1.651 1.292 1.056 0.742 0.549 0.439 0.274 0.181 0.154 0.104 0.060 0.016 0.003 4.731 3.989 3.169 0.005 5.398 4.658 3.971 0.007 5.061 4.574 3.941 3.157 0.001 4.704 4.422 1881 3.143 2.669 2.331 1.907 1.650 1.291 1.055 0.02 4.033 3.956 3.649 3.135 2.643 2.314 1.900 1.649 0.03 3.629 1656 3.407 3.029 2.613 2.296 1.891 1.642 1.288 1.052 0.05 3.083 3.146 3009 2.796 2.494 2.232 1.860 1.623 1.280 1.048 0.438 0.07 2.748 2.796 2.721 2.574 2.363 2.150 1.820 1.602 1.270 1.043 0.737 0.438 0.10 2.399 2.412 2.389 2.294 2.163 2012 1.744 1.554 1.250 1.031 0.436 0.273 0.060 0.15 1.998 2.019 2.001 1.944 1.903 1.786 1.600 1.454 1.205 1.004 0.723 0.540 0.433 0.272 0.180 0.153 020 1.720 1.738 1.720 1.689 1.663 1.591 1.454 1.350 1.150 0.968 0.708 0.532 0.429 0.271 0.060 0.30 1.335 1.349 1.343 1.323 1.323 1.291 1.216 1.134 1.029 884 0.670 0.513 0.417 0.267 0.060 0.40 1.080 1.099 1.073 1.073 1.073 1.046 [.002 0.962 0898 0.793 0.624 0.487 0.402 0.261 0.059 0.50 0.889 0,904 0.893 0.888 0.893 0.886 0.819 0.709 0.574 0.459 0.385 0.255 0.173 0.149 0.098 0.059 0.60 0.742 0.754 0.755 0.744 0.743 0.765: 0.714 0.696 0.686 0.63 1 0.526 0.428 0.364 0.248 0.058 0.80 0.535 0.539 0.543 0.534 0.535 536 0.521 0.517 0522 0.495 0.433 0.367 0.321 0.057 1.00 0.798 0.409 0.409 0.395 0.403 0.396 0.390 0.390 0.392 0.393 0.355 0.311 0.277 0.212 0.151 0.137 0.090 0.056 0.016 1.20 0.301: 0.309 0.309 0.295 0.307 0.304 0.299 0.295 0.314 0.313 0290 0.261 0.240 0.192 0.131 0.055 1.40 1.50 0.231 0.204 0.238 0.210 0.245 0.212 0.229 0.202 0.239 0.212 0.243 0.217 0.233 0.206 0.236 0.206 0.248 0.221 0248 0.224 0.236 0.217 0.219 0.201 0.205, 0.173 164 0.124 0.124 0.121 0.081 0.053 0.052 0.016 1.60 0.181 0.186 0.193 0.179 0.185 0.187 0.184 0.184 0.198 0.189 0.193 0.184 0.173 0.155 0.117 0.051 1.80 0.144 0.154 0.151 0.142 0.148 0.151 0.146 0.145 0.156 0.165 0.161 0.155 0.150 0.140 0.111 0.049 2.00 0.115 0.114 0.122 0.114 0.119 0.124 0.117 0.116 0.130 0.136 0.136 0.131 0.126 0.125 0.098 0.104 0.071 0.047 2.50 0.070 0.072 0.075 0.067 0.074 0.075 0.072 0.078 0.083 0.085 0.086 0.087 0.086 0.113 0.075 0.088 0.061 0.042 3.00 0.045 0.046 0.048 0.042 0.048 0.050 0.049 0.050 0.052 0.056 0.058 0.060 0.060 0.083 0.061 0.046 0.050 0036 0.015 8.50 0.031' 0.031 0.035 0.027 0.033 0.034 0.033 0.035 0.036 0.039 0.04Ï 0.042 0.040 0.061 0.048 0.036 0.041 0.031 4.00 0.022 0.022 0.023 0.018 0.023 0.023 0.024 0.024 0.023 0.026 0.030 0.031 0.030 0.046 0.037 0.028 0.033 0.027 4.50 0.016 0.016 0.018 0.012 0.017 0.017 0.018 0O18 0.018 0.019 0.022 0.022 0.023 0.035 0.031 0.021 0.027 0.022 5.00 0.012 0.012 0.012 0.008 0.013 0.013 0.014 0.013 0.014 0.015 0.017 0.017 0.018 0.027 0.025 0.016 0.023 0.020 0.008 6.00 0.007 7.00 0.005 10.00 0.002 0 0 0 0 001 0 0.001 0.001 0.001 0.002 0.003 0.003 0.005 0.004 0.004 12.00 0.001 20.00 0.001 0.002 00 0 0 0

