Higher-order asymptotic expressions for the velocity field of a propeller duct or an array of tipvanes in axisymmetric flow

ARCH IEF

Technische Hogeschool:

Dem

DELFT UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF AEROSPACE. ENGINEERING

Memorandum M-280

HIGHER-ORDER ASYMPTOTIC EXPRESSIONS FOR THE

VELOCITY FIELD OF A PROPELLER DUCT OR AN.

ARRAY OF TIPVANES IN AXISYMMETRIC FLOW

by

Th. van Holten

DELFT - THE NETHERLANDS

June 1977

Memorandum M-280

HIGHER-ORDER ASYMPTOTIC EXPRESSIONS FOR THE VELOCITY FIELD OF A PROPELLER DUCT OR AN ARRAY OF TIPVANES IN AKISYMNETRIC FLOW

Th. van P3lten

Deif t - The Netherlands June 1977

The report is an extension of ref. I.

Closed form expressions are developed for the velocity field associated with a cylindrical ring on whose surface vorticity is distributed. The distribution

is rotationally sytmuetric, and involves only vorticity components in

circum-ferential direction. It is assumed that the chord c of the ring is small compared with its radius R. In contrast to the analysis of ref. 1., the

asymptotic solution developed includes terms of order O (dR)2. The reeult is, that not only the axial velocities at the ringehord ara corrected with respect

Notations 3 Introduction

¿e. Coordinate systems

5»±he boundary value-problem and its' ymptotic

approximation -,

6. Solution f o the first Birnbaum distibution Zero'th order soludon

First-order solution

e) Secon1-order solution 12

7. Potential of general vortex distributioi 13

8. Radial and axial velocities on the duct surface 15

9. Mass flow induced by the vortex cylinder ¡6

10. Comparison with numerical results 18

11. References 19

Appendix

Some far field expansions

Alternative far field expressions

Stream functions of several ring singularities

2. Notations

a constant

a a , a _{unit vectors perpendicular to surfaces of or}

z = constant

Ak, A constant in series expression

b constant

c chord length

c _{constant in series expression}

n

d _{constant in series expression}

n

D(k) integral efined in eq. (A-13)

e _{constant in series expression}

n

E(k) _{elliptic integral of the second kind}

f _{constant in series expression}

n

F flux

g _constant

i _{unit vector in x-direction}

j unit vector in y-direction

k _{unit vector in z-direction}

k integer

k k _{functions of r and z deflined by eq. (A-8) and (A-23) resp.}

K(k)

elliptic inte'ral of the first, kind

m

_{dipole strength per unit length}

n integer

r _{circular cylinder coordinate defined in fig.} I

R _{radius of the cylindrical ring}

u _{velocity component in x-direction}

y _{velocity component in y-direction}

V _{axial velocity component}

mean axial velocity along the ring chord y in the point P

p

Vr radial velocity component

V _{total velocity vector}

w _{velocity component in z-direction}

w _{w in the point P}

p

X X _{Cartesian coordinates defined in f ig.}

z z r ci Y

r

e

ri p p D1, 2 X Indices common cOmp f ar near -.. r....

Cartesian coordinates defined in fig. 1.

Cartesian coordinates defined in fig. I.

angle

vortex strength per unit length

total vortex strength integrated along the chord

small quantity

running coordite in z-direction

ellipticàl cobrdinate defined by eq. (4-4) circular cylinder coordinàte defined in fig. I

radlüs vector

...

distances defined in eq. (4-8) and (4-9) elliptical coordinat& defined by eq. (4-4)

velocity potential

circular cylinder coordintc defined in fig. I.

circular cylinder coordinate defined in fig. I

common field composite field far. fiei:d

3. Introduction

In ref. I,analytical expressions were deriv4 for the velocity field of.a cylindrical ring 9n.whose surface vorticity is _{distributed in such} away that the ring represents a propeller duct or an infinite aray of tipvanes, The.expressions were derived using a."matched asymptotic. expansion" technique based on the assumption that the _{chord C of the}

ring is small compared with its radius R.. The analysis of ref., I neglected terms of a relative order Q.{.(tR)2}. It was found that this theory led tp. axial velocity components çorrecting the flow of the

zero'th order theory where the ring chords _{are treated by two-dimensional}

strip theory.

