Department of Aerospace and Mechanical Engineering, Second University of Naples, Via Roma 29, Aversa, 81031 Caserta, Italy
The velocity induced by a plane, uniform vortex is investigated through the use of an integral relation between Schwarz function of the vortex boundary and conjugate of the velocity. The analysis is restricted to a certain class of vortices, the boundaries of which are described through conformal maps onto the unit circle and the corresponding Schwarz functions possess two poles in the plane of the circle. The dependence of the velocity field on the vortex shape is investigated by comparing velocity and streamfunction with the ones of the equivalent Rankine vortex (which has the same vorticity, area, and center of vorticity). By changing the parameters of the Schwarz function (poles and corresponding residues), rather complicated vortex shapes can be easily analyzed, some of them mimicing an incipient filamentation of the vortex boundary.
1. Introduction
The present paper investigates the self-induced
velocity field of a uniform vortex having Schwarz function of its boundary with
two simple poles on a suitable transformed plane. This analysis is viewed as a
first step towards the study of the dynamics of such a vortex in terms of the
related Schwarz function.
The use of the Schwarz function in searching vortex
equilibria and also in investigating their stability properties dates back to
the eighties of the last century (e.g., see the elegant discussion in [1, Section 9.2]). As well known, the Schwarz function of an analytic curve in the plane of the complex variable satisfies the request (a definition is indicated by joining the two
symbols “” and “”, with the first one on the side in which
the new quantity appears) in any point and is defined via analytic continuation on a
suitable neighbourhood of the curve
In general, singularities of the function are found inside and outside
An extensive theoretical background on the Schwarz function can be found in
[2], while an attempt
to summarize the original ideas on the use of such a function in searching
vortex equilibria may be the following one.
The curve is taken as the boundary of a uniform vortex (with unitary vorticity, for the sake of
simplicity) and it is assumed that can be rewritten as the sum of two functions: ,
where and are analytic inside and outside ,
respectively. It is worth noticing that the behavior of for follows just from the above splitting of
Indeed, consider the integral on the path of Figure 1 of the function ,
for fixed away from the vortex. It gives the value
of in terms of the moments of the Schwarz
function: with the integral having in place of vanishing, because is analytic inside
By accounting for that the integral for gives times the area of the vortex, the asymptotic behavior of follows as
Figure 1: Integration path for studying the asymptotic behavior of the function
It can be noticed that the conjugate of the velocity is analytic outside ,
while is analytic inside the vortex (both cases will
be unified in terms of a Cauchy integral of in (2.9) below); it follows that must have the form Indeed, the continuity across
the vortex boundary follows by that on the , behaves at infinity as and the corresponding vorticity is uniform and
unitary inside and vanishes outside. At this stage, it is
also easy to show that the asymptotic behavior of the velocity is related to
the (complex) moments of the Schwarz function, through the power series of (1.1).
The formula (1.2) relates velocity and Schwarz function
in a very appealing way. However, the identification of the functions and is often a quite complicated goal, which is
reached by firstly identifying through its asymptotic behavior (1.1) and then
by taking as
In the present paper, an approach which is alternative to the use of the
relation (1.2) is proposed: and are related through a Cauchy integral, which
is evaluated without splitting the Schwarz function. The two approaches are
completely equivalent: if the Schwarz function is not too complicated, the
splitting (1.2) is still more convenient, but in other cases the present approach
can give some advantage. As an important example, for the class of vortices
investigated in the present paper the use of the integral approach results
in being much easier than the one of the splitting
(1.2).
When the shape of the uniform vortex is not far from a circular one (quasicircular vortex), the identification of and is rather simple and the analysis can go ahead
[3]. The vortex
boundary belongs to a ring inside which the Schwarz function is evaluated via a
Laurent series, with and being the analytic continuations of its
regular and singular parts, respectively. Due to the small differences from a
circle, it is also convenient to use a Laurent series in order to define the
vortex shape through a conformal map between the unit circle and
The coefficients of the series for are nonlinear combinations of the ones of the
map. Configurations of absolute equilibrium in presence of external rotation
and stationary strain (intensity and principal axes of which are kept fixed in
time) have been investigated by imposing that the normal velocity of the fluid
at the vortex boundary vanishes:
This relation is enforced to be identically satisfied in ,
that is, any power of is multiplied times a vanishing coefficient: a nonlinear system in the map coefficients follows, to which conditions concerning the area of the vortex and the position of its center of vorticity are added. The system is numerically solved
via a multidimensional Newton solver. Finally, the linear stability of
stationary solutions also is investigated with the above approach. The
coefficients of the conformal map are taken as functions of time, and
it is enforced that the normal components of the
velocity of the fluid and of the vortex boundary agree:
The above constraint is linearized around a stationary solution, time
dependence is fixed in the form (with being a complex number, the real part of which
determines the stability properties of the vortex) and an eigenvalue problem
is deduced. Several interesting cases of bifurcations of equilibria are
discussed, with particular regard to the energy conservation. Also the dynamics
of quasicircular vortices has been investigated [4], but by using a quite
different approach. The vortex boundary is described in terms of a Fourier series and an equation of evolution for the Fourier coefficients is deduced.
It is also found that the derivatives along the vortex boundary of the velocity
and of the Schwarz function are related through a Cauchy integral.
In the last ten years, a large research
activity has been devoted to find and analyze
stationary solutions of the Euler equation by using the Schwarz function. The
vortex shape is searched by starting from the following form of the
streamfunction in a corotating frame of reference: The form (1.3) is assumed valid
outside or inside the vortex, while is taken as vanishing on the complementary
part of the physical plane. Multipolar equilibria have been analyzed in
[5], where symmetrical
configurations of point vortices are placed into a uniform one, in such a way
that the total circulation vanishes. In this paper, the form (1.3) is built as an
extension to general shapes of the streamfunction for the shielded Rankine
vortex (see also [6],
in which the discussion is carried out in the framework of the quadrature
domains by using a fundamental theorem of Aharonov and Shapiro). The stability
properties of the above equilibria have been analyzed in [7], both in the framework of a
linear theory and through fully nonlinear numerical simulations with a contour
dynamics approach (see (2.2) below). Quadrupoles, pentapoles, and higher have
been found to be stable equilibria, while the tripoles result in
being linearly unstable. Other interesting equilibria involving a
doubly connected uniform vortex and an internal set of pointwise ones are found
in [8], mimicing the
overlapping of shielded Rankine vortices. An irrotational region remains
trapped at the center of the vortex. In [9], a class of stationary solutions consisting in a
uniform vortex surrounded by a certain number of point corotating vortices
placed on the vertices of a regular polygon is
built and analyzed. This vortex pattern depends on an integer number ,
which specifies the number of satellite vortices, and on a parameter belonging to a lower bounded interval of the
real axis. In correspondence to the minimum value of and for a given ,
called ,
the central uniform vortex exhibits a nonanalytic boundary in which cusps are present. For growing ,
the central uniform vortex becomes smaller and smaller (its area decreasing as ), with its shape going to a circular one. In
[10], the
streamfunction (1.3) is used to generalize to finite-area vortices the ideas of
Aref and Vainchtein [11]
who search asymmetric equilibria of point vortices by inserting new vortices on
points of relative rest (in a corotating reference frame). By such an approach,
growing uniform vortices are inserted in a corotating vortex pair until the
Rankine vortex is reached. The stability of such one-parameter family of
equilibria has been also tested through numerical simulations with a contour
dynamics approach. Other equilibria depending on two parameters and involving
uniform vortices and point vortices are found in [12], still starting from the streamfunction (1.3). A central
uniform vortex is surrounded by an alternate distribution of pointwise and
uniform vortices. Also in this case, vortex shapes with cusps are found for certain
critical values of the parameters and numerical simulations show the formation
of filaments in configurations having large satellite vortices with cusps, as
well as in perturbed (by displacing point vortices) equilibria.