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0.0

0 0 0

Table 1.3. V, (n/ri <1) N z/nj O OEl 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.85 0.90 0.93 0.96 0.98 0.99 0.995 o 0 0 0 0 0 0 0 0 O 0 0 0 0- O 0.003 0 - 0.070 0.193 0.285 0.531 0.005 0 0.116 0.235 0.452 0.170 0.007 0 0.161 0.323 0.598 0.929 0.01 O 0.003 0.008 0.016 0.039 0.055 0.086 0.127 0.228 0.443 0.760 1.079 0.02 0 0.007 0.015 0.031 0.077 0.108 0.170 0.247 0429 0.754 1.075 1.269 0.03 0 0.010 0.022 0.037 0.115 0.160 0.250 0.361 0.592 0.927 1i90 1.325 0.05 0 0.007 0.016 0.024 0.038 0.077 - 0.188 259 0.395 0.546 0.805 1.084 1.277 1.34$ 0.07 0 0.010 0.023 0.035 0.053 0.076 0.107 0.153 0.257 0.352 i5i3 0.684 0.929 1.152 1.291 1.338 0.10 0 0.014 0.033 Ó.052 0.075 0.150 OE213 0.350 0.465 0.643 0.810 1.023 1.183 1.286 1.307 0.15 0 0.022 0.048 0.078 0.110 01-53 0.215 0.305 0.471 0.590 0.773 0.900 1.066 1.167 1.239 1.244 0.20 0 0.029 0.062 0.099 0.142 0.194 0.271 0.375 0.548 0.668 0.827 0.926 1.057 1.129 1.184 1.182 0.30 0 0.042 0.085 0.135 0.195 0.258 0.353 0.463 0.631 0.725 0.846 0.906 0.994 1.037 1.080 1.067 0.40 0 0.052 0.103 0.164 0.231 0.306 0.400 0.500 0.647 0.721 0.812 0.851 0.916 0.945 0.990 0.966 0.50 0 0.058 0.t14 0.196 0.251 0.319 0.419 0.512 0.632 0.68S 0.762 0.790 0.841 0.868 Ö.893 0.877 0.60 0 0.062 0.120 0.182 0.253 0.324 0.419 0.502 0.602 0.648 0.708 0.729 0.771 0.801 0.810 0.798 0.80 0 0.120 0.182 0.252 0.304 0.391 0.450 0.527 0.558 0.603 0.615 0.647 0.685 0.682 0.663 1.00 0 o.Ô59 0.111 0.168 0.229 0.274 0.347 0.388 0.452 Ò.4730.511 0.520 Ó.557 0.583 0.565 0.556 1.20 0 0.052 0.099 0.144 0.236 0.300 0.329 0.386 0.400 0.434 0.437 0.474 0.498 0.477 0.469 1.40 0 0.047 0.087 0.120 0.174 0.202 0.257 0.288 0.329 0.335 0.371 0.374 0.406 0.429 0.4l6 0.400 L50 0 0.082 0.163 0.238 .305 0.343 0.399 0.383 0.370 1.60 0 0.041 0.077 0.109 0.152 0.173 0.321 0.239 0.282 0.290 0.318 0.315 0.350 0373 0.353 0.343 1.80 0 0.Ó35 0.067 0.085 0.125 0J49 0i92 0.208 0.243 0.247, 0.275 0.272 0,304 0.326 0.305 0.295 2.00 0 0.029 0.058 0.077 0.116 0.127 0.161 0.184 0.211 0.219 0.239 0.238 0.266 0.288 0.268 0.256 2.50 0 0.023 0.044 0.050 0.086 0.085 0.121 0.131 0.152 0.150 0.173 0.167 0:205 0.217 0.199 0.185 3.00 0 0.018 0.034 0.033 0.061 0.064 0.089 0.096 0.112 0.107 0.131 0.121 0.159 0.170 0.152 0.138 3.50 0 0.015 0.028 0.023 0.047 0.046 0.068 0069 Ò.08 0.082 0.102 0.091 0.128 0.138 0.119 0.106 4.00 0 0.013 0.023 0.016 0037 0.031 0.048 0.052 0.068 0.063,0.081 0.070 0.107 0.116 0.097 0.084 4.50 0 0.012 0.020 0.010 0.030 0.026 0.038 O.Ö41 Ô.054 0.048 0.067 0.054 0.091 0.100 0.081 0.068 5.00 0 0.011 0.017 0.007 0.025 0.017 0.031 0.030 0.043 0.036 0.056 0.044 0.080 0.088 0.064 0055 6.00 0 0.030 0.041 0.069 0.072 0.039 7.00 Ò . . 0.021 0.031 0054 0.062 0.028 10.00 0 0.007 0.009 0006 0.007 0.0Ì2 0.017 0.043 0.047 0.015 0.014 12.00 0 0.013 0.038 0.043 0.009 20.00 0 0.006 oo O O O O O O O O O 0 0 0 0 0 0 0 182 M 1WASAKI Vt nr1

< I

-v

_r/r

i

Tablet 1. 4. Vt,, (rrj 1)