The present report also includes effects of Of(C/R)2}, which _leads to the addition of radial velocity components as well.

The higher-order extension given in the _{present réport was neèded to} gaIn some insight into the required shape _{of sócalled tipvanes (ref. 2)} The general procedure of the analysis is explained in _{detail in the} chapters 4 to 6 for the first Birnbaum ("flat plae") _{type of vortex} distribution. Expressions valid for general distributions _{are given in} chapter 7. A comparison with the numerically calculated _{tables given} in ref. 3, showing the range of applicability _{of the results is given in}

4. CoorMtate-system.

The coordinate systems used are

depicted in figs.

i

and 2. They are

defined as follows.

The basic reference system is the

circular cylinder system (r,,z). The

zxiSpoiñtSii1 the diréction Of the f ree-strearn velocity'UwheraS

the boupdary conditions for the veloci.ty

will be applied to the ring

surface r=R, -c/2 < z< c12, wheré

R=D/2 is the radius of tle rìng.

Inhevîcinity of the ring surf aèe it' is

conrenie'nt to refer the position

of apoint to a local two-diniensional Cartesian sytem (x,y). Other

local reference systems are the circular

cylinder systen (p,x) and

theellipticalcYliùder coordinate system

(n,cp)(fig; 2)

the làter

system related to thè lbcal Cartean

coordinate

(x,y)by

x= c/2 cosh r

cos

p

y/2 si'rth

in

.'

.

. .

(4.-2)

The value of r. ranges. between

O and , with n..

constant repreSefliflg

ellipse

degenerating for=.Q intp, the duct chord.

The :dte p

ranges between -ir and +rr,

the lines cpconstant

representing hyperbolae

orthognal..with the ellipses. O.ntl.e,duçt s.urfce

the variables

:

and x are. related by

.

.

.cp

arccos (--)

(4-3)

the positive sign refers to the outer

duct surface, the negative

sign

to the inner surface.

At large distances from

the duct section, i.e. n -'

or p/c +

the elliptic and circular

coordinates are related by

-= x +

(C)2

sin 2x

(4-4)

(4-5)

The inverse transformation

of the elliptical cylinder

system is given by

n = arccosh

(4-6)

21

P-

arccos

_c

5. The

boundary value problem and itsasymptotic approximation..

The perturbation velocity potential associated with the vorticity

distribution on the cylinder r=R, -c/2 < z < c/2 must satisfy Laplace's equation

div grad O = O (5-1)

and the boundary condition

um

- -

urn _{- = (z) on the duct surface} (5-2)

r+R

Z

r4R.

where y(z) denotes the vortex strength per unit length in chordwise direction. In order to exclude any other singularities than vortices on

the duct surface:

11m - 11m = O on the duct surface (5-3)

r+R

r+R

and finally

2 2

0 + O for r + z + (5-4)

This boundary value problem may be approximated for small values of cID as follows. In terms of the circular cylinder coordinates (r,p,z)

Laplace's equation (5-1) reads in the case of circular syimnetry (see e.g. ref. 4):

2

rr

2

(5-5)

Substituting r = RO + , eq. (5-5) may be written

c(1 c y

- 0 (5-6)

2 D D 2

on condition that cID is small, and that we only consider the part of the field close to the duct surface (the near field), where -. and

are of order unity. Since the partial derivatives occuring in (5-e are all of the same order of magnitude in the near field, the equation shows

that, for very small cID, the perturbation potential 0 satIsfies a

unexpected looking at'oü'problein onth scale of the dut :Ç1Prd c, c/D + O means that D is increased indefinitely so that indeed a two-dimensionaistripproblem remains. This leads us to assuiaean asymptotic seriessolution for the, near field potential ønr:

0near0OD0l

()202+

...

forc/D+O

(5.-7)

where 00 is the purely twcrdimensional field obtained in the limit c/D0, whereas the other terms describe the way in which the field becomes two-dimensional when cID + 0, It will appear later that also terms occur

behaving like c/D ln(c/D)9 (cID)2 ln(c/D), etc. for c/D+ O. For

convenience, such terms have not been written explicitly in (5-7), but

are assumed to be included n the corresponding terms having an asymptotic

behaviour like cID, etc. Substituting the assume1 type of solution (5-7) into eq. (5-6) and equating terms of an equal order finally shows that 0, 01 and ₂ satisf,, the following equations:

a20 °

=0

(5-8)

a(X)2

a()T)2 a20 aø I 0 (59)

x2

a ()

a 2 a

'a20 " a20 aø

______ +

'x2

a 2 a

whereas the boundary cnndit ions become

30 30

hm

._2.

-

him _..2. - y(z) on the duct surface (5-11)

r+R

Z

_r+R

'r

11m

s. -

11m ---- - O on the duct surface (k=1,2) (5-12)

r+R

uZ

_r'R

'30..."

a0

' .

um

-

hm -a--- =0 on the duct surface (k=0.,1,2) (5-13)

r+R

r

r''R

r

The boundary condition at infinity (5-4) does not apply to

_oj2'

since

the asytaptotic approximation for 0 discussed so far is only välid in the

near field. This causes the problem to be undetermined as yet. The

indeterminacy can be removed by derivirg an approximation for the far

field as well, and matching the near and far field a's will now be explained. (5-10)

The far field is the field at distance of the order D from the duct

surface. Looking at our--prolilemon _{the scale.. of the duct diameter D,}

the limit c/D -* o means that c is decreased indefinitely, which means ttiátthe far field

₀

will become-in the symptocappràximátion the field of a discrete ring of singularities with diameter D. In order .t findaunifornly valid approximaion for the field

which is

valid

at small as well as large distance from the duct, we construct a socalled "composite" field

₀

as follows:

comp

0

=0

+0

-0

comp near far common (5-14)

where 0 is called the "common part of ₀ and O . The common

common near far

part 0 i-s really a correction field, chosen such that it cancels common

(to the required order of accuracy) the far field clos to the duct surface

so that only the near field remains there. At the same time ₀

common cancels ₀ at large distance, so that at distancof the order of the

near

diameter D only 0far remains. This t3me of structure of the fitial,

uniformli valid composite dield O implies the condition

comp

nearp + order D T

farp

order c = 0common

tO the required order of accuracy (5 15)

Or, rewritten in terms of the respective "characteristic coordinates

hm

0 = hrn

0far for c/D - 0 (5-16)

near

p/D+O

which condition, the socalled "matching condition", complètes the

6, Solution for tie. first Birnbaum distribution.

a) Zeroth order solution. If we neglect all the terms of order

O(c/D) and smaller in eq. 7) 0near will consistofØ, the

dimensional field,.only. in:terms of the elliptic coordinates (n,cp) the

two-dimensional Laplace:equatioa(5-.8) reads (see e.g. ref. 4).

o o

csh2

L

flCOScp

2 2 + 9cp2 - 0 (6-1) grad O = + (x,O ) = -I o cf2lsin

p I W

(O,q)

+ (grad - _{/2lsin pl}

where = arccos (j-.). The corresponding point on the inner surface is given by -p , and a here points in positive x-direction, so that

(x,O ) = a . (grad ø)

= c/2 sin

-p

Boundary condition (5-Il) thus transforms to

= y(cp) p) (6-3) (6-4) (6-5) (6-6)

having the general solution (ref. 4)