The present paper deals with the self-induced velocity
of any vortex, the boundary of which possesses a Schwarz function with two
simple poles (on a suitable transformed plane). This vortex shape appears to be
the simplest, but nontrivial, possible one: the velocity can be evaluated
analytically by using an integral link with the corresponding function of its boundary. For this reason, it has been
selected by the authors as the starting point of an analysis of the vortex
motion through the dynamics of ,
which satisfies the evolution equation [13]: In order to investigate nontrivial
sample cases, the analytical forms of the velocity are needed in (1.4). The
present analysis achieves the first step of such a way. At the same time, it
shows the advantage of using an integral link between and
The paper is organized as follows. The integral
relation between Schwarz function and velocity is presented in Section 2. In
Section 3, an overall view of the geometrical properties of the vortices having
Schwarz function with two simple poles is given. A classification of this kind
of vortices is proposed in Section 4, while a discussion of several sample
cases follows in Section 5. In Section 6, the inverse map (from the physical
plane to the transformed one) is built and the self-induced velocity is
analytically evaluated. The different velocity fields (together with the
corresponding streamfunctions) are shown and discussed in Section 7, where
we also propose the use of an equivalent Rankine
vortex in order to investigate the dependence of the velocity on the vortex
shape. Conclusions are offered in Section 8, together with a sketch of the
principal research lines under investigation at the present time.
2. An Integral Relation between Schwarz Function and Velocity
The velocity induced by a uniform vortex (inside which the vorticity is unitary, for
the sake of simplicity) depends only on the shape of its boundary :
if it is smooth and its length is finite, then in any point of the plane the velocity is given by where is the Green function of the Laplace operator,
that is, (the modulus of a vector is represented with
the same symbol, but without using the bold character, e.g., ), and is the curve element. The motion of such a
kind of vortex is defined by the time evolution of its boundary, starting from
a smooth and finite-length boundary at the initial time (). In order to numerically investigate that
motion, the form (2.1) of the velocity is used as briefly described below.
Consider a parameter on at time
With the vortex motion being a material one, it is
possible to write as a function of the corresponding parameter at the initial time () and of the time itself:
The Lagrangian representation of the position on the vortex boundary becomes natural and the velocity (2.1) evaluated
on that point, that is ,
gives the Lagrangian velocity
It follows that the motion of is the solution of the Cauchy problem: The above approach is known as contour dynamics [14, 15], a powerful tool to
investigate the inviscid, incompressible two-dimensional vortex motion.
Obviously, in the numerical practice a certain number of nodes are selected on
the initial boundary and their motion is followed in time. In the discrete
framework, the velocity (2.1) is evaluated through an interpolation procedure
which rebuilds an approximation of
By starting from the contour dynamics form of the
velocity (2.1), in the present section an integral relation between Schwarz
function and velocity is deduced, which is equivalent to the form (1.2) but it
does not require the splitting of the Schwarz function in the sum
The key observation about (2.1) lies in considering that if the point is not on the vortex boundary, an integration
by parts enables us to write with the dot indicating a scalar product. By
conjugating both sides of the above relation, it may be rewritten in the
following complex form: Notice that the first integral
in the right-hand side of the above equation holds if lies inside the vortex (), while it vanishes when is external to the vortex (). Equation (2.4) relates velocity and Schwarz
function in any point which does not lie on the vortex boundary,
while if the point belongs to that curve, the discussion must be
carried out in a more sophisticated way.
The tangent derivative of the conjugate of the
velocity on the vortex boundary has been investigated in a previous paper
[4], where it has
been shown that it is related to the tangent derivative of the Schwarz
function, that is, to the function ,
with being the unit vector which is tangent to
The relation is through a Cauchy integral (see [4, equation (18)], rewritten here for reader's convenience): By changing with ,
an integration by parts gives in which the second integral in
the last side holds ,
it follows that In this way, the conjugate of
the velocity in the point belonging to the patch boundary becomes
Equations (2.4) and (2.8) are summarized by introducing
the characteristic function of the domain ,
which holds inside , outside ,
and on the boundary, in the following new form of
the self-induced velocity: with the position that the
integral must be a Cauchy one if the point lies on the curve
It is worth noticing that the velocity (2.9) is a function of both and inside the vortex, while it depends on only on (through the Schwarz function) and outside the
vortex. The velocity (2.9) is a continuous function across :
indeed if approaches from the inside of the vortex, goes to and the integral to plus continuous terms (given by the
singularities of the Schwarz function), while if reaches from the outside, the first term vanishes and
only the above continuous contribution remains. On the vortex boundary, holds ,
while the integral must be considered as Cauchy's
one and it leads to a contribution plus the continuous contribution due to the
singularities of
The above form of the velocity gives also the correct asymptotic behavior: for going to infinity, being the area of the vortex. It is also
important to notice that once the splitting is inserted into the integral at the
right-hand side of (2.9), the original formulation (1.2) is recovered.
As a first sample case for the use of (2.9), consider an
elliptical vortex having center of vorticity on the origin and semiaxes along and () along (the related quantities , , ,
and will be also used below). By using the angle ,
the curve is parametrized as ,
which is rewritten in terms of as Equation (2.10) defines a
conformal map between the plane of the unit circle () and the physical one (). Due to the fact that the ellipse is a
simple curve, the equation in cannot have another solution on unless Indeed, it also possesses the solution (notice that ). The solutions and are expressed in terms of in the following way: If the point belongs to the vortex,
then
and the same holds also for ,
while for , lies outside the unit circle and
By conjugating both sides of (2.10) for ,
the Schwarz function follows naturally as the function of : The velocity is evaluated by
rewriting (2.9) in the transformed plane : If ,
then both poles and are internal to and the above equation gives while if , lies outside the unit circle and ,
so that the velocity becomes However, in the present case the
splitting of the Schwarz function (2.12) can be easily performed: which still gives the fields
(2.14) and (2.15) through the relation (1.2). As discussed above, when the functional
form of the Schwarz function is rather simple, as in (2.12), the old formulation
(1.2) is convenient with respect to the new one (2.9). This is not the case of the
class of vortices investigated in the present paper.
