J . Fluid Mech. (1967), vol. 30, part 1, p p .

177-196 177
Printed i n Great Britain

The flow field near the centre of a rolled-up

vortex sheet
By K. W. MANGLER AND J. WEBER
Royal Aircraft Establishment, Farnborough

(Received 4 January 1967)

Most of the existing methods for calculating the inviscid flow past a delta wing
with leading-edgevortices are based on slender-body theory. When these vortices
are represented by rolled-up vortex sheets in an otherwise irrotational flow, some
of the assumptions of slender-body theory are violated near the centres of the
spirals. The aim of the present report is to describe for the vortex core an alterna-
tive method in which only the assumption of a conical velocity field is made. An
asymptotic solution valid near the centre of a rolled-up vortex sheet is derived
for incompressible flow. Xurther asymptotic solutions are determined for two-
dimensional flow fields with vortex sheets which vary with time in such a manner
that the sheets remain similar in shape. A particular two-dimensional flow
corresponds to the slender theory approximation for conical sheets.

1. Introduction
Observation shows that, when a slender delta wing is placed at incidence in a
stream, the fluid separates from the surface along lines near the leading edges.
Shear layers, springing from these lines, form in the fluid and roll up (in opposite
senses on the two halves of the wing) into spiral vortices which lie above the wing
and inboard of the leading edges. Although viscous effects play some part in this
flow, mainly in determining the position of the separation lines and in the inner
part of the core, various attempts (Legendre 1952; Mangler & Smith 1959;
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

Smith 1966a, b ) have been made to calculate this flow on the basis of an inviscid
flow model, in particular for a wing with sharp leading edges where the position
of the separation lines is known.
The theory developed by Legendre (1952), Mangler & Smith (1959), Smith
(1966a) is based on slender-body theory. However, in the vortex core velocities
have been measured (seee.g. Earnshaw 1961) which differ considerablyfrom those
of the main stream which implies that it is doubtful whether slender theory is
applicable near the centre of the core. We therefore aim in this paper to examine
the flow near the centre of a coiled vortex sheet by a method which avoids the
assumption of small perturbations.
Experiments suggest that a delta wing of aspect ratio less than two produces a
vortex sheet which even in subsonic flow is approximately conical over the for-
ward part of the wing, away from the trailing edge and outside the immediate
neighbourhood of the apex. Therefore, it was decided to consider the flow as
12 Fluid Mech. 30
 178 K . W . Mangler and J . Weber
conical, i.e. the velocity components are assumed to be constant along rays from
the apex of the delta wing.
I n the present paper we consider only a narrow region around the vortex
core and derive an asymptotic expansion of the velocity field near the centre,
based on conical incompressible flow. We avoid thus the difficulties? which
would arise if one were to try to calculate the entire flow field past a delta wing
by conical incompressible theory.
We further derive an asymptotic expansion for the velocity field near the
centre of a conical vortex sheet when the assumptions of slender theory are made.
The two results will be compared to show the consequences of the slenderness
assumption. The equations derived by slender theory for conical sheets are the
same as for the two-dimensional flow field of a vortex sheet which grows linearly
with time. This is a special case of the two-dimensional flow fields of endless
rolled-up vortex sheets which vary with time such that their shapes remain
similar. A certain family of two-dimensional similar vortex sheets which grow
with time has already been investigated by Prandtl (1922). Another family,
including the one with linear growth, is considered in this paper.
We ignore viscous effects and assume that the vorticity is concentrated on a
thin sheet so that the flow between the turns of the sheet is irrotational. The task
is then to find solutions of the equation for the velocity potential which satisfy
the boundary conditions at a free vortex sheet.
We attack the problem by introducing two stream functions and transforming
to co-ordinates X , Y such that the two faces of the vortex sheet become lines
Y = const., say Y = 0 and Y = y. The problem is then to solve three non-linear
first-order partial differential equations subject to the boundary conditions at
Y = 0 and Y = y . We derive only the leading terms of formal asymptotic
expansions in terms of the distance from the centre of the sheet. The conver-
gence of these expansions is not investigated, nor are the solutions they represent
necessarily unique. The conical and the time-dependent problems are treated
separately.
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2. Conical vortex sheets