= aq.+ b

+

n1

.(ce"cos

na+ de

sui

np+

cos ncp+ fe sin np ) .+ g. (6-2)

wtiere the constants ab,c ,d,ef.and:g are to be determined by the

boundary conditions. In the elliptic. system grad 0 takes the form (ref. 4):

3- 0

2 . 2

1

+ a

c/2(cosh

n -

cos p)

where is the unit vector rpendicular to the surface fl=constant, and

a is defined likewise. On the outer surface of the duct (fl=O), the

value of SO/ax then corresponds to

Likewise, boundary condition (5-13) transform to

-d

(O, p) +

-s- (°,p) = O

As an example, we will take here the special vorticity distribution (flat:plate, or first Birnbaurndistribution, see e.g ref. 5)

,2r c/2-x

2r

I-coscp

y(x).

"c/2+x

or

y(q)

= ;

_(sin

cp(

where r is the total circulation. Application of the conditions (6 6) and (6-7) shows that eq. (62)

must

take the form

. sin

¿)+ 2

(ecosh nn.cosn+ fnSit

nr.sin nq,)

+ g

(6-9)

where the

terms

involving côsh n and smb n- although becoming

infinité far from the duct hord (

)

carnot

be ruled out

immediately, since there is no condition at infinity. The last boundary condition to be satisfied is the matching condition (5-15) or (5-16). Using the expansion formulae (4-4) and (4-5), the'behavíour of the

field (6-9) at large distances from the duct surface (ri -* ) caxi be

shown to be

{x

I (c)2 c/4 O = sin 2X - sin X ...}

+ g +

o 2Tr p + 2e1

-9-r

cos X + 2f1 -9. sin X

.. (-a

)

(610)

For p of the order of magnitude D, i.e. p = ctD, it is seen that a term

c/4

.

like - sin

_{x has an asymptotic behaviour of the order 0(cíD) for}

c/D 0. In the present approximation where we neglect terms of such

orders and smaller, we may thus simplify eq. (6-IO) to

where the terms with e and f must be kept, since the order of magnitude cf these coefficients is a priori not known. Nevertheless, eq. (6-11) shows that to the present order of approximation the behaviour of the near

field at large distances involves just one type of singularity, viz.

a vortex type singularity associated with the term X

The far field of the zer&th order solution thus consists of a discrete

vortex ring only. Formulae for the field of such a ring are given in

the appendix, where it is shown that the behaviour of the vortex ring field

close to the ring is given by

0far = 0vortex ring = + I(P)2

sin 2x

_-&

ln(2.)}+ ...!

( - 0) (6-12)

For p of the order of magnitude c, i.e.p = c, it is seen that to the present order Of app ximain all the te of (6.12)may be neglected except thefirst:

'1 r

far

r..

(ØfR# 0Y. (6-13)

The thatching conditior thus shows, on-comparing (6-Il) id(6-1), that

g = e = f = O so that is completely determined now:

r

00 = '

(qre . sin

The zeroth order approximation for the composite field e(0) is:

e(o) r cotnp ZTr sín in

(Ç)

cos _{x +}

r

vortè- ring

:X;

(6-14) 6-15) y to;rder (c-ID)°

b. First-order .solutio.. _{If we. include terms of order (c/DY in (57),}

we will have to determine the field 01 by eq. (5-9), the boundary conditions (5-12), (5-13) and the matching _{condition (5-16). Using the}

relation

2

2

(coshnsinq-+ sinhrìcôs

cosh fl-cos q) ç

and the Laplace equation (61), _{eq. (5-9) transforms to}

-

(coshnsinp sinhn cos

n ç n q)

(sinhr cosç*

e

cog 2cp)

r

_-2n

0partic

= -

(n cosh cos cp+ cos 2cp + e )

(1)

:0 E0

_L

near o D I 2i1

-e sin p) +

(6-16)

(6-Ì7)

where, the expression..(6-14) has been substitutèdfor

0.