3. The Schwarz Function with Two Pole Singularities
The present paper deals with a uniform vortex having
Schwarz function of its boundary with two pole singularities
(it is worth noticing that the vortex having
Schwarz function of its boundary with only the pole : is a circular, with center in and radius ) on a suitable transformed plane, with the aim
to investigate the corresponding self-induced velocity field. For ,
the Schwarz function is considered. The residues , in (3.1) are assumed to be nonvanishing complex
numbers (notice that one of them, e.g. ,
can be assumed unitary without loss of generality), while the poles , are chosen outside and the origin (). It is also assumed that their conjugates
with respect to ,
that is, and ,
satisfy the two conditions and
The Schwarz function (3.1) gives the position in the point as The point given by (3.2) can move counterclockwise or
clockwise along ,
even if runs always counterclockwise on
In order to specify the direction in which moves on ,
a parameter is introduced, which holds when runs counterclockwise and in the other case. It will be evaluated in
Section 4.
Analytic continuations of the Schwarz function (3.1)
and of the map (3.2) outside will be considered in the following, so that
the definition of the inverse map outside the vortex boundary needs also to be
discussed (see Section 6). It is shown that exists almost everywhere, unless on a closed
curve ,
the inside of which has a vanishing area (see Figure 2).
Figure 2: Sketch of the invariant circle in the planes of the variables and and its image in the physical plane. The points and their images are represented with green dots. The same
colours (blue and red) are used for the arcs that are images through the maps and of the two arcs in which is divided by the points
For running on ,
the point moves on the closed curve ,
first in a direction and then in the reverse one. is composed by two superimposed arcs of the
circle (yellow line)
between the points and : its inside has a vanishing area.
3.1. Constants , , and
In the
following, the poles and the corresponding residues of the Schwarz function (3.1)
will often appear combined into the constants: moreover the related quantities: , , and will be also employed. In terms of the above
constants, for example, the compact form of a vector which is tangent to the vortex boundary on its
point : is obtained. The function (3.4)
suggests a first constraint on the Schwarz one (3.1): cannot vanish on
The nonvanishing zeros of are given by ,
with being the branch of the square root of for which (see also Appendix
D. Notice that and its modulus verifies the relation ,
with being the phase difference ). As a
consequence, in order to enforce on the unit circle, the points are hereafter assumed external to
Other important constraints will be discussed in the next section.
3.2. Map and Constraints on Schwarz Function
The Schwarz function (3.1) cannot be assigned in an
arbitrary way: for the curve is the boundary of a uniform
vortex, it must be simple. This constraint can be enforced by requiring that
for any ,
the equation in has no other solutions on ,
unless
The only solution of that equation which differs from is so that the above condition
results to be equivalent to the one: for each
Equation (3.5) implies also that the points and go on the same point :
the analytic continuation of the function (3.2) cannot be defined on the whole -plane, but only on a suitable subset of it
including (see Section 5).
It is worth noticing here that the function (3.5) maps
the points and (on which goes to the infinity) one in the other one and
viceversa: for this reason these points and will be called hereafter as “conjugate”
through the map
Moreover, goes in and viceversa is mapped in ;
the same occurs for ,
which goes in and viceversa is transformed in
More details about the map can be found in Appendix A, in particular the
“viceversa” parts of the above statements are trivial consequences of the
property (A.1). As well known and summarized also in that appendix, transforms any circle in another circle. In
the important example of the unit circle, its image is for the circle having center and radius given by the following relations: while becomes a straight line in the case of
As discussed below, the position of relative to will be one of the key-points in order to
understand the analytical structure of the velocity (see Section 4).
Coming back to the constraint on ,
the condition with leads from the definition (3.5) to the
following equation in : which has no solutions if and
only if where
If the constraint (3.8) is verified, cannot intersect
:
this fundamental property, implying that is a simple curve, depends only on the
function (3.5), or on the constants , ,
and (3.3). Hereafter, it will be assumed always
satisfied.
3.3. Invariant Circles for the Map
An interesting issue about the map (3.5) lies in searching a circle which is
transformed in itself by :
such a circle will be called invariant. Two families at one parameter () of invariant circles have been found. In the
first (and most important) one, center and radius, indicated with , in the -plane and with , in the -plane, are the following
functions of : (the branch of the square root affects the sign of ,
but not the definition of the center). Centers (3.9)
move along a straight line parallel to and passing on the point for running on the real axis. The importance of
the circles of the first family lies in the fact that one of them (called hereafter, see Figure 2) will be used in order
to define the inverse map
It divides the complex plane in two parts which are mapped one in the other by
the function (3.5), one of them being the image of the
physical plane. As shown in Figure 2, another interesting feature of is that its image ,
called ,
results from the overlapping of two equal arcs of a circle (see Appendix E for details): for running on , moves along first in a direction and then in the reverse
one. The branch of the inverse map, as well as the geometry of the two arcs on (unless their endpoints, which do not change
with ), will depend on the value of the parameter (see Appendix D).
Centers of the second family lie along the straight
line orthogonal to the one of the first family, but still passing through the
point
Center and radius of a circle are in that case the functions of : being larger than
An inspection of the formulae (3.9) and (3.10) shows that circles of the first
family intersect on the points
Notice also that is divided into two complementary arcs by the
points : if the point runs on from to in the counterclockwise sense, its image moves on the same circle and between the same
points, but in the reverse direction. Finally, it is found that the function maps the points in the endpoints of the two arcs of which form
4. Vortex Classification
A classification of the vortices having the Schwarz
function of their boundaries of the form (3.1) will be proposed below. It is
based on two important properties of the maps (3.5) and (3.2) and allows us
to build the inverse map and then the analytical velocity field.
The first property specifies the relative position of
the curve with respect to the unit circle
(hereafter, the case ,
in which the circle becomes a straight line, is
excluded): can be internal to (the vortex is classified of kind ), or external and including (kind ), or external and not including (kind ). Vortices of the first kind have the inside
of ,
say ,
naturally decomposed as joined with an annular set external to
In order to use compact notations, this annular set or the whole (for vortices of kind or ) will be hereafter indicated with
In turn, vortices of the second kind have the inside of naturally decomposed as joined with an annular subset external to ,
that will be indicated with
For vortices of kind ,
the same symbol will indicate the unbounded region external to both circles and
Appendix B is devoted to a comprehensive discussion of the first property, the
results of which are summarized in Table 1. The above classification enables us
to specify where goes through the function : it is found (see Appendix C for details) that is mapped onto itself for vortices of kind ,
while it goes onto the outside or the inside of for vortices of kind or ,
respectively.
Table 1: Behaviour with respect to the first property,
consequences on the position of the circle
and corresponding constraints on
,
and
The quantities
and
are defined in (
B.1).
A second property specifies where goes via the other map : it can go onto the inside of the vortex (which is classified of kind ) or onto its outside (kind ), depending on the values of the constants , and on the positions of the poles and with respect to the unit circle. This issue is
investigated by evaluating if the image of the origin in the -plane (which is still the origin in the
physical one) lies or not inside
To this aim, the logarithmic index of the point with respect to the curve is considered. The residues
(times ) of the integral (4.1) on the poles and (if they lie in ) hold ,
while those on and (if they lie in ) hold (notice that both pairs of points are
conjugate through the map ). Results of the discussion of the
logarithmic index (4.1) are summarized in Table 2.