I

Nr/ri

N

z/ri N

1.00 1.005 1.01 1.02 1.04 1.07 1.10 1.15 1.20 1.30 1.40 160 1.80 2.00 2:50 3.00' 3.50 4:00 5.00 10.0 o 1.57 3.126 3.111 3:080 3.021 2:936 2.856 2.732 2.618 2.417 2.244 1.964 1.746 1.571 1.257 1.047 0.898 0.186 0.629 0.314 0.003 3.676 3.410 3.232 0.005 1.588 3.926 3.594 3:340 0.007 4.096 3.742 3.433 0.01 1602 4.261 3.953 3:563 3.281 3.089 2.972 2.803 2.677 002 1.629 4.517 4.283 3:910 3.516 3:234 3.077 2.877 2.701 i 003 165346244443413337143376 317629492786 0.05 1.696 4.134 4.631 4.385 4.004 3:612 3.371 3.090 2.891 1.621 1.285 0.07 1.735 4:800 4.711 4.525 4.195 3.808 3.555 3.221 2.995 2.676 2.436 2:088 1.754 1.640 0.10 1.788 4:873 4:809 4.660 4.385 4.018 3.753 3.401 3.141 2:795 2.521 1.669 1.314 0.15 1.866 4.967 4:921 4.795 4.586 4.261 4:014 3.651 3.373 2:958 2.651 2.229 1.940 1.716 1.342 0.938 0:20 1.936 5:045 5.001 4.896 4.716 4.420 4:209 3.838 3.551 3.111 2.772 2.312 2:002 1.763 1.371 0.654 0.30 2.058 5.175 5.141( 5048 4.900 4.644 4.459 4.104 3.824 3.36Ï 2.993 2.472 2.121 1853 1.427 0.40 2.164 5.245 5.166 5.033 4.792: 4:627 4.289 4.005 3.564' 3.192 2.616 2.232 1.944 1.481 0.680 0.50 2:256 5.377 5.345 5.269 5.140 4:924 4.747 4.432 4.168 1704 3.319 2.744 2.336 2.030 1.534 1.235 1:031 0:878 0.693 0.60 0.80 2.337 5.458 5:429 5.354 5.231. 5006, 4.852 4.545 4284 3.843 3.448 2.854 2.471 5.594 5.569 5.506 5.379 5.166 5.019 4.717 4.465 4.041 3.648 3.037 2.430 2.109 1.585 2.592 2.254 1.680 0.705 0.730 1.00 2.571 5:704 5.674 5.616 5.493 5.284 5.140 4.846 .4.593 4.182 3.796 3.179 2.725 2.365 1.767 1:404 1.155 0.972 0.755 1.20 2.660 5.792 5.761 5.703 5.581 5.380 5.235 4.947 4.683 4.297 3.904 3.293 2.833 2.474 1.845 1.200 0.779 1.40 2:730 5.864 5.849 5.774 5:658 5.458 5.32! 5.027 4.781 4.381 3:988 3.388 2.921 2.558 1.914 1.519 1.243 0.802 1.50 2.760 5.894 5873 5805 5.690 5.480 5.353 5.061 4.812 4.419 4.012 3.421 2.959 2.592 1.946 1.543 1.264 1056 0814 0.362 160 2787 5922 5901 5834 5718 5513 5385 5092 4845 4451 4073 3461 2994 2630 1976 1283 0825 180 2835 5970 5949 5881 57665564 51454904450841303519 3055 2686 2029 1618 1320 0847 2.00 2:874 6.010 5.977 5.922 5806 5.605 5.499 5.189 4.943 4.556 4.178 3.571 1107 2.740 2.076 1.657 1.355 1.120 0.867 2.50 2:941 6.083 6.045 5.993 5:882 5.685 5.576 5.270 5.023 4.646 4.266 3:669 3.204 2.843 2.171 1.743 1.430 1.196 0.915 3.00 2.996 6.130 6.095 6.039 5.932 5.728 5:606 5.324 5083 4.703 4.317 3.735 3.272 2.913 2.240 1.813 1.490 1.262 0.957 0.408 H 3.50 3:028 6.162 6.125 6.072 5.965 5.767 5:621 5.3605.119 4.743 4.360 3:775 3.319 2.963 2.291 1.863 1.539 1.306 0:994 4.00 3.051 6.184 6.149 6:096 5.988 5.795 5.653 5.386 5.144 4.775 4.389 1811 3.354 3.002 2.330 1.903 1.578 1.346 1.026 4.50 3:068 6.200 .6.167 6.115 6.005 5:808 5.683 5.404 5.161 4.789 4.411 1843 3.382 3.029, 2.359 1.931 '1.609 1.376 1.053 500 308062126174612560185820568654195174479544293851 340330472382 1958 1635 1402 1077 0459 600 3:097 6.229 2.414 1.114 7.00 3.107 6.239 . 2:436 1.142 10.00 3.122 6.254 6.217 6.167 6.062 5.709 5.465: 5.177 . 4.484 3:930 3.485 3.134 2.470 2.063 1.753 1.526 1.192 0.544 12.00 3.126 6.258 2.482 20.00 3.133 1.774 1.571 0:584 3.142 6.252 6,221' 6.160 6.042 5.872 5.712 5.464 5.236 4.833 4.488 3.927 3:4913.142 2.51312.094 1.79511.57J)]1.25710.628