Eq. (6--17)

is a two-diienionai Poisson equation, having thé paricula 'solution

0

partic

(6-18)

The complete solution of _{(6-- 17) contains the particular solution}

(6-18) as well as-the general solutin (6-2) of the _{two-dimeTisiolial Laplace}

equation. The boundary conditions (5--12) and (5-13) read in

_terms

of the elliptic cOordinates (ì-,ç)

01

(O,q) +

_{(0,-q)) =}

0 _(6-19)

(O,cp) +

_{(o,-cp) =}

_6-20)

showing that the first-order solution for _{0 should have}

_{the form}

near

re

----

(n cosh cos cp+

en

cos + cos +

e2

_{+ iiI) +}

+ {g + 2

nI

Expanding this for, large .va.l-ues _{./e;aad retainingon'ly Eth te:} up to order Q(c/D)..: T'

r./4

_r

ønear=t1X)

ln(-)} +

(g + 2e1

/2

+ 2 f1 2 X + ) fOr p/c. -,co _6-22)

Eq. (622) shosihat, to the

accuracy aáhievadùow, the"bèhaviour of the near field at large distanceinvolves _{two types of singularities, viz.}

a

vortex type singularity and _{a d.ipole typé singularity associated with}

the term - siny . The far field will thus consist

of a discrete 2 _p

vortex ring as well as a discrete dipole ting, and can be shown (see the

appendix _{to beh'v, negLecting order O(c/D)2, for}

small p/R like

0_far

=0

_{vortex ring}.

+0.

=

dipole = _{{x lti(} P/R) _cos

r

_C.jSifl

21

l(P1) +

21 cos 2x} for P/R -* 0 (6-23)

Matching (6-22) and

(6-23)

according to eq. _{(5-16) then shows that the} unknown coefficients g,e and f must have the following values

= f = O, except e1 =

-+

from

_which.

=

- {ncosh fl cos + eT1, cos cp + ., cos 2 q):.

-2

r

comp _2r' sin )

r

4ir D

- Pc/4.

rc

p

-

(X ---sìnx)

cosh fl COSt:p+ cos cp+

n)

₊

0vortex ring .'dipole

ln(_.9.) cos +

+ ). cos +

ln-)}

up to oder O(c/D) (6-25)

Note that the constant terms and the term w-tth cash r cos =

occu!ng in the near field (6-24) have cancelled against the identical

constant terms and the term with -e-- cas

_{x =}

_-4--

_{in the common field}

ci2 ci2

(6 22), as a result of the subtraction of the common part from the sum of

the near and far field. Explicit expressions for . and _O

vortex ring dipole

are given in the appendix, where for the

dipole

strength m has been

substituted tu =

C) cdôrr o]tioj. The procedure for solving the secoid order

problem where terms of order (cID)2 are taken. into account, is identical to the procedure outlined in

occurrence of the term

_{-j1z- ()}

the far -'field has :t ....e úppl

ring 0 . The final expressions will not be given here explicitly,

audr

.

but are included iñ-the gnérai expressions given in sction 7. sction 6'. In this case; duet the

2sin 2 in the expansion of p(see eq.(4-4)),

7. Potential f. general vortex distributions.

From eq.. (6-2) it may be derived that the most gèneral vortex distribution

has a zero'th order near field of the form 00 = Aq + A e sin .icp

n

n=1

representing

according

to (6-6) the following distribution of vorticity

- (A

+ V _{n A cos ncp}}

1nq o n= IL n

where p on the surface is given by eq. (3) Naturally, for a given value of x, ,expressiàn (7-2) yields, as it should be, the same result for y

on the outersurface (cp> O) and on the innersurf ace (cp< O). Therefore, it is

convenient

to introduce a variable

6 = = arccos

c72 (O <

e <

(7-3)

which. is,uniqucly related to x on the duct surface, in contrast to

cp which is double valued. If the vortex distribution is given in the

form of the standard Birnbaum series

1-cos8

y(9)

= + y sin. nO (7-4)

n=I

the coefficients A(n=O,1,2,...) are related to y (n=O,l,2,...) by A

=.Ee

+

h'1)

0 4 o A1 =

*(v0

+ (75) A = .-

_{- y1)}

(n=2,3,....)

Using the type of analysis outlined in section 6 it is found = _{cos pe + cosh}

..

[n +

ln(-Ç)} 1 +

and = A sin 2cp1

+s±nh2-{ '+

- A1 sinh sin cp e + cos 2cp + + 2n + 2

+

n2

A e (n - n cosh 2 - sinh 2rì). sin(n-2)cp 2n sin ncp sin(n+2)ç

n-I . 2...

n-I

The near field up to the order (cID)2 is

= 0 + 0

(c)2

2

As is shown in the appendix, a convenient representation of the corresponding far fe1d is

(2) r1

F{r,l,z

far L

1=1

+ (2-i) *, where the function

j. F(r,i,z) denotes the potential!n the poiut(r,t4,z) due

to a vortex ring with strength r The values of r are given by eq (B-4)

The corresponding common field is

mori =

_i1

F+0{rPz + (2-i)*}

(7IO)

whe the functiànF

(r,ii,z) is up .to order (c/D)2givtiby (A-9),

using the transformations

(p/R)2 = (z/R)2 + (r/R-1)2

r / R- I

tanx

-z/R

The composite potential thus becomes

(2) (2) (2) (2) comp near far common

(7-7)

(7-8)

(7-9)

8. Radial and axial velocities on the duct surface

Using eq. (6-4) and (6-5) to;ether with (7-8) the axial velocity on the

duct surface: j . (x,0)= A + n A cos ne - +

c/hsln9o

n n=1 - J-

(.)

_{A { I} + ln(EÇ-)} - A1 -

n2

An cos ne (8-1)

Likewise, the radial velocity on the duct surface is found to be

(x,O) =

- c/2 sin e fl A sin ne *

n= I

c/D

+ 3 2

A cos B + ln(_j.) } - A1 + cos 2e+ln(.EÇ) } + 4c D o

A

+ _{-- {n sin n& sin e + cas nB cos e}} (8-2) n2.n -1.

These expressions show that the terms of order (cID)0 yield the

vel'ôcitiés of a two-dmensional tnin aerofoil. The terms of order (cID)1 add a self-induced axial velocity, whereas the terms of order (cID)2 add

9. Mass flow induced by the vortex cylinder.

The volume rate of flow V through the surface z0, r< R may,, according to Gauss's theorem, be written as

V =

2TJ

z=o r dr = 27r

Rf

r=R (9-i)

in which expression should b substituted

0=0

_near

+0

_far

-0

_common

It is convenient in this case, to write the far field again in the form

(B-i), and to use the corresponding expressions (A-9).., (A-II) and (A-12)

for the common field. The integral (9-1) taken of each field component

0 0 and ₀ separately, is divarc'ent. To avoid this

near .. far common

difficulty, expression (9--i) is writtenIike z ro ø . . -,

V = 2rR hm

I. .

(

near) +. V

(R,E.),+ y

- (R,c) + r r=R vortex dip L. J o O

e+o

.

:.ZO.

+ Vquad(RE) - .1

COfl)r=R

where the function V (r,z) denotes the volume rate of flow, vortex

through a circular surface of radius r at axial position z.

An exaression for V (r,z) for r -- R and z + O has been derived in vortex

appendix C, together with similar expressions for V . and V dip quad

The near field 0near is given by eqs. (7-i), (7--6), (7-7) and (7-8) in terms

of the elliptical coordinates (fl,p). Therefore, the integration hiit z0 in the first integral in (9 2) is first replaced by r. The integration

result, which is a function of n, is afterwards written in terms of z

using the expansion (3ee eq.(4-5)):

I e 2

+ O{(c/D)4} n0

= ln(---)

Ï

(e---)

o

Eq. (9-2) then yields

dz1 (9-2)

+ V

_vortex

(Rc) +

A

dip

= zIi1oo (

near)

_dp

(

near)

c-O

z

RE) + Vquad(RE)

10 C +

(C)2

₊

. 1n(-) }

+

(_1)k

A2k r -

(52

k2-I

I

(9-4)

IO. Comparison with ¿umerical res.ilts

In order:'to aiv aiÌ impression of the. accuraçy. of the asymptotic expressions

the.' figures 4 through 9 have been prepare1. These figures show for the first three Birnbauin distributions the axial velocity V and the radial velocity Vr on the vorticity surf-ace,. in1 comparison with' the tàbulated values of ref. 3.

The axial velocities shown do not include the component

h'(0).

It appears

that, although the asymptotic; -theory wa dveloed under the assption

CID « 1, the asymptotic theory does not lead to large errors until cID reaches the Order of magnitude unity s

II. References

1. Th. van Holten: 'symptotic expressions for

th

velocity field due to a

cylin.rical ring distribution of vorticity', Deift

Univer-si.ty

Aerospace Eng., Memorandum

M-223, Odtober 1974.

2. Th. van UoLten: t1Performancc

analysis of a windmill with increased power

output due to tipvane induced diffusion of the airstream,'

Delf t Univ. of Techn., Dept. of Aerospace Eng., Memorandum

M-224, 1974.

3. D. Kiichemann and J. Weber: "Aerodynathcs of propulsionvi, LLcGraw-l{ill Book

Cy., 1953, New York.

Appendix A. Some far field expansions

Expressions for the axial and radial velocities induced by a vortex ring

lying _{ir. thé pläne z=O with radius R and strength r are derived in ref. 3:}

vortex ;. r z/R . . I . '

2 nR

- _{2 2} k(k) -{I _{+ 2} {(I+r/R) +(z/R) .}2_. (1-nR)

+(z/R)

2 where k = 1-k which yields vortex r cos

- 2irR

PTX_

+

p/RcosX{ -&+ln(Ç)

+sin2}

+

O{(p/R)2}

(i-7)

vortex

r [sin

+

ln(Ç)

+

cos2x +

p/R sin x {

---inP4

+ _{sinx]-+ 0{(p/R)2}j} (A-8)

From (A-7) and (A-8) it follows that the velocity potential near the vortex ring is given by

vortex 'r

1tK(k)

-elliptic rr/2

2(i/k-Í)...

(A- I)

E(k)

-(A-2)

z . where K(k) and

¶12

2 . (I+/R) +(zfRy}2t

E(k) denote corrplete

{1 + 2 2 (1-nR) +(z/R) integrals defined by K(k) = with dc = J , o

(1-k2sin2a) dOE (A-3)

2 4

(1-k sin c)

4r/R

(A-4)

(1+r/R)2

+

(z/R)2

In the vicinity of the ring one may expand (A-l) and (A-2) using

K(k)

=

ln(A-)

+

{ln(r) _1}

+

(A-5)

p,'R

0

= -.x-

p/R

ln(-vortex 2

..

The potential field of..a vortex ring has the fOrm

0vortex (r,i,z) _r.._2ir F(r,i,z)

0quadi'

(z/R)2

- sin 2< . . cos X cos 3y 3, . . .3,

= n - + .

-

Ir

' -

2 + --'sin + - ( /R)' 1L. +o{(p/R)}J (A-12) cs.x +

(p/R)2 sin2x{+.1n(Ç)}

+ 1 + O{(p/R)} (A-9) (A-IO)

where F is a non-dimensional

function

of the coordinates. The field of

.a dipoe- and quadrupole ring are obtained by differentiation of (A-10), and their behaviour for p-3-O is derived from (A-9):

0dip =' -

m(Z,R)

- m

+ in2--. + cos2x +

r

Appn4ix.B

.

Alternative far field e4r

s ions.

The far fiethüp to order (c/D)2consists in general of a vortex-,

a dipole- and a quadrupole ring, and. my:be written: ccording.to (A-10),

(A-II) and (A-12)

as

(2)

4- -- (A

far 'vortex 'dipole 'quadr =

(r,) -

n

/R)2'V'

(B-I)

where r ,m and n associated with a general vortex distribution (see eq. (7-l)) are determined by.he.matchig cdndition ,from which:

c2

A2)

Up to order

(cID)2

the far field may, as will be shown, a'Iso

be represented by three discrete vortex rings (fig. 3), which is a somewhat more conven].ent represent.ation in actual calculations _Using

this representation and expanding the field in a Taylor series: (2) F(r,,z+c/4) +

(r,tp,z)

+ F(r,,z-c/4) = far 2ir

Ic

= (r + r + r3 F(r,,z) + (rl - r3)

(zIR)

(r,,z) +

2rr 1 2

+ .-)._ (.)2(r+r)

(z/R)2

(r,tb,z)

+ O{(c/R)3}

In order that (B-3) equals (B-I) one should choose

r1 = 27r(A - A1 +

A2),

rf2lr(-A-2A2), r3=27T(A--A1+A2) (B-4)

The corresponding common field is given by

r,

--F

(r,i,z)

0cmmon = F

0(r,,z+c/4)

+ 2îr p O 2ir 0(r,tp,z-c/4) (B-2)

I

-. (B-3)

V . (r,z) = 2rrr

J

dip

z

Appendix C: Stream functions of -several ring singularities.

Formulae for vortex rings maX be found in ref. 3. The volume rate of flow V through a circular surface of radius r at axial position z is

there given as 'j r

V4(r,z) = 2ir

V.'_ (r',z)

r' dr' = o = -2rR r/R 2 {K(k) - 2 K(k)

_E(k)}

(C-1)

f(1+r/R) +(z/R) }2 k

If z/R« I and IrRI IR « I one can, using (A-5) änd (A-6), expand eq.(C-I) in terms of the coordinates p and _x (fig. I):

O/R sin

x {i +

ln2--)}

+ Vvortex(PX) = - FR [2 _{+ ln_ã.) 2} R + 2 + 3 in

Ç) }

-

()

p 2 sin. .2_x +

ln(-)} +

+ O{(p/R)3}: (C-2)

In the case dipole ring, one may use Gauss's theorem to write

dip

r (r,z')dz' (C-3)

Using

0dip =

_3(7R

(A-11)

where F is defined by (k-10), one may integrte. (C-3):

Vd (r, z) = 2rrr in (r,z) (C-4)

so that, using eq (A-7), for p - O the following expression is found:

Vdi(P,X) = 27rR in COS X + sin 2X +

c

cos 3x +

Likewise:

f

= 2irr n

V

(rz) = 2iîr

quadr

J

(z/R)(r/R)

z

(C 6)

Using (A-7):

V ₍

X) = 2irR n

cos 2

sin

cos 2

+ i

quadr

'

L.

(p/R)2

p/it

16

Fig.1::. Coordinate systems.

Yr

\

4

i

.3 2 o -1 o

i -

---cosO

C/2

Fig. 4. Cpmar.ison present theory

_.()

_{jth ref. 3 (---)} First Birnbaum distribution; axial velocity.

twb.- dime nsionäl

/1

2

i

two-dimensional cw=

0.0 0.25

o-

- -- -

---

t -1 0

1 --

_=cose

C/2

twò-d irnénsionot

i

i

.-.,

C/2 Fig. 6: Idem, 2nd Birnbauxn distribution; axial velocity.

two C/D;1.O

Fig.. 7: idem, 2nd Birnbaum distribution; radial vèlocity.

-i--,. cos e

.0.5 Oh Ö3 0.2 Ö.1 '-0.1

0.2

0.3

J

two- dimensional +1 _=CQs8 CID =0.25 10 0.5

2.0 1.0 - 0.1

0.2

Fig.

two- dimensionaL

Idem, 3rd Birribauna distribution; radial velocity. C/2