Table 2: Behaviour with respect to the second property,
consequences on the map and corresponding constraints on the poles ,
In our classification, the vortices belonging to a
given class are of the same kind with respect to
both properties: a class is identified by the couple of numbers which indicate
the kinds with respect to the first and second properties. As an example,
vortices in the class have the circle external and not including and is the outside of the vortex.
The last issue about the map concerns the orientation of the vortex
boundary, that is, the evaluation of
To this aim, the logarithmic index of the point with respect to has to be evaluated: it holds when is oriented counterclockwise and in the other case. In the last integral, the
residue (multiplied times ) on the point holds ,
the ones on and (if they lie in ) give and the residue on (if it lies inside ) holds
By accounting for the behaviour with respect to the second property summarized
in Table 2, it can be easily shown that any vortex of kind has a counterclockwise () oriented boundary, while the orientation of
the boundary of any vortex of kind is clockwise ().
5. Geometrical Discussion
In the present section, the information about the maps and given in Sections 3 and 4 are joined to give a
comprehensive picture of their geometrical properties. In order to reach an
intuitive representation of the behaviour of these functions, in Figures 3, 4,
and 5 families of circles will be transformed through these two maps. Hence,
the circles and their images are drawn with the same colour.
Figure 3: Planes of the
variables: (first column), (second), and (third) for vortices of kind with respect to the first property. In the
first row, a vortex of kind (, , , ) and in the second one a vortex of kind (, , , ) are investigated. Two families of circles
are considered in the -plane: one inside (yellow lines) and the other one outside (blue lines). Image curves (that are still
circles in the -plane) are drawn with the corresponding
colours. Circles , (a), (b), (d), and (e) are drawn with red
lines, as well as the vortex boundaries in (c) and (f). Invariant circles are drawn with green lines in (a), (b), (c),
and (d). The same line is used to draw in (c) and (f). The centers of and (squared filled symbols), the poles and (solid black circular symbols) and the
conjugate points (empty triangular symbols) and are also drawn.
Figure 4: As in Figure
3,
but for vortices of kind
(first row, with
,
,
,
and
) and of kind
(second row, with
,
,
,
and
).
Figure 5: As in Figure
3,
but for vortices of kind
(first row, with
,
,
,
and
) and of kind
(second row, with
,
,
,
and
).
Our analysis starts from vortices of kind with respect to the first property, having
For such vortices, the function maps onto itself, while it goes onto (e.g., , outside, see Table 2) or onto (, outside ,
see Table 2) through the map
In Figure 3, samples of the planes (Figures 3(a) and 3(d)), (Figures 3(b) and 3(e)), and (Figures 3(c) and 3(f)) for vortices of kind (first row) and (second row) are shown.
In order to determine a branch of the inverse function ,
a neighbourhood of in the -plane must be defined, such that its image
through the map is just its complementary set. This goal is
reached by defining two subsets and of which go one in the other one through that
map: they are necessarily separated by an invariant circle of the first family ,
so that and lie outside and inside ,
respectively. The value of is chosen in the following way. As shown in
Appendix D, two circles of the first family, corresponding to the values and (D.4) of the parameter, are tangent to and then to ,
at the same time. It follows that every invariant circle for lies inside and then is a good candidate for our aims. In
Figure 3, the parameter is chosen as the mean value
Once the subsets and have been introduced, by joining with the outside of a neighbourhood of the unit circle is
obtained, on which the behaviour of the maps and is investigated. To this aim, in Figures 3(a)
and 3(d) two sets of circles are considered, one outside and the other one inside
Their images through the map are shown in Figures 3(b) and 3(e),
respectively. Circles inside go inside ,
while circles outside are transformed in circles lying inside
The behaviour of the map is then shown in Figures 3(c) and 3(f): in the
first row, circles in are mapped in curves lying inside the vortex,
while circles outside are transformed in curves external to the
vortex. The contrary occurs in the second row. In other words, the branch of
the inverse map corresponding to the present choice of the
parameter is such that the image of is (first row) or the outside of (second row), while the image of is the outside of (first row) or (second row).
In Figure 4, the above analysis is extended to
vortices of kind (first row) and (second row): as discussed below, the main
difference between this case and the other ones lies in the boundness of the
image of the physical plane through the inverse map
Being the vortex of kind with respect to the first property, the image
circle includes
In the first row, both poles belong to (see Table 2), so that and ,
while in the second row and lies outside (see Table 2), so that and is external to
The set is mapped by the function onto the outside of the image circle ,
while the annular region is transformed onto itself by the same
function. As above, two complementary subsets and of (one going in the other one and viceversa
through the map ) are identified as the parts of inside () or outside () an invariant circle of the first family
belonging to (its is still chosen as the mean value between the
ones corresponding to the tangent circles). In order to explain the behaviours
of the maps and
(Figures 3(a) and
3(d)), two sets of circles have been considered in Figures 4(a)
and 4(d): one inside (yellow dashed
lines) and the other one inside (blue). The image circles through the map are drawn in Figures 4(b) and 4(e). Finally,
in Figures 4(c) and 4(f) the image curves through the function are shown: the inside of the unit circle is
mapped in the inside (first row) or in the outside (second) of the vortex and goes onto the complementary set. A
neighbourhood of the curve is determined, with its image through the map filling the whole physical plane.
Vortices of kind (first row) and (second row) are investigated in Figure 5:
they have image circles external and not including
In Figure 5(a), the poles and belong to the inside of the unit circle so
that and ,
while in Figure 5(b) and lies outside the unit circle: as a
consequence, and
Also in the present case, is divided in two subsets and through the introduction of a suitable
invariant circle: the choice of its parameter is not obvious, but comprehensive details can
be found in Appendix D. The subset is defined as the outside of the unit circle
which lies inside (if ,
as in Figure 5) or outside it (if ). With the neighbourhood of being the image of the physical plane through
the inverse map ,
two sets of circles: one inside (blue dashed lines) and the other one in (yellow), are introduced in order to give a
picture of the behaviours of the maps (Figures 5(b) and 5(e)) and (Figures 5(c) and 5(f)). As discussed before, goes onto and onto through the function
Moreover, the function maps the inside of the unit circle onto the
inside or the outside of the vortex of kind or ,
respectively.
We are now ready to use the analysis of the maps and to obtain the analytical form of the velocity
field induced by the vortex.