2.1. General equations
We investigate the flow field near the core of a rolled-up conical vortex sheet. For
steady flows the sheet must be a stream surface, which means that the velocity
vector is tangential to the sheet. We assume constant total head for the core
region. Then the condition that the sheet cannot sustain a pressure difference
between its two faces requires that the magnitude of the velocity is the same on
opposite faces of the sheet, but the directions of the velocity vectors may differ.
The velocity field must satisfy the continuity equation. We ignore compressi-

t Germain (1955) has shown that the assumption of a wholly conical incompressible
flow must lead to the existence of singularities in the flow field outside the wing and the
vortex sheets originating a t the wing; as a consequence the solution is not uniquely
defined.
 Flow near the centre of a vortex 179
bility effects, then the continuity equation can be written in spherical polar
co-ordinates (R,$, 8 (see figure 1)) as

a
a
(R2%sin $) + -
a
alC. (Rv$
a
sin $) + - (Rv,) = 0.
ae
This equation is satisfied automatically by the velocity components which are
derived from any pair of functions Y,(R, $,19)and Y 2 ( R$, , 8 ) by means of the
following equations :

It follows from ( 2 ) to (4)that

so that Y,and Y 2are constant along streamlines.

There is some freedom in the choice of stream functions Y, and Y , for a given
three-dimensional flow. We make use of this to choose them for the conical prob-
lem so that
Yl = R2f($, 61, (5)
y 2 = S(P,8). (6)
q($, 8) = const. represents the intersection of the conical stream surfaces with
the sphere R = 1. The vortex sheet is thus represented by a curve g($, 8) = const.
From ( 2 ) to (6) we obtain the following relations between the velocity compo-
nents and the functions f and g:
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The subscripts to v represent the velocity components; subscripts to all other

letters denote partial derivatives.
Since we assume that we are dealing with a potential flow, a function @(R,$, 8 )
exists such that
aa,
V R = a '

1 a@
"$=%@,
I a@
v, = - 7 -
R sin $ 80 '
12-2
 180 I ! . W . Mangler and J . Weber
For conical flow, the potential function is of the form
0 = W(@,8)
and vR = $3

vy? = &3

1
VO = 7$0.
sin @
We introduce the variable
t = -log tan &@,

i.e.

sin $ cosh 6 = 1.
With t and 8 as the independent variables we obtain from (7) to (13) three partial
differential equations for q5, f and g:

I n order to obtain relatively simple equations for the boundary conditions

at the vortex sheet, we introduce new independent variables X,Y so that
e = e(x,Y ) ,
t =x
Then

Now we choose
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

so that

The three equations (14) to (16) take the form

OF$ = -cosh’Xf,,
6F-X = 2fP+ (e,>’I,
(6F = ZP,.
We note that this sytem of differential equations is equivalent to equations (14)
to (16) if and only if 8, is neither zero nor infinite. This condition is equivalent
to the statement that no conical stream surface is either tangential or normal to
a circular cone @ = const. In this way the range of possible solutions is restricted.
 Flow near the centre of a vortex 181
By the transformation of the $, 0 variables to the X , Y variables, the region
between the turns of the spiral vortex sheet (projected on the unit sphere R = 1)
is mapped on a strip 0 < Y < y in the (X, Y)-plane so that the lines Y = const.
correspond to the projections of the streamlines on the unit sphere, with Y = 0
and Y = y corresponding to the two faces of the sheet (see figure 1). The lines

FIGURE
1. Notation.

X = const. correspond to the projections of the cones $ = const. We consider

the special case where 9 = 0 , i.e. <+coy represents the centre of the vortex sheet;
then the vortex core, $ < $, say, is represented by the semi-infinite strip
_ _ _0 < Y < y. The region ABCD in the (X, Y)-plane is mapped
X > -log tan _+$o,
on the region ABCD in the ($,8)-plane,with the straight line B D representing a
circle $ = const., and the curves A D and BC representing lines 8 = const. For
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

two points B and B on either side of the vortex sheet, [ = X has the same value
but the angle 6' differs by 2n. The geometric boundary condition at the sheet
reads therefore
AO(X) 8 ( X ,7)- O(X,0) = - 27~. (20)
When this condition is satisfied, then we obtain a one-to-one mapping between
the ($, 6') and the ( X , Y)-planes.
For the velocity components we obtain the equations:

OR = $2 (21)
1
~~=-cosh[$~=-coshX---
1 + (8,)s 4x, (22)
 182 K . W . Mangler and J . Weber
The magnitude J' of the velocity is given by
v 2 = v&+v$+v;
1
= $2 + cosh2 X ($XY*
1+
The pressure condition, A P ( X ) = 0, leads thus to the condition

Since it follows from equation (20) that As, = 0 , we may replace this by

2.2. Derivation of a n asymptotic expansion for the vebcity JieZd

We intend to obtain an asymptotic solution of the equations (17) to (19) which
satisfies the conditions (20) and (24) for the inner part of the vortex core, i.e.
for small @ or large X. The presence of the term cosh X in equations (17) and (24)
suggests that we try to obtain an asymptotic expansion of a solution in powers
of ex. We therefore seek a solution in which the functions O(X, Y ) ,$ ( X , Y )and
f ( X , Y )for large X are expressed as series in ex:
19 = B,(X, Y )ex + 8,(X, Y )+ 0 2 ( X ,Y )e-x + ..., (25)
$ = $,(X, Y )+ $,(X, Y )e-x + ..., (26)
f = f o ( X ,Y )e-2x +f , ( X , Y )e-3x + .... (27)
We assume that the coefficients and their first derivatives are bounded for large
X or tend to infinity like Xm. This excludes such functions as sin ex.
It follows from equations (17) and (18) that

This equation together with the assumptions about the coefficients in the
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

series (26) and (27) require that the series for O begins with 8,(X, Y)eX.The
boundary condition (20)requires that O0(X,0 ) = 8,(X, y ) . This and the assump-
tion that 8, must not vanish for finite X and 0 < Y < y are satisfied if So is
independent of Y .
We satisfy the boundary condition if
Y
B,(X, Y ) = -2r-+h,(X)-f (28)
Y
and B,(X, 0 ) = B,(X, y ) for n 1. + (29)
From equations (18) and (19) it follows that 4, is a function of X only and that
$ly does not vanish identically if $oxdoes not vanish identically. It follows from

t There exists a multitude of functions 0,(X, Y ) which satisfy the conditions. The
particular choice in equation (28) has no influence on the velocity potential when this is
expressed in terms of $ and 0.
 Plow near the centre of a vortex 183
equation (17) that f(X, Y )behaves like e-2x if + ( X , Y )behaves like an algebraic
function.
To determine the functions Bo(X),h , ( X ) , 8 , ( X , Y ) , ... $ o ( X ) , $ l ( X , Y ) , ...
we insert the expansions (25) to (27) into the differential equations (17) to (19)
and the boundary condition (24) and consider in each equation the leading terms.
Then the following set of equations is obtained:

2fo(Bo + 00x1 = + l W
4(e0+ e 0 A 2 W + 1= +o&91 -AAX).
It can be shown that these equations are satisfied by
f - - - 4n
C2(X+6+4),
0 -
Y

where C is a non-zero constant, 6is an arbitrary constant, and ho is a constant

related to the origin of the &scale. It can be shown that, with the assumptions
made about the coefficientsin equations (25) to (27), the solution is unique.
To determine the function g l ( X ) above and h l ( X ) in ( 2 8 ) we consider in the
differential equations (17) to (19) and in the boundary condition (24) the terms
which are one order smaller than the leading terms. This procedure leads to the
following set of equations:
P ( X )= ( X + 6 + * ) 4
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(30)
877
fix - 3f1 = - 2C2(2F2- 1)82p+ -
Y
(- 4n
Y
C2FY +gl) ,
-

fl = 2 ~ 2 ~ 2 -8 , ~
Y
f i _ 8 n C 2 $ 7dh
21=
F y dX Qf2y'

AB, = 0,

+nC2 ( 4 F 2 + k ) ] = 0.
 184 K . W . Mangler and J . Weber
It can be shown that this set of equations is uniquely satisfied by
g, = 277C2F,

(7
h, = 71,

fi = Tan- C2F(2F2+ 1 ) Y -n),

~ (Y-n-) +7
6 F 2 + 1 271 2
02 = h,(X).