Table 1. 5. -Ve, (n/ri < 1) N nr1 Z/r1N. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.85 0.90 0.93 0.96 0.98 0.99 0.995 0 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 3.142 0.003 3.300 3.693 0.005 3.402 3.622 3.946 0.007 3.499 3.782 4.116 0.01 3.174 3.204 3.213 3.230 3.265 3.310 3.415 3.635, 3.965 4.281 0.02 3.06 3.319 3.386 3.475 3.664 3.977 4.319 4.534 0.03 3.239 . 3.360 3.408 3.502 3.618 3.869 4.197 4.486 4.639 0.05 3.298 3.298 3.303 3.306 3.362 3.362 3.404 3.497 3.573 3.719 3.885 4.178 4.451 4.657 4.749 0.07 3.368 3.480 3.417 3.449 3.510 3.629 3.720 3.908 4.085 4.384 4.597 4.751 0.10 3.444 3.444 3.464 3.578 3.654 3.815 3.943 4.132 4.324 4.572 4.733 4.847 4.890 0.15 3.621 3.717 3.784 3.886 4.086 4.227 4.419 4.588 4.766 4.876 4.961 4.986 0.20 3.742 3342 3.775 3.808 3.892 3.966 4.104 4.312 4.452 4.623 4766 4.894 4.980 5.049 5.075 0.30 4.042 4.042 4.067 4.108 4.217 4.305 4.462 4.653 4.771 4.917 5.003 5.075 5.137 5.188 5.209 0.40 4.318 4.318 4.334 4.392 4.492 4.585 4.741 4.896 4.999 5.098 5.178 5.21Ö 5.260 5.300 5.319 0.50 4.562 4.562 4572 4.625 4.742 4.8 15 4.946 5.083 5.165 5.245 5.296 5.321 5.364 5.395 5.413 0.60 4.766 4.766 4.781 4.822 4.942 5.005 5.112 5.229 5.297 5.361 5.404 5.413 5.452 5.483 5.495 0.80 5.118 5.118 5.123 5.162 5.252 5.293 5.369 5.455 5.499 5.540 5.569 5.562 5.596 5.622 5.633 1.00 5.380 5.380 5.378 5.422 5.480 5.498 5.553 5.614 5.658 5.673 5.693 5.677 5.710 5.733 5.737 1.20 5.582 5.582 5.567 5.602 5.622 5.648 5.681 5.735 5.767 5.776 5.792 5.770 5.799 5.821 5.825 1.40 5.741 5.741 5.708 5.724 5.777 5.762 5.777 5.827 5.855 5.857 5.865 5.844 5.871 5892 5.895 1.50 5.765 5.808 5.866 5.891 5.875 5 902 5.922 5.924 1.60 5.843 5.843 5.814 5.825 5.847 5.849 5.855 5.900 5.927 5.921 5.928 5.904 5.949 5.951 1.80 5.901 5.901 5.895 5.904 5.927 5.917 5.908 5.958 5.982 5.974 5.990 5.953 5.996 5.998 2.00 5.962 5.962 5.958 5.962 5.992 5.970 5.949 6.004 6.031 6.017 6.021 5.994 6.018 6.035 6.038 2.50 6.062 6.062 6.062 6.062 6.092 6.062 6.049 6.088 6.110 6.095 6.096 6.068 6.090 6.106 6.110 3.00 6.128 6.128 6.126 6.128 6.153 6.118 6.106 6.141 6.162 6.145 6.146 6.116 6.138 6.156 6.158 3.50 6.168 6.168 6.166 6.158 6.190 6.155 6.139 6.177 6.206 6.179 6.183 6.548 6.170 6.188 6.190 4.00 6.194 6.194 6.189 6.194 6214 6.180 6.165 6.201 6.225 6.202 6.206 6.170 6.192 6.211 6.213 4.50 6.214 6.214 6.205 6214 6.234 6.198 6.181 6.218 6.242 6.219 6.223 6.187 6.209 6.227 6.229 5.00 6.226 6.226 6.219 6.226 6.248 6.211 6.194 6.231 6.255 6.231 6.235 6.199 6.221 6.240 6.242 6.00 6220 6.248 6.248 6 238 6.256 6.258 7.00 6.226 6.259 6.258 6.227 6.248 6.267 6.269 10.00 6.270 6.270 6.269 6.270 6.292 6.233 6.250 6274 6.292 6.273 6.283 6.243 6.264 6.282 6.284 12.00 6236 6.279 20.00 6.282 6.239 6282 6.285 6.250 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.383 6.283 6.283 6.283 6.283 6.283 184 M. IWASAKI

-vz

rin < 1

Table 1. 6. V,, (nr1 1)