6. Inverse Map and General Form of the Velocity Field
In the present section, the inverse map is explicitly built and then the velocity (2.9)
is analytically evaluated. Before calculating the inverse function, the
following quantities: have to be introduced for the
sake of simplicity of the algebraic calculations. They enable us to write the
quantity without square roots. Indeed, ,
so that the roots (6.1) can be chosen to give
The inverse map is then built by solving the second degree
algebraic equation in
From the definition (3.2) of the map one obtains in which the zeros of the discriminant have the following
form: The branches of the square roots
in the inverse (6.2) must be carefully chosen through the considerations about
the map that are discussed below. However, from the
inverse point of view, it is only important to know if the branches are equal
(i.e., both roots use or both ) or different: an integer is introduced, which holds in the first case and in the second one. It will be found that changes in passing from the inside to the
outside of unbounded regions as the ones in Figure 6, obtained by joining the
arc of the circle superimposed on with two half straight lines parallel to the
real axis and departing from the points
Figure 6: Qualitative
behaviour of the phase difference
Notice that if the point moves counterclockwise on to reach ,
then
The angle holds identically (a) or (b) on (bold line).
The index on a given point is calculated by performing the following two
steps. In the first one, the parameter is related to the phase difference between the vectors and for (see Figure 6). This particular phase
difference will be indicated with and does not depend on the position on : as shown in Figure 6, if is the nonoriented and positive angle (in turn, can be related to via the function (E.14)) between the vectors and in a fixed , is identically given by in Figure 6(a) and
by in Figure 6(b) for
any
The second step consists in relating the phase difference in the given to the corresponding value of the index ,
by replacing with the proper function of (6.5).
The starting point is the equation which is obtained by enforcing
that for
By introducing the arguments , of the vectors , and the angle ,
it leads to the relation
The second step consists in evaluating the index in the given point
It is achieved by reconsidering the -planes in Figures 3, 4, 5 and the
classification with respect to the first property. Indeed, the inverse map for
vortices of kind
(or with ) satisfies the relation
On the contrary, the same map for vortices of kind (or with ) verifies the opposite condition
By introducing the function ,
the conditions together with the relation (6.5) lead to the
inequalities for ,
or to the opposite ones for
On the basis of the relations (6.6), is defined by the sign of ,
which in turn follows from the behaviour of the phase difference inside and outside
In the case of Figure 6(a), belongs to intervals inside and outside : the function is positive inside and negative outside. The
contrary occurs in the case of Figure 6(b): belongs to the intervals inside and outside : is negative inside and positive outside. The
present discussion can be extended to cover the more general case in which the
half straight lines departing from have other intersections with the curve
Results of such analysis are summarized in Table 3, in terms of the values of
Table 3: Branches to be used for the square roots and in any ,
expressed through the index ( equal, different). They are specified in terms of the
kind of the vortex with respect to the first property (first column) and of the
phase difference (second column). Here it has been assumed that ,
while if the complementary value has to be considered.
Once the index has been calculated, the final issue about the
inverse map (6.2) concerns it asymptotic behaviour, which
can be written in the following form: with being the index measured inside
For the sake of simplicity and without any loss of generality, the poles and and the corresponding residues and in the definition (3.1) of the Schwarz function
are hereafter renamed in such a way that for
The velocity in follows from the general form (2.9), via the
change of variables from to : with being .
Fixing a point as different from
or ,
the velocity contribution of the corresponding inverse point ,
if it lies inside ,
follows as which is evaluated by inserting
the inverse value (6.2) inside the definition (3.1) of the Schwarz
function .
On the contrary, if lies inside ,
an analogous contribution is produced with replacing .
Moreover, the velocity contribution of a pole (, ), if it lies inside ,
has the form of a point vortex/source singularity: where the (complex) intensity of
the point vortex/source is On the point (, ), that is, or ,
the pole becomes of multiplicity two. In both cases,
the contribution to the velocity () is as follows: Finally, the contribution of the
point (, ), if it lies inside ,
is constant: On the point ,
which corresponds to and ,
the following form of the velocity is obtained: Also in this case the
contribution of a pole (, ) assumes the previous form (6.13), while the
one of the pole is ,
according to the formula (6.9) in .
In the next section, the velocity is explicitly built for each kind of vortices
and comparisons with suitable simple vortices are carried out in order to
explain the dependence of velocity and streamfunction on the vortex shape.
7. Analysis of the Velocity Field
In the present section, the velocity and
streamfunction fields for the six classes of vortices introduced before (see in
particular Section 5) are built, by accounting for the different contributions
evaluated in the previous section and the orientation of the vortex boundary
established in Section 4. All the fields, with the only exception of the
external velocity field for the vortices of kind ,
involve the inverse map which has been analyzed in the previous
section. It is also found that vortices of kinds , have the same analytical structure of the
velocity of the ones of kinds , ,
respectively.
7.1. Vortices of Kind
Our analysis
starts from vortices of kind ,
which have the pole inside ,
while lies outside: as a consequence lies outside and inside .
The velocity in a point of the physical plane is the following one: the behaviour of which in
neighbourhoods of the point ,
of the closed curves and ,
as well as at infinity must be discussed. First of all, it is worth noticing
that the field is regular in a neighbourhood of : the singular behaviour of for is corrected by when (), or by when () or ( external to the vortex); see (6.10). The
continuity across the vortex boundary is enforced by the definition of the
Schwarz function: it implies that for going to any point on from the inside of the vortex, while that term
vanishes outside ,
due to the presence of the characteristic function of the vortex .
Another issue concerns the continuity of the velocity
(7.1) across the closed curve ,
which lies entirely inside the vortex in the present case. Indeed, the Schwarz
function evaluated in or in gives rise to a composite function of which is discontinuous across that curve, due
to an analogous behaviour of the inverse function .
As discussed in Sections 3 and 6, a point on (different from the endpoints and ) must be considered as the superimposition of
two points (say and ) which come through the map from a point and from its image .
With these points on being different, the limit value
of in a point from one side of that curve () or from the other side () is also different. The continuity of the
velocity (7.1) is proved by noticing that the sum of the Schwarz functions
calculated on both points ( in the above discussion) and (that is ()) is in fact present into the velocity for
any and that sum results to be continuous across .
Finally, the field (7.1) has the right behaviour at
infinity. Indeed, the inverse function goes to (and then ) when ,
as a consequence: where the coefficients for have been introduced (notice that and are real and positive, so that and ,
moreover ). By inserting the above asymptotic expansion
inside the one for the velocity (7.1), one obtains in which it is worth noticing
that the area of the vortex is given in the present case by the following
relation: Once the above value of the area
is accounted for, the asymptotic expansion (7.3) becomes the expected one: .
On the basis of the velocity (7.1), the streamfunction
which behaves as for is now evaluated. To this aim, the velocity is
rewritten in the following form: in which only the inverse
function appears. Notice that the right-hand side of
the above equation is an analytic function of in its domain of definition (being ,
it is an analytic function of ,
too) called hereafter. As a consequence, a complex
potential (the imaginary part of which is just the
streamfunction ) exists and is defined by the differential
relation in which apices indicate
derivatives in .