To determine the functions g 2 ( X )and h,(X), we have again to go back to the

differential equations and the boundary conditions and solve the set of equations
derived by considering the terms of the next lower order. After some lengthy
calculations we obtain the following solution for g2(X)and h 2 ( X ) :

where ei(h) = JA m e-t

h2 = .J?T
3-eFzerfc(F)
2 1
where

We have not derived further terms in the expansions since the algebra involved
is very complicated.
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2.3. Discussion of the expansion

Summarizing the results, we quote the equations for the shape of the streamlines
and for the potential function:

(34)
where F ( X )is definedin ( 3 0 ) ,h,(X) andg,(X) are given in (31), ( 3 2 ) .X i s equal to
< which is related to the cone angle q9 by equation ( 1 3 ) and a conical stream sur-
face is a line Y = const. and the two faces of the sheet are given by Y = 0 and
Y = y.
 Flow near the centre of a vortex 185
The potential difference across the sheet is

+-2n2
F
e - 3 X + ...
[ ( 4 F 2 +l)e2Fzei(2P2)-e4Faei(4P2)]

= 4ntan$$(-logtan+$+6++)*+0($-3)
I
= 27.44 -log$$+S+$)~[1+0($2)]. (35)
The e-3x term has been obtained from the set of equations from which the func-
tions h 2 ( X )and g 2 ( X )were derived.
By eliminating Y from equations (33) and (34) we can express r$ as a function
of 8 and of X , i.e. of $. To do this, we introduce the notation 8,($) for the value
of 6' which the inside face of the vortex sheet has for a given $.8,($) results from
equation (33) for Y = 0. B,($) is a single-valued function in accordance with our
assumption that conical stream surfaces g = const. are never tangential to the
cones $ = const.
The relation between the velocity field and the potential function is given by
equations (21) to ( 2 3 ) .We obtain:

-'R = -logtan$$+&
C2
- 2 tan $$( -log tan $$ + 6 + +)+[6,($) - 8 - 771 + O($21og $)
= -log $$+6- $( - log $$+a+ +)* [o,($) - 8 -TI + o($'lOg$), (36)
3 = - 2 ~ 2 e - x - 2 ~ ( 2 ~ 2 - 1-) ~
C2
= -$(-log+$+S+&)
(? 1 - e-zx+
n ...

- $2( -log
3 = F + ( ~ F z 1-)
C2
(7
$11.+ 6 )( -log
Y - n ) e-x
$$ + 6+ &)$ [8,($) - 8 -4 + 0($3(10g $y), (37)

+ F 3 + 2 F + -7) - ( Z F 3 + 2 F - & ) EY-n) 2

6P

- (n2/F)[(4F2+ 1) eZF2ei (2 P 2 )- e4Faei (4F2)] e-2x

1 + ...
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= ( - log +$ + 6+ $14+ II.(- log +$+ 6 )LO,($)- 8 -TI+o ( p ( i o g9)". (38)

In these formulae 8 varies between 8,($) - 2n and 8,($).
To simplify the comparison with other work, we express the velocity field in
terms of cylindrical polar co-ordinates x , r, 8.
v, = vRcos $ - v@sin $,
_ -- - l o g -2x
vz
C2
r+ + ) * ~ s ( ~ ) - 8 - n j + . . . ,
r+ 6 - ~ (X- l o g - + 62x
(39)

_ -- 2 2
c
2 ----r)
v, = vBsin $ + v$ cos $,
v,, 1r
x
2 r
[-log5+8+l] ( - l o g r~ + d + $ ) d ~ s ~ ) - 8 - ? r ] +
..., (40)

! = (-log-+&+$
E
C2
r
22 )$
+-:[ - l o g - r+ d ] [ B , ~ ) - 8 - n j + . . . .
2x
 186 K . W . Mangler and J. Weber
The shape of the section of the vortex sheet by a plane x = const. is given by