I

I- Ql rin N. z/rt LOO 1.005 1.01 1.02 1.04 1.07 1.10 1.15 1.20 1.30 1.40 1.60 1.80 2.00 2.50 3.00 3.50 4.00 5.00 10.0 o L.15710 0 0 0 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0.003 0.533 0.288 0.104 0.067 0.005 -1.588 754 0.456 0.223 0.111 0.007 0.912 0.592 0.350 0.155 0.01 -1.602 1.059 0.757 0.452 0.219 0.112 0079 0.048 0.019 0.02 -1.629 1.261 1.04. 0.730 0.413 0.223 0.156 0.096 0.03 -1.653 1.319 1.16 0.910 0.569 0.332 0.230 0.148 0.058 0.05 -1.696 1.339 1.256 1.07 0.776 0.511 0.363 0.231 0.163 0.095 0.064 0.013 0.07 -1.735 1.280 1.131 0.893 0.643 0.472 0.312 0.225 132 0.090 0.045 0.019 0.10 -1.788 1.299 1.267 1.158 0.981 0.760 597 0.412 0.304 0.183 0.124 0.064 0.04 0.027 0.013 0.15 -1.866 1.235 1.193 1.146 1.019 0850 0.720 0.531 0.412 0.260 0.182 0.095 0.040 0.010 0.20 -1.936 1.161 1.104 0.997 0.880 0.774 0.604 0.487 0.322 0.231 0.123 0.078 0.052 0.025 0.30 -2.058 1.055 1.050 1.012 0.938 04860 0.793 0.657 0.563 0.405 0.301 0.174 0.112 0.076 0.036 0.40 -2.164 0.956 0.951 0.921 04870 04819 04770 0.661 0.595 0.444 0.350 0.213 0.097 0.50 -2.256 0.867 0.862 0.840 0.801 0.764 0.731 0.640 0.590 0.453 0.374 0.240 0.162 0.115 0.057 0.030 0.019 0.012 0.006 0.002 0.60 -2.337 0.789 04783 0.766 0.734 0.702 0.685 0.608 0.569 0.451 0.386 0.257 182 0.130 0.066 0480 -2.471 0.656 0.654 0.617 0.595 0.538 0515 0433 0.379 0271 0.204 0.152 0.081 1.00 -2.571 0.550 0.550 0.534 0.521 0.504 0.513 0.474 0.453 0.391 0.354 04269 0.211 0.163 0.091 0.052 0.033 0.022 0.012 1.20 -2.660 0464 0.461 0.446 0.440 0.429 0441 0.413 0.399 0.349 0.323 0.257 0.207 01098 1.40 -27300.394 0.387 0.375 0.374 0.364 0.381 0.361 0.350 0.309 0.291 0.242 0.199 0.162 0.102 0.062 1.50 -2.760 0.364 0.358 0.350 0.345 0.334 0.354 0.337 0.328 0.290 0.275 0.232 0.193 0.103 0.062 0.045 0.030 0.007 1.60 -2.787 0.337 0.330 0.319 0.330 0.315 0.306 0.273 0.266 0.223 0.188 0.158 0.103 1.80 -1835 0.290 0.284 0.275 0.274 0.267 0.288 0.276 01270 0.241 0.232 0.204 0.175 0.151 0.103 0.065 2.00 -1874 0.251 0.233 0.236 0.234 0.252 04243 0.235 0.213 0.208 0.187 0.162 0.140 0.100 0.066 0.049 0.034 0.019 2.50 -2.9490.180 0.171 0.199 0.166 0.164 0.186 0.175 0.173 0.159 0.159 0.148 0.133 0.119 0.091 0.065 0.0490.037 300 -2996 0.133 0.126 0.161 0.120 0.119 0.141 0.134 0.132 0.121 0.122 0.117 0.109 0.097 0.081 0.058 0.047 0.036 0.023 0.012 3.50 -34028 0.101 0.089 0.107 0.088 0.088 0.111 0.104 0.101 0.093 0.095 0.095 0.090 0.081 0.070 0.052 0.044 0.035 0.023 400 -3O51 0.079 0.067 0.070 0.066 0.062 0.089 0.080 0.082 0.074 0.076 0.078 0.075 0.068 0.06 I 0.046 0040 0.034 04023 4.50 -3O68 0.062 0.051 0.045 0.050 0.050 01074 0.065 0.059 0.063 0.064 0.063 0.057 0.053 0.041 0037 0.033 0.022 300 -3.080 0.050 0.041 0.031 0.038 0.037 0.062 0.053 0.067 0.048 0.(57 0.055 0.054 0.049 0.046 0.035 0.033 0.027 0.021 0.017 6.00 -3.097 0.033 . 0.021 0.046 0.033 . 7.00 -3.107 0.023 . 0.011 0.036 01023 10.00 -3.1220.007 0.020 0.008 0.021 0.008 0.008 0.011 0.015 0.011 0.016 0.014 0.011 0.010 0.0100.016 12.00 -3.126 0.003 0.016 0.003 20.00 -3133 . . 0.009 0° -3i42 O O O O O O O O 0. 0 0 .0 0 0 0. 0 0 0 0.

Consequently the author performed the integration of U with respect to Zi/Ti along

the column of =1 without - any particular consideration through the point (zJri = 0, rIn =1) assimiing that U was continuous through this point. _{And the}

results obtained on this assumption did nOt showéd any irrationality, for, example, Vz-u,/rj=1 and z/rj=20'

*ir12 ±

U2 d(zi/r1)

.3 13

while the value of the present function which results from the consideratiOn of

section 7.2) is ir _{3.14. Therefore agreement betWeen these two values is very good,}

and consequently the assumption that U2 is continuous through the pointU seems reasonable. _{For tha remaining cases, 'in which. singularities are thoúght to raise}

discussions, the author found no irrationality by applying the ordinary method of

integratiàn.

7) Iwasaki, 1. e. ante p 159.

* _{On both sides of the point (n/rj=l z/r1=oo) the}

ortex cylinders of infinitely large

Ienth extend. _{This can be considered as the case in which the. distance between the}

base of thó vortex cylinder in figure 5 and the point P becomes infinitely large. In this case, therefore, the v2lues ot U,. etc. due to the vortex elements near the point P (l,00) may have much effect on the values of. Vr etc at point P.

rin z/ri 0.999 0.9995 -1.00 1.0005 . -, 1.001 -1.002 0.0 0.003 0.007 96.60 35.30 17.00 50.63 3.444 --3.189 3.021 57.48 103.4 41.62 23.00 158.0 72.03 186 M. IWASAKI

of V and V for these cases are very small, so that the decrease in significant

figures does not affect the accuracy of the final results of the calculations of v/ V and va/V by equation (2). In making figures 6.1-.-6.7, however, the consideration

of the counter plan against the decrease in, accuracy was given. _{For example,} when we integrated equatiOns (22) and (23), the total range of integration which extended from z/rj=0 tò infinity was divided as follows:

0-.0.0l, 0.007-05, 0.4-.-2.5, and 2'oe

for r/rj= 0.99.