In this way, the complex potential is calculated via an integration of the
equation ,
which gives in any point external to the vortex (dependences on are omitted, for the sake of simplicity) the
streamfunction: The points , ,
and lie inside the unit circle, so that the
branches which are discontinuous on the segments joining with and with must be chosen for the first and the second
complex logarithms, respectively. Moreover, by taking into account the
asymptotic expansion of the inverse function (6.7), one can observe that the modulus behaves as for growing : as a consequence, (7.7) has the right behaviour, that is, ,
for going to the infinity. In any point belonging to the vortex the streamfunction has
the form with the constant being
evaluated by calculating in a point of the vortex boundary the external
streamfunction (7.7). In (7.8), the points , are external to ,
while the ones , are internal to .
This fact enables us to adopt the branch which is discontinuous on the straight
segment joining with for the logarithm ,
while the branch to be used for depends on the positions of the points , with respect to : if the segment joining with intersects ,
the branch which is continuous on that segment is used. On the contrary, the
branch which is discontinuous on the same segment must be employed.
The streamlines for the vortex of kind in Figure 3(c) are drawn in Figure 7(a). From
that figure, it appears rather hard to investigate the dependence of (and then of ) on the shape of the vortex, mainly for the
presence of the prevailing isotropic component of such field, responsible for
the isotropic behaviour at infinity. In order to overcome this
intrinsic difficulty, a Rankine vortex of unitary vorticity with center on the
center of vorticity of and area is considered: the induced streamfunction and
velocity are indicated with and ,
respectively. In Figures 7(b) and 7(c), the circular boundary of such an equivalent vortex is drawn with green dashed lines. Now, the
differences , are much more meaningful than the fields , ,
with the dependence on the vortex shape of such quantities being easily
identified, due to the linear behaviour with respect to the vorticity of
Biot-Savàrt's law. The streamfunction ,
as well as the differential velocity ,
can be interpreted as generated by a vortex having only two levels of
vorticity: in the parts of that are external to Rankine's vortex () and in the regions of the equivalent vortex
outside (), while it vanishes in the central region of
the vortex () and outside .
For the present vortex, the isolines of as well as the vectors on a finite set of points (Figures 7(b) and
7(c)) exhibit a four-lobed structure of both fields, more evident in the
streamfunction rather than in the velocity. Two intense regions are located
near the tip of the vortex, generated by two adjacent zones having opposite
signs of the vorticity. As a result, strong ingoing velocities are produced
near the tip. From Figure 7(c), it can be also seen that the velocity vanishes in a rapid way at infinity ( for ).
Figure 7: (a), (d) The
streamlines (blue lines, (a)
and (d)
) are drawn for the vortices in Figures
3(c)
and
3(f), respectively. (b), (c), (e), (f) The boundaries of the vortices are
drawn with thick red lines, while the ones of the equivalent Rankine vortices
with green dashed lines. The differential streamlines and velocities (with a
scale factor of
) are shown in (b) (
), (e) (
), and (c), (f), respectively.
The above kinematical analysis enables us to
investigate the onset of the filamentation of the vortex boundary in the region
near the tip. According to Pullin [16], the filamentation consists in the ejection of thin
streams of vorticity which “may form patterns of ever increasing complexity
and apparently ever decreasing minimum spatial scale as they are convected and
strained by the irrotational flow outside the vortex.” A heuristic explanation
of the filamentation is based on the presence in a neighbourhood of the vortex
boundary of hyperbolic critical points of the velocity which “rapidly distort
the vorticity and convect filaments away from the vortex core.” In the present
case, the evaluation of the relative streamfunction in a corotating frame,
obtained by using the diagnostic ellipse approach [17] and the analytical form of
the streamfunction (7.7), (7.8) shows the presence of one hyperbolic critical
point near the tip: as a consequence, the vortex has high probabilities to
develop filaments in that region. Even if different paths to filamentation have
been proposed, the analysis of the time behaviour of the Schwarz function in a
neighbourhood of the tip could offer other interesting interpretations.
7.2. Vortices of Kind
A vortex of
kind has both poles ( and ) in the Schwarz function (3.1) outside ,
so that and .
The velocity is evaluated as the behaviour of which in
neighbourhoods of the curves and ,
as well as at infinity, must be investigated. The velocity (7.9) is continuous
across due to the definition of the Schwarz function
of that curve .
Indeed, if from the inside of the vortex, the sum of the
first two terms (the other ones are continuous across ) gives and the same limit value is obtained from the
outside. By accounting for the property ,
the above limit agrees also with the value assumed by the sum on the point .
About the continuity across ,
the same explanation which has been given for the vortices of kind holds also in the present case, with the only
difference that lies outside the vortex. The behaviour for is evaluated by considering that and and by using the asymptotic expansion: together with the above one
(7.2). It follows that ,
with the area of the vortex assuming in the present case the
form once the clockwise orientation
of the vortex boundary has been accounted for.
The streamfunction in any point outside the vortex follows from the velocity
field (7.9) as where the branches of two
complex logarithms must be chosen. The first one is : it is evaluated by using the branch which is discontinuous on the segment joining the point with ,
lying that segment in .
In order to evaluate the second logarithm, that is ,
two cases must be distinguished: if the segment joining the point with intersects the unit circle, the branch which
is continuous on that segment must be used and viceversa, if the above segment
does not intersect ,
the branch which is discontinuous on the same segment must be adopted. The
streamfunction in a point is in which the above
constant is
calculated by evaluating (7.12) in a point .
The complex logarithm is evaluated by considering that the points , are internal to ,
while and, as a consequence, .
It follows that the branches of the above logarithms to be used are
discontinuous along the segments joining with and joining with .
The isolines of the streamfunction (7.12), (7.13) for the
vortex of Figure 3(f) are drawn in Figure 7(d), together with the boundary of
the vortex itself (red thick line). A certain anisotropy of such field appears
from the figure, but in order to quantify such behaviour it is more convenient
to analyze the differential streamfunction ,
and the isolines are drawn in Figure 7(e). Four regions in which the differential
vorticity holds or can be identified, while it vanishes in the
region and outside .
As a consequence, two regions of the vortex boundary experience intense inward
velocities, while in other two regions act strong outward velocities (see
Figure 7(f)).
7.3. Vortices of Kind
Vortices of
kind have both poles inside : as a consequence, and .
The velocity is given by in which the singularities in
the points and are only apparent, due to the form (6.10) of the
term .
About the form of the field (7.14) outside the vortex it is worth observing that
it coincides with the one due to a couple of vortices/sources placed on the
points and .
With the area
of the vortex being given by (7.11), the
relation follows: the total source intensity vanishes (due to the incompressible nature of
the fluid).
The streamfunction in any point external to the vortex has the form The first complex logarithm,
that is, ,
is evaluated by using the branch which is discontinuous on the segment joining with (internal to the unit circle), while for the
second logarithm, that is, ,
the branch is chosen by considering the relative positions of the points , with respect to the invariant circle .
For example, if the segment joining the point with intersects ,
the branch which is continuous on that segment must be used. Inside the vortex
the streamfunction holds: where the constant is evaluated
as above. The complex logarithms and are evaluated by using suitable branches which
depend on the relative positions of the points , and , with respect to the unit circle. For example,
if the segment joining with intersects ,
the branch which is continuous on that segment is used.