We note that the axial and the circumferential velocity components tend to
infinity when we approach the centre of the spiral whilst the radial velocity
component tends to zero. The first term in each of the equations for the velocity
components is independent of 8. The velocity field arising from the first term of
the asymptotic expansion is thus continuous across the sheet and axially sym-
metric.
It is interesting to compare the velocity field with that obtained for an axially
symmetrical conical flow with continuously distributed vorticity. Hall ( 1961)
and Ludwieg (1962)have investigated this flow for slender cones, i.e. for small
values of r/x. Their solution reads
v, = C,{ - log ( r / x )+ C,],
v, = -+C,(?./z),
++
v g = Cl( -log ( r / x ) -tC,)*,
where C, and C, are arbitrary constants. A comparison of this solution with
equations (39) to (41) gives the noteworthy result that for our particular case the
leading terms of the velocity components for a potential flow with the vorticity
concentrated along a vortex sheet are the same as for an axisymmetric flow with
distributed vorticity.
We note that all the terms that have been found in the present asymptotic
solution are determined by the choice of the two constants C and 6, just as the
axially symmetric solution is determined by the two constants C, and C,.
The present approach of term by term evaluation of the asymptotic expansion
cannot provide information about its convergence. The expansion is certainly
not valid when a stream surface g = const. (e.g. the vortex sheet) is tangential to
a cone @ = const. and therefore the derivative go = l / O , vanishes. The trans-
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

formation to the X , Y = g co-ordinates breaks down in this region. Experimental

evidence (see e.g. Earnshaw 1961) and numerical solutions in Smith (1966a)
suggest that the vortex sheet from the leading edge of a delta wing is tangential
t o such a circular cone at certain points well away from the centre of the sheet.
Whether infinitely many further points of tangency arise as the centre is ap-
proached cannot be determined by either means.

3. Time-dependent two-dimensional vortex sheets

3.1. General equations
We investigate the flowfield near the centre of a vortex sheet in two-dimensional
incompressible flow which varies with time. The sheet is a discontinuity of the
velocity field which is carried along by the flow. For a free vortex sheet in in-
viscid flow the vorticity of a fluid element does not change with time, therefore,
 Flow near the centre of a vortex 187
the sheet consists of the same particles at all times. This statement about the way
in which the sheet moves with the fluid leads to the geometric boundary condition.
The second boundary condition is again derived from the requirement that no
pressure force must act on the sheet. For a time-dependent flow, Bernoulli’s
equation for the static pressure contains the derivative of the potential function
with respect to time. Therefore, the condition that the pressures on the two faces
of the sheet be the same does not require that the magnitude of the velocities be
the same.
The velocity field must again satisfy the continuity equation. Let r and 6 be
polar co-ordinates and t denote the time co-ordinate, then the continuity equa-
tion reads
a(rv,) +-3% = 0. (43)
ar a6
Since we consider only irrotational flow, a potential function @ ( r ,6, t ) exists, such
that
v = -a@ (44)
ar ’
1 a@
vg=--. (45)
r a6

Equation (43) becomes the Laplace equation

1 1
Qrr+- Qr+- @‘eg = 0.
r r2

It is possible to solve the problem by means of the theory of complex functions

since @ is the real part of the complex function @ + iY where Y is the stream
function.
We intend to solve the problem by a method which is similar to the one used in
6 2. Since the problem is concerned with particle paths, we introduce two func-
tions Y,(r, , t) which are constant along particle paths. If Y,and
6, t ) and Y z ( r 6,
Y2satisfy the equations
r = ylrY20-Y10Yzr, (46)
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

rvr = Y10Tzt-YltY28, (47)

VO = ~lt~2r-Ylr~209 (48)
then equation (43)is satisfied identically. Furthermore,

i.e. Yl and Y Zare constant along a particle path. The values of Y,and Y, can be
taken as ‘labels’ which identify a particle. If we give to the particles forming the
vortex sheet at one time t = to one common label T2= const. then the sheet is
at all times defined by Y 2= const. The inside face of the vortex sheet is denoted
by Y z= 0 and the outside by Y,= y.
We consider here only self similar flow fields for which the streamline pattern
at any time t is similar to the pattern at time t = to. This means that the velocity
 188 K. W. Mangler and J . Weber
components can be written as functions of r/h(t)and 8 with a scale factor which
depends only on t.
We introduce the variables r = t and r* = r/h(t) and choose Yz to be inde-
pendent of r. Then it follows from equations (44)to (48)that Y,, Y, and @ are
expressible in the form?
Y
l= h2@)f@*,
(3, (49)
y, = g(r*,8), (50)
0 = h(dh/dt)[&*, 8 ) + +r*2]. (51)
From equations (46), (49), (50) we obtain
r* = f,*go-fog,*, (52)
from equations (44),(47), (49) to ( 5 2 )
r*$v = -2fgs, (53)
and from equations (45), (48) to (51)
W*)$@
= 2f9F (54)
To obtain simple relations for the boundary conditions we introduce new
independent variables F , Y so that
8 = O(F, Y ) ,
r* = i;