Next, each range of the ingegration was drawn on sperate section_papers

and between these ranges the regions of overlapping were assured. By this method

all the contributions of Ur, U and Ug weré treated with the same weight of accuracy.

In the calculations of V etc. the singularities at the points on the vortex

sheet itself must be carefully considerd _{Now we shall inspect, for example, the}

case of V at r/r = I

to estimate the contribution of the values of U of the

vortex ring, which are very near the vortex mg corresponding to the singular point to the value of V of this point As shown m the table of U, it does not show any very large change in its value (table 2).

8) Jewel, J. W. and Heyson, H. H. calculated the vertical induced velocity v2/V around a helicopter rotor with constant radius of slipstream in the case of uniform disk lea-diñg with the cutout of 15 per cent R. Their method seems to be simliRr to the one shown in referece 1). Cf Jewel, J. W. and Heyson, H. H., Charts of the Induced Velocities near a Lifting Rotor, NASA Memo 4-15-59L, May 1959.

VORTEX THEORY OF A!RSCREW ₁₈₇

§ 4. Equation of Continuity, vortex System etc.

For the. calculation of airscrews, the fundamental equation (2), the equation

of continuity and the consideration on the boundary conditions are necessary. In

the case of airscrews, the equations of continuity are composed of two equations,

the one of which corresponds to the axial component and the other to the rotational component. The former is written as

(V+ v0)orodro= (V+ v)rdr. (26)

This condition will be used in Section 5. The latter, though the author has called

it the equation of continuity, is intrinsically the equation of motion or the equa-tion of continuity of angular momentum in the slipstream. This is

wr2= constant along stream line in slipstream, (27)

where w is induced angular velocity. If we consider wr=v, this equation of conservation of angular momentum is written in nondimensional form

Vt F

- . = constant. . (28)

How this condition is satisfied in our calculation will be shown in the later

sec-tion. Next we shall consider the vortex systems of propellers, windmills etc. by a numerical example and we shall discuss the effect of a boss on the flow field

around the airscrew used in this exampleNow we shall consider a windmill having

its aerofoil from r/R=0.27 to 1.0,8) and posessing the donstant circulation of r =

-0.0225 throughout this region of the radius. This circulation is the same magnitude

with the circulation around the aerofoil sectiOn at r/R=0.75 f the windmill whi-ch we calculated by the present theory considering the effect of the deformation

of its slipstream. The vortex system is shown in figure 7.1 ®. We shall assume

that the cylindrical vortex sheets springing from the tip and the root section have the constant radiuses throughout the slipstream and the helical vortex filaments over both of the cylindrical vortex sheets have the constant pitch-diameter ratio of H/D=rt/QR/V=3.14212.5 = 1.26. Then we have

R/r1.cot1 = R/r1 . 2ri/V= .QR/V= 2.5. (29) The formulas of equations (40) and (41), which will be expained later, are

written in the present case as follows:

v/V=(r) (.cot

_r1

.v) ±r(cotcti. y0)

_{tu) T1 Coot}

RO

©

1/ k/ore - vorteo sy wivJIi y; o oiponIth of vortex Sysim v,atfthork of wiod,.11I _{vr fr,(0)} ¡t,a ok jvI1h ret'

Fig. 7. 1. Various vortex systems and their flow fields

(constant circulation T' along blades)

0.0563 . (Vztip_Vzroot)

v/V= r. V tip(R)

T _27L) U11 . d (ro/R)

+ . y,

root

()root

0.0255 V1 _tip- 0.0833 V1_noL

-

'

R'2

± 0.0225 _{.íO2l(-) Ur,Rod(ro/R).}

And from equation (2)

v7/V(1) (.cott . Vr)tip ± -

(!.cot,

.Vr)

= 0.0563 (Vr tip

In equation (31) the term concerning to is zero, because iñ this calculation the author assumed the radiuses of the cylindrical vortex sheets are constant with

respect to z,'R. V7, ,. in these equations have the meaning of V7, , when

we calculate the induced velocities in front of the windmill. By using dingrams

(6.l)(6.7) and figures (2.l)'-(2.6) of the reference 1) fOr U7 the equations (30), (31) and (32) are calculated, and shown in table 3 with the values without round

brackets.

When we inspect the table of vj V, we can find that the circulatory flow is almost completely confined in the slipstream of the windmill (nR 0.27 1.0 and z/R = 0-'- infinity), and outside this slipstream v,/ V is almost eliminated. This

w,,J 'sf

ai

wir4a;11kHs1 r r,

IWASAKI-windmill disk (z/R = 0)

tip vortex (r/R = 1.0) surface of doss (nR = 0.27) axis of windmill (nR

0)

1±

Table 3.

Results of calculation for constant circulation

B1'

'4it2VR

= r

= - 0.0225 = const.

The values without round brackets ( ) correspond to the cose of Vi of Fig. 7.