The isolines of the streamfunction (7.15), (7.16) are
drawn in Figure 8(a): they exhibit a high level of isotropy, at least
qualitatively. This is due to the small amount of circulation with respect to
the total one which characterizes the vortex tip and it still suggests the analysis
of the above differential fields. Indeed, the differential streamfunction ,
the isolines of which being drawn in Figure 8(b), enables a more quantitative
analysis: an intense region generated by a differential vorticity level appears in correspondence with the tip on the
vortex boundary, while other regions are much less intense and then they can be
discarded in a first analysis. As a consequence, intense outward and inward
velocities appear before and after the tip (see Figure 8(c)), while the
remaining vortex boundary appears to be quasistationary.
Figure 8: As in Figure
7,
but for the vortices of Figures
4(c) and
4(f). The steps between two
streamlines are (a)
,
(c)
,
and (b), (d)
,
while the scale factor in drawing the differential velocities
is
.
7.4. Vortices of Kind
Vortices of
kind have external to the unit circle and ,
as a consequence and lies outside the image circle .
The velocity assumes the following form: the behaviour of which in
neighbourhoods of the point and of the vortex boundary needs to be discussed. The apparent singularity lies outside the vortex and, as before, it is
compensated by the term .
Moreover, if from the inside of the vortex, the sum of the
first two terms goes to ,
while if from the outside, the term with the Schwarz
function goes to the same limit and the first one vanishes. That limit agrees
also with the value of the sum on the vortex boundary due to the property of
the characteristic functions: for .
The behaviour of the velocity (7.17) at infinity is the right one: and it follows from the asymptotic expansion
(7.10) of ,
by accounting for that the area of the vortex is still given by (7.4).
The streamfunction in any point external to the vortex
is given by The branches used for the
complex logarithms and depend on the positions of the points , and , with respect to the unit circle. For example,
if the straight segment joining with intersects ,
then the branch which is continuous on that segment is used. In any point the streamfunction is given by the following
formula: with the constant being
evaluated as described before. The first complex logarithm, that is, ,
is calculated by using the branch which is discontinuous along the segment
joining with (internal to the unit circle), while the
branch employed for the second logarithm, that is, ,
depends on the positions of the points , with respect to the invariant circle .
The isolines of the streamfunction (7.18), (7.19) are
drawn in Figure 8(d): a certain anisotropy appears that becomes strongly
evident in the differential streamlines of Figure 8(e). Also in this case a
four-lobed structure of the field appears with a very intense region generated
by a negative level of vorticity near the center of vorticity. The central
region is predominant on the other ones: indeed, if this picture is reread in
velocity (Figure 8(f)), a strong differential clockwise velocity is found near
the center of vorticity. Two less intense regions generated by counterclockwise
differential vorticity are identified on both tips, leading to a very
complicated distribution of normal velocities on the vortex boundary. The
analysis of a relative streamfunction in a suitable rotating frame shows the
presence of two hyperbolic critical points near both tips of the vortex, just
outside the vortex. As it is shown by Figure 8(f), the normal differential
velocities are oriented inward on the left tip and outward on the right one, so
that filamentation could occur in the region near the right tip only.
7.5. Vortices of Kinds and
Vortices of
kind have both poles and inside ,
so that and .
The velocity assumes a form identical to the one of (7.14), which holds for
vortices of kind .
Finally, for vortices of kind the pole lies outside ,
while is internal to the same curve. As a
consequence and and the form assumed by the velocity is
identical to the one of (7.17) which holds for vortices of kind .
The streamfunction induced by the vortex in Figure 5(c) is shown in Figure 9(a), where its isolines are drawn, together with the
vortex boundary (red thick line). The streamfunction appears to be
quasi-isotropic, due to the small amount of circulation, with respect to the
circulation of the vortex itself, which is contained into the vortex tip. This
suggest to investigate the differential streamfunction, the isolines of which
are drawn in Figure 9(b), together with the boundaries of the vortex (red thick
line) and of the equivalent Rankine vortex (green dashed line). From that
figure, the role of the tip is clarified: it induces an intense differential
streamfunction and corresponding counterclockwise velocities (see Figure 9(c)),
while two regions before and after the tip are characterized by clockwise
velocities. As a consequence, strong outward and inward normal velocities are
experienced by the vortex boundary before and after the tip.
Figure 9: As in Figure
7,
but for the vortices in Figures
5(c) and
5(f). The jumps between two
consecutive streamlines are (a)
,
(b)
,
(d)
,
and (e)
.
The scale factors in drawing the differential velocities are (c)
and (f)
.
The streamfunction for the sample of vortex of kind which is shown in Figure 5(f) is investigated
in Figure 9(d). The isolines are quasi-isotropic, while the ones of the
differential streamfunction (see Figure 9(e)) exhibit a strong region near
the vortex tip, which generate intense clockwise velocities (Figure 9(f)). As a
consequence, strong inward normal velocities appear before the tip, while
outward velocities are
generated just on the tip. The other part of the vortex boundary appears to be
quasistationary, unless small tangential velocities in
counterclockwise direction on the right and on the
left of the tip, while clockwise velocities are experienced by
the region of the boundary which is opposite to the tip.
8. Concluding Remarks and Perspectives
The present paper consists in a first application of
the relation (2.9) to the study of the dependence of the self-induced velocity on
the shape of the vortex. Although this shape is
described in terms of only three complex parameters (, and ), the corresponding family of vortices
appears to be very rich and of great interest. In particular, vortices of
shapes very far from the circular one can be obtained, the self-induced
streamfunction and velocity of which being analytically calculated, without any
approximation.
A classification of all the vortices having the
Schwarz function with two simple poles is proposed, by stressing the different
ways in which the inverse map is built and their consequences on the
self-induced streamfunction and velocity fields. Six kinds of vortices have
been identified, on the basis of the relative positions of the circles and and of the global behaviour of the map .
The velocity in a point is found to be a linear combination of Schwarz
functions evaluated in and in and of pointwise source/vortex singularities,
together with the contribution of the solid body rotation inside the vortex. Depending on the kind of
the vortex, as well as on the fact that lies inside or outside the vortex, one or more
of the above terms disappear. Vortices having only Schwarz function contributes
are found (kind ,
see Section 7.2), as well as vortices the external velocity of which is due
only to two pointwise source/vortex singularities (kind ,
see Section 7.3). Tools to build inverse map ,
streamfunction, and velocity for the vortices here analyzed are available on
the website http://www.meccanicadeifluidi.it/. Numerical tools (using the contour
dynamics approach) for a comparison with the present analytical results may be
also found.
The qualitative understanding of the analytical
streamlines and velocities is strongly simplified by introducing the
differential fields , which assume the equivalent Rankine vortex as the reference one. These fields are viewed
as induced by the regions and
where the vorticity holds and ,
respectively. The remaining part of the plane does not contribute. In a lot of
vortices, a qualitative inspection to the differential fields enables us to
estimate normal velocities on the vortex boundaries, which is the first step to
understand their small-time dynamics.
As briefly discussed in Section 1, the final aim of
such an analysis lies in representing the vortex motion by following the time
evolution of the corresponding Schwarz function. At the present time, the forms
(7.1), (7.9), (7.14), and (7.17) of the velocity have been used in the right-hand
side of (1.4), which becomes a sum of rational functions of .
Several important issues can be now addressed. First of all, does a vortex
among the ones here analyzed that retains in time the structure of its Schwarz
function exist? In other words, does the left-hand side of (1.4) have the same
algebraic structure of the right one? If this is the case, the
evolution equations for poles and residues can be easily deduced. Moreover, why the two sides have different poles?
Were new singularities born at the initial time? and so on. In the opinion of
the authors, some of the above questions could find their answers in the near
future.
Appendices
A. Some Properties of the Map
The function (3.5), known in literature as Möbius map,
plays a crucial role in the present analysis.
Fixing an arbitrary point in the physical plane, this
point maps the corresponding inverse point in a new one, that is ,
on which the function (3.2) still assumes the value .
The most important property is that the inverse of the
map coincides with the map itself: so that the function is a self-inverse one. A picture of the global
behaviour of the map is obtained by rewriting it in the more
general form: (here and ). As it is well known, the form (A.2) is
relevant, showing that maps a circle of radius and center on in another circle when the function (A.2)
remains bounded, that is, does not belong on .
First of all, it is proved that a redefinition of the
coefficients and enables us to reduce the mapping of any circle
to the one of the unit circle .
Indeed, consider the circle having center on the point and radius ,
a point of which is written as for .
It follows that the function can be rewritten as ,
that is, in the form (A.2), by changing its coefficients and in and ,
respectively. For this reason, it is sufficient to prove that the function (A.2)
transforms the unit circle in another one, for .
This property is verified through the search for a point (the center of the transformed circle) and a
positive constant (its radius) such that the following equation
(where the constant stays for ): holds identically for any .
In order to reach the independence of of the third side, or must be assumed. But the first choice implies ,
so that only the second one is really possible, leading to the following
specification of the center and of the radius of the transformed circle :
In order to specify the above relation for the
function (3.5), its values on the origin () and at infinity () are introduced, the coefficients and are changed in and ,
and the relations between , ,
and and the constants , ,
and (3.3) are used. A circle of radius and center on the origin of the -plane is mapped by the function (3.5) in another circle having center on the point: the second side of which shows
that this center moves from to the infinity along the direction for growing from to the critical value .
From the third side it follows that the center moves from the infinity to the
point along the direction when runs from to .
this way, the center results to be external to the line joining the points and for any .
The radius of the image circle is given by which results to be greater than the distance from the center
and the point for and from the center and the point for : all the circles cut the segment joining with .
Moreover, two circles and do not intersect
if .
In what curve the function (A.2) maps the unit circle
for ?
By assuming and in the definition (A.2) one obtains from which it appears that the
circle is transformed in a straight line. By introducing the constants , ,
and (3.3) inside , ,
and ,
the above relation implies that the circle having the critical radius is transformed by the function (3.5) in the
straight line: The line (A.8) is orthogonal to
the segment joining with and it cuts that segment in its middle point .
B. Position of the Circle with Respect to
In the present section, the relative position of the
two circles and is investigated, by showing that it depends on
the constants , and and on the phase difference .
For the sake of clearness, some issues about the position of the center and of the radius (3.6) of the transformed circle need to be preliminarily discussed. An
important information about concerns if it lies () or not () in .
By accounting for that ,
the modulus holds for or for with In terms of the above
quantities, results to be larger than for and smaller for or .
About ,
it is larger than for and smaller for , .
Our analysis of the relative positions between the two
circles and starts by searching the
conditions in which
lies inside .
As stated above, it is needed that or ,
so that .
In this hypothesis on ,
the inequality leads to the relation that is satisfied via the
constraint (3.8) for , or for ,
but it can be shown that the second case cannot be verified. It follows that when , and : the vortex is said to be of kind .
Instead, the image circle is external and not including the unit circle if , and or if , and ,
the condition leading also in this case to the inequality
(B.2) and implying .
However, it can be shown that the first case (, ) cannot be verified. It follows that is external and not including for , and : in this case the vortex is said to be of kind .
Finally, the conditions to have the image circle including the unit one are deduced by solving the inequality ,
which can be rewritten as It is satisfied for by using the condition (3.8). The same
condition inhibits the other possible solution ,
because and cannot be both smaller than .
It follows that is external and including for : in this case the vortex is said to be of kind .
C. Global Behaviour of the Map
As summarized in Appendix A, the global behaviour of
the map is easily understood by considering its action
on a circle with center on the origin and growing radius
and by accounting for that the image of such a circle is still a circle with
center in (A.5) and radius (A.6). For growing from to the critical value , moves along the straight line passing through
the points and , say, in direction from () to the infinity (). At the same time, grows from () to infinity (). Indeed, the circle is mapped by the function in a straight line which is orthogonal to and cuts the segment joining with in its middle point. When grows above ,
the curve becomes still a circle: moves on always in direction from the infinity () to (), while decreases from the infinity () to (). The behaviour of the map is now specified on the basis of the
classification with respect to the first property.
A vortex of kind has (see Table 1) and its corresponding set does not fill the whole .
The image of each circle with goes outside through the map ,
even if its position is still not known with
respect to .
It can be deduced as follows. With the circle lying inside the unitary one, select an
arbitrary radius (in case, it approaches the unity from below)
such that .
The image of through the map cannot lie outside ,
because for it goes on .
Moreover, it cannot intersect
(or ). Indeed, suppose that two intersection
points exist. For the map
is self-inverse
(see Appendix A) and ,
it means that these points come from two points of the -plane lying on (or ), which is absurd. As a consequence, the
image of the circle still lies in .
It follows that is mapped onto itself, while goes onto the outside of .
For this reason, a circle with and that intersects
is mapped in a circle which has an arc inside ,
the complementary one outside ,
and does not intersect
.
The arc inside is the image of the arc of that lies outside ,
while the arc lying outside is the image of the arc inside .
Consider now a vortex of kind ,
for which (see Table 1) .
The set goes in that case onto the outside of the
circle .
Indeed, the other possibility () cannot occur, because for the image of through the map does not remain bounded. As a consequence, is mapped onto itself. Finally, a vortex of
kind has (see Table 1): its corresponding set goes onto the inside of ,
for the same reason about the behaviour of in a neighbourhood of the critical radius.
Also in this case, is mapped onto itself.
D. Invariant Circle Tangent to and
In the present appendix, the values of the parameter for invariant circles (3.9) tangent to both and are calculated: they are needed in order to
define a branch of the inverse map .
D.1. Invariant Circle Tangent to
Consider an
invariant circle which is also tangent to .
That invariant circle can lie inside () or outside it, in this latter case can include () or not () the unit circle. At this stage, one does not
know which case among the above ones holds. The position of its center and its radius satisfy one of the following relations: Due to the form (3.9) of the
functions and ,
it is convenient to evaluate the quantity and then