and

We choose Y=g

so that
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

This change of independent variables is permissible if and only if go and 0, are

neither zero nor infinite. As in $ 2 , we consider only this restricted problem. The
differential equations ( 5 2 ) to (54) take the form:
re, =fF,
F&g+ = -2f[l+P(8,)”,
= -2fr0;..
+f There is some freedom in the choice of Yland Y 2and .therefore inf and g . If
= G(9)
then Y , = h+-
dGldg
satisfies the equations (46) to (48).
 Blow near the centre of a vortex 189
We need not follow the paths of the individual particles therefore the value of
TI,
i.e. off, is of no direct interest. However, we do not eliminate the function f
but retain it as an auxiliary function for solving the equations.
The static pressure for constant total head can be determined from Bernoulli's
equation:
2+
P at
+ gvr"+ v;) = B(t).
With the above equations this can be written as

In the ( p , Y)-plane the flow field is represented by the semi-infinite strip

0 6 Y 6 y, 0 < F , where the region near the centre of the core is represented
by the neighbourhood of i; = 0. For two points on opposite faces of the vortex
sheet the values of 8 differ by 27r; therefore the geometric boundary condition

For the second boundary condition (that the pressure is continuous across the
sheet), we obtain from equation (58) the relation

3.2. An asymptotic expansion for the velocity jieEd

We restrict ourselves to time laws of the form

h ( t )= I ~ t l ' ~ ,
where k and m are constants and
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An asymptotic solution of equations (55) to (57), (59), (60)has been found by

expanding the functions O(?, Y ) ,Fal-WL)#(F,
Y), F-2f(F, Y) into power series of
R = n-Fm/Cwhere the coefficients are functions of Y only and C is an arbitrary
constant.
+
For 0 < m < 2 and m 1, the first terms read
e - _i_ _
-
7~ mR
Y 4-m2
2y+[x(2y-1)
Y (2-m)(4-22m-m2)
3m2 1
R+ ..., (62)
 190 K . W . Mangler and J . Weber
For the special case m = 1, the functions e(T,Y ) , $ ( F , Y)+C210gR and
f(7,Y ) were expanded in power series with respect to R = Tn-/C. The result
reads :
e = c,+ ( 1 1 ~-A) + ( ~ A z + R -
- (+43 - +A)~2
77

c
2

Y
with A = 2--1,
Y
C, and C2are arbitrary constants.
The vortex sheet is thus given by
C 1
0, = 7+~(cl-
11772
i)+--~-(g+4++4q --,~3+0(~5)
r 6C C
and the potential difference across the sheet by

For the velocity components we obtain? the relations:

7T- 7J2 -
= -A -r2--(A2+2)-r3+...,
C c2
dh F o g $ ;
ve =
l+(r~i;)2
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

It is of course possible to use equation (64) to express A as a function of 8 - B,(R)

and R and then to write v, and vo as functions of F and 8.
We have stated before that the time-dependent problem for m = 1 is equiva-
lent to the conical problem treated by slender theory; we have only to substitute
r/(lcx)for F = r / ( k t )(see e.g. Mangler & Smith 1959; Kiichemann & Weber 1965).
Maskell has also examined the asymptotic behaviour of the conical leading-
edge vortex sheet by slender-body theory and given some results at the
I.U.T.A.M. Symposium on Vortex Motions a t Ann Arbor in July 1964 (see Smith
t I f the centre of the vortex core were not a t the origin of the co-ordinate system then
we would add the velocity components produced by a uniform flow which is parallel to
the line joining the centre of the vortex core and the origin of the co-ordinate system.
 Flow near the centre of a vortex 191
19663). He has derived the leading term for the derivatives of A@ and of
r,/x with respect to 8 where r, gives the distance of a point on the sheet from the
centre of the sheet. The result for A 0 is the same as given in this paper. For r,
his result reads :
= - 5 (1+ q1 sin 28 -pl cos 20) + ...,
dB 82
where Ic,, q1 and p 1 are constants. The important difference from the solution of
the present report is the appearance of the sinusoidal terms. We have excluded
such a type of expansion when we subjected the coefficientsin the expansions to
certain restrictions.
When introducing a more general type of expansion, we have also tried to avoid
the restriction implied in the assumption 8, + +
0 , 00 by choosing as independent
variables [, 7, e.g. the functions $(r, 8) and g(r,8). It is not yet known whether it
is possible to solve the resulting differential equations for the coefficients in
closed form.
A different approach to the asymptotic solution for a slender core was made by
Mangler & Sells (1967)who obtained (by conformal mapping) a solution with more
degrees of freedom. The leading terms agree with the present cases. The free
parameters cannot be determined locally but are used to fit the core to the ex-
ternal flow past the wing and the outer turns of the vortex sheets.