I. The values in ( ) correspond to the cose of © of Fig. 7. 1.

Nz/R

N

n/RN. -0.55 -0.05 0.05 0.55 1.25 0 -0.005 0 0 1.025 -0.006 -0.003 -000! -0003 0.975 0.010 -0.006 0.145 0.148 0.75 -0.008 -0018 0.195 0.188 0.295 -0.010 -0.011 0.472 0.486 0.245 -0.018 0 0 0.016 O O O O 0

Nz/R

N

r/R

N

-0.55 -0.05 0.05 0.55 1.25 0.972 (0.972) 0.992 (0.992) 1.008 (1 008) 1 028 (1 028) 1.025 0.956 (0.956) 0.944 (0.944 1.056 (l.056) 1.043 (1.044) 0.975 0.953 (0.952) 0.891 (0891) 753(0.753) 0.680(0.680) 0.75 0.952 (0.948) 0.842 (0.842) 0803(0.803) 0.719(0.717) 0.295 0.924 (0.920) 0880 (0.869) 0.764(0.776) 0.722(0.725) 0.245 0.924 (0.921) 0.925 (0.902) L075 (1.009) 1.075 (0S99) 0 0.927 (0.923) 0.977 (0.940) 1.023 (0.969) 1.074(988) 1.25 1.025 -0.028 -0044 (-0.028) (-0.044) -0.008 (-0.008) -0.056(-0.056) 0.008 0.056 (0.008) (0056) 0.028 (0.028) 0.043 (0.044) 0.975 -0.047 (-0.048) -0.109 (-0.109) -0.247 (-0.247) -0.320(-O.320) .0.75 -0.048 (-0.052) -0.158 (-0.158) -0.197 (-0.197) -0281(-0.283) 0.295 -0.076 (-0.080) -0.120 (-0.131) -0.236 (-0.224) -0.278(-0. 275) 0.245 -0.076 (-0.079) -0.075 (-0.098) 0.075 (0.009) 0.075 (-0.001) 0 -0073 (-0.077) -0.023 (-0.060) 0.023(-0.031) 0.074 (-0.012) 1.25 1.025 0.023 (0.023) 0.041(0.042) 0.032 (0.032) 0.156 (0.158) 0.032 (0.032) 0.156 (0.158) 0.023 0.041 (0.023) (0.042) 0.975 0.041 (0.041) 0.156 (0.159) 0.156 (0.159) 0.041 (0.042) 075 0034 (0035) 0.075 (0.078) 0.075 (0.078) 0.034 (0.035) 0.295 0.008 (0.010) -0.062(--0.041) -0.062 (-0.041) 0.008 (0.010) 0.245 0.009 (0.010) -0.082(-0.056) -0082 (-0.056) 0.009 (0.010) o 0 (0) 0 (0) 0 (0) . . (0)

9) Jowkowski, N. Theorie Tourbillonaue de I'Helice Propulsive, Gauthier-Villars, Paris, 1929.

There may be the case that the wall of the stream tube is connected with the vortex tube at, some place a little distance inside of the lip of the vortex tubc as the case of

stagnation pint of an aerofoil section. But the error due to this- stepped connection

will be small 'in the present consideratiom

190 M. IWASAKI

result corresponds to Joukowski's result9 expressing that, for the case of the two-bladed propeller having aerofoil from r/R=0 to 1.0 and constant circulation aro-Und every aerofoil section along the blades, the rotational component of induced

velócities is confined in the slipstream. Jowkowski's results were derived by formulas

without numerical calculation, while in the present case the similar results are

derived by numerical calculation. Next we shall look over the table of

v/

V and

I

+v/

V. Then we can see that in the region between r/R=0 and 0.27 there are the deceleration of velocity in front of the windmill and the acceleration of it

behind the windmill. Therefore the stream tube, which contains the flux between

r/R==0 and 0.27, has a definite radius of r/R_00 at negative infinity of z/R where the velocity in this tube is vi V= V/V=l. Then the tube expands to the radius

r/R_Z,R in front of the windmill and contracts to the radius of r/Rzj,: at some distant place hehind the windmill. Of course there is no flow across the wall of the stream tube in front of the windmill. And as the vortex element moves to

the direction of the velocity governing the point where the vortex element is

situated, there will, be no normal flow across the wall of vortex sheet. Therefore

these stream tube and vortex tube may be considered as a kind of stream tube. Therefore we may replace this tube with a rigid wall. Accordingly the case of

® of figure 7.1 can be also the flow field of a windmill with a boss, which is

a curved circular cylinder which has. the radiusés of r/R_r/R..(R'---r/R+Z/R. Next in order to Obtain a vortex cylinder of constant radius of r/R=0.27, the

following procedure was tried The author thought that if the axial velocity of

I ±

vj y

is nearly equal to one in the vortex tube stated above, downstream from

the windmill, then the radius of this tube had approximately constant value of r/R=0.27, so that thi wall might be replace by the wall Of a boss which had the

radius of 0.27. But by this procedure the flow in front of the windmill is more

retarded than in the case of ® (see table 3, thé values in round brackets), so that there the radius of stream tube has greater value of r/R_ z/R than in thé case

of.®. Thérefore the the case of © or the flow field around a windmill with a boss

of constant radius cannot be realized. But the flow field around a windmill, whiöh has the boss that has the form of (r/R=O.27'r/P_Z,R'--0.27), will be realized by

the procedure stated above. The author used, therefore, thi way of cOnsideration to know how the boss of constant radius of 0.27 placed behind the windmill has

effect .on thé flow field around the windmill.

Later in this section it will be seen that by making the values of

v/

V in

the boss equal to zero the values of Vr! V on the surface of this cylinder db

not approach zero in so favorable degree as the case of v/ V agaiñst the supposi-tion illustrated in figure 7.2 (see Dr/V in table 3). This unfavorable result may

(tz)

¡ &arIs.4 of yor/,vs,v

a2ILtafl$ cida viJe

--to 5 '

bet,4. r'Tfr a.,d

r-74e iokdI;',es cones7d

ja me Caft'/ wia

ada& valac.t,oa

iU Aazii Wades at fat

V0y/

a ma

vtoX

-± L (R/rj. V2)0 -Jot z/R=O.05

cot &wIth bo,,

-2

Ftg. 7.2. The vortex system (dotted lined), by which the - cylindrical surface of radius rs becomaes a wail

of stream tube (surface of a boss). (Constant circulation r along blades)

equal to zero. Anyhow paying attention exclusively tó the values of _Vt!V, we shall

proceed to the further study of the present case. In equation (30) y is constant and R/r1 also approximately constant, while V2 is the function of the relative

coordinate z/r1 and nr1. Therefore this is constant too, so that the only one

vaiable which is changeable is cot qS. As shown in the later description of this

section, a little change in vortex system near the boss cannot have any appre-ciable effect to the flow field near the tip cylindrical vortex sheet, (concerning

v2/ V and v,J V). Consequently the value of cot i- at the tip vortex is not

chang-ed in the present case. Then the variable is only the cot q5 on the suiface of the

boss. Accordingly the author determined this value of cot i, so that the value of v/ V at the points (n/R=0.245, z/R=0.05) and (0.245, 0.-55) became

approxi-mately zero. Thus Ìnaking equation (30) equal to zero, we have

E(R/ri.cot .V2)111

L (Rin. Vz)bo -itt z/R=O 55

0.481 ± 0.526

= 5Ó, :. q'i

= 63.5°,

=

.2

while

Cot without

(- f)

= 0.68, :. Çiwiuout i,oe = 55.8°.

This value of cot 0 with . is 0.74 times as large as the original value of cot

1wut

As stated above by .the present calculation the flow field

correspon-* _{For z/R=oo, v2/V=0}

_(V2

_{tIp= V2boss =-2n), so that 01}

boss=55.8° here. The present

results, therefore, can be applied behind and near the windmilL The complete infor-mation about i for oo<z/R<+oomay be obtained by giving comOlete cOnsidera-tion to the peformacOnsidera-tion of the slipstream with respect to its pitch and form

192 M. IWASAKI

ding to the case of © of figure 7.1 cannot be explained in the strict sense, but the effect of the cylinder of cônstant radius or the boss attached to the baók side of the windmill will be explained qualitatively, and consequently the case of ©

of figurO 7.1 may be expressed in. approximate manner. In this rather not so exact

meaning the author will treat the present case as the case Of ©. Then the result of calculation just described above may be expressed, in other words, as follows: in the case of © of figure 7.1 the pitch angle of i over the boss becomes larger than the corresponding value in the case of ®. Using this pitch angle of 63.5°

for the helical vortex filament over the boss the induced velocities calculated again by the equations (30)(32). The results òf calculation are shown in table 3 in

the round brackets. Seeing the table of v/ V and 1 + v/ V, one will be able able

to know that the velocity behind the windmill in the cylinder of r/R = 0.27 is

approximately équal to I and vj V is approximately zero, while the reduction in the velocity in front of the windmill appears more clearly than for the original

case of (.

Therefore in the case of the vortex cylinder of constant radius

be-hind the windmill the expansion of the 'stream tube in front of the windmill becomes

larger than for the case ® the previously discussed.

As seen in the table the

difference in the induced velocities between the two cases corresponding to ® and

© is very small. As the limiting case the author calculated the induced velocity v/ V neglecting the contribution 'of the cylindrical vortex sheet at r/R=0.27 or

assuming = 'r/2 there. Being selected from the results of this calculation,- the

values of i,/ V at r/ R = 0.45 are shown in the followings:

The values without round bracket were obtained in consideration of the cylindrical vortex sheet and the values in the round brackets are the ones corresponding to the case of the negcection of this cylindrical vortex sheet.

The position of r/R=0.45 is the innermost position, where that main

calcu-latiOn of the induced velocities whiOh will be described in the following sections,

is ma4e. Therefore it will be uñderstood that evOn in this position the 'effOct of

the vortex system over the surface of the boss on the values of v/ V may be neglected without producing any. appreciable errors.

By the present discussions we have been able to flíid that the pitch angle of the helical vortex filament over the boss becomés larger than the onO in the case without the boss and its value approaches ir/2 The vortex system 'corresponding

to an axial flow fan or a turbine without' guide vanes will be the system shown

in of figure 7.1. This vortex system was derived from the consideration stated

above as the limiting case corresponding cot =0 for both of vortex sheets of tip

and boss. By this vortex system only vt/ V is induced in 'the region.extending from

r/'R= r/Rb09, to r/R= 1 and between z/R=0 and infinity, while the remaining field feels no change in velocity. This flow field does' not contradict with the one of

/V at r/R=Ó.45

nR O . '0.05 0.55