4. Discussion of the results

It is of interest to compare the results derived with and without the slenderness
assumption. In order to compare the differential equations, we introduce the
variable X = -log? into (55) to (57). Equation (55) reads then
fly = -e 2 X f x (70)
and equations (56) and (57) become identical with (18) and (19). The important
difference between equation (70) and ( 1 7 ) is the factor $ on the left-hand side,
since $ tends to infinity for X -+ co.
The geometric boundary conditions equations (20) and (59) are the same. I n
order to compare the pressure conditions, we introduce for conical flow the arc
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

length elements ds and da defined by

+
(as), = (dB)2 BS(da)2,
= (all.),+ (sin $d8)2.

Along a streamline Y = const. we have

du =
1
cosh X
~ +
(1 (OX)2)*dX.

We can therefore write the pressure condition of the conical problem, equation
(24), in the form

Introducing the arc length d a along the cross section of the sheet in a plane
x = const., for which da = a?(1 + (r8;)2)+,
 192 K . W . Mangler and J . Weber
we may write the pressure condition for slender theory, equation (60) with
h = t , in the form

We note that the difference between equations (71) and (72) is a factor # in the
first term.
We note that in slender theory the circumferential velocity, equation (69),
tends towards a finite value whilst the circumferential velocity derived without
the slendernessassumption, equation (41),has a logarithmic infinity at the centre
of the core.
Since the flow considered in slender theory is a two-dimensional flow free of
sources, the mean value of the radial velocity taken along a circle r / x = const.
is zero. For conical flow, the mean value of the radial velocity taken along a circle
T = const. in a plane x = const. need not vanish [according to equation (40)it
is - i C z ( r / x +
) O ( ( ~ / X ) ~since
) ] , the mass entering the circle can escape in the
x-direction. To obtain an approximate solution of the conical problem, Roy (1966)
has suggested introducing a fictitious source distribution in the 'pseudo-trans-
verse ' (i.e. pseudo-two-dimensional)flow in a plane x = const. in order to account
for this three-dimensional effect.
According to equation (51) the potential function (I, in slender conical theory is
related to 4 . by . the equation

(I,=x[#(;,0)+;(32]

The streamwise velocity component v, is therefore (to a first order) equal to 9.

We note from equation (65) that the velocity component parallel to the axis of
the vortex core calculated by slender theory tends to infinity in the same way as
when it is calculated without the assumption of slenderness, equation (39).
For the vorticity distribution along the sheet y = d ( A # ( / d rwe obtain for the
slender solution'

and for the conical solution

https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

The first term of the slender theory was derived in Mangler & Smith (1959) by
a somewhat incomplete argument.
If we compare the derivative

from equation (66), with the value in equation (42), we note that for the conical
sheet )d#,/d(r/x)1 increases less steeply with decreasing r / x than for the slender
sheet, i.e. the spiral is not quite so tightly wound. The slender theory result
agrees again with that of Mangler & Smith (1959).
 Flow near the centre of a vortex 193
Summarizing, we can say that the introduction of the slenderness assumption
has not changed the main character of the asymptotic behaviour near the centre
of the vortex sheet. Both theories produce tightly wound spiral sheets for which
the vorticity distribution tends to zero a t the centre.

2. Notation for double branched vortex core.

FIGURE

Both theories give logarithmically infinite velocities at the centre. The occur-
rence of these infinite velocities raises of course some doubts about the relevance
of results for incompressible flow. However, for the flow past delta wings at low
speed we can expect from the work of Brown (1965) that compressibility effects
are important only for a very narrow region of the core. Brown has investigated
the compressible axially symmetric conical flow with distributed vorticity and
shown that for small values of the Mach number a t the edge of the core the flow
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

can be treated as incompressible except for a narrow subcore. I n a similar in-

vestigation about the effects of viscosity Hall (1961) has shown that the flow
can be approximated by an inviscid flow except for a narrow subcore. A compari-
son with experimental results (see e.g. Kuchemann & Weber (1965)) shows that
an inviscid incompressible theory can predict some of the important features of
the flow.

5. Double branched vortex cores

With the method derived for a time-dependent vortex sheet, we can also de-
termine the flow field past two (or more) similar vortex sheets which have a com-
mon centre and a common time law h ( t ) . Stream functions Yl and Y2can again
be introduced such that both sheets are represented by curves
Y,(r/h(t),6)= g(r*, 6) = const.
13 Fluid Mech. 30
 194 K. W. Xangler and J . Weber
We assume again that go does not vanish, so that between the sheets, g is for
constant r* a continuous monotonically varying function. We may choose
Y = g = 0 to represent the inside face and Y = y the outside face of one sheet.
For the position of the second sheet any value of Y in the interval 0 < Y < y
can be chosen. We select the particular case of a symmetrical two-branched sheet
(see figure 2 ) by choosing Y = +y and a constant value for the leading term in
8,. We denote by the suffix I the functions for 0 < Y < &y and by the suffix I1
the functions for +y < Y < y. The differential equations are the same as before
and with the time law given by equation (61) the boundary conditions read:

+
For 0 < m < 2 and m 1, the first terms of an asymptotic solution, come-
sponding to equations ( 6 2 ) and ( 6 3 ) read:

( 2 - m) (4 - 2m- m2)
-K
-
( 2 - m ) ( 6 + 7m 7m2- 2m3)
+ 3m2 3m3

+K2-
+
( 2 - m)2( 6 2m - m2)-K3 (2 - m)3(2+ m)
______ ]R+ ..., (73)
2m2 2m3
https://doi.org/10.1017/S0022112067001363 Published online by Cambridge University Press

(l-m)(Z-rn) 1 ( 2 - m) ( 6 - 2m - 4m2+m3)
K
+ 3m 273-2 3m2

+KZ-------- +
( 2 - m)2( 4 m - m2)- K3( 2 - m)3( 2 + m)
]R2+*..), (74)
2m 2m2

where the upper sign applies in the interval 0 < Y < &y and the lower sign in the
interval &y < Y < y. The constant K determines the potential difference across
the second sheet. If we consider only the terms of lowest order, then we note that
 Flow near the centre of a vortex 195
the ratio between the potential differencesfor the second sheet and the first sheet
is K/(1- K ) . Equations (73) and (74) are applicable in the case m = 1, except
that the term ?2(1-nz)/2(m-1) in equation (74) is replaced by -log R.

6. Conclusions
Some asymptotic solutions for incompressible flow near the centre of a rolled-
up vortex sheet have been obtained for conical steady flow and for time-depend-
ent two-dimensional flow. Once the rather special form of the expansions is de-
cided, the solutions are unique and the number of free constants is very small.
In both cases the shape of the sheet is that of a tightly wound spiral. For conical
flow the axial and circumferential velocity components tend to infinity in the
same way as for axially symmetrical rotational conical flow (Hall 1961).
The results have been used to show the differences between the solutions
derived with and without the assumptions of slender theory. The circumferen-
tial velocity tends, with decreasing distance from the axis, logarithmically to
infinity for the conical solution and to a finite value for the slender solution. The
velocity component normal to a circle around the centre of the spiral has for the
conical solution a finite mean value which tends to zero with the distance from
the centre; for the slender solution this mean value is always zero. The axial
velocity component tends to infinity for both solutions.
Comparisons of an asymptotic core solution based on inviscid flow with experi-
ments made in a real flow are very difficult. Kiichemann & Weber (1965) gave
some examples which showed that the tightly rolled spiral and the large axial
velocity components predicted by the calculation seem to be consistent with
experiments.

This paper is Crown Copyright, reproduced with the permission of The Con-
troller, Her Majesty’s Stationery Office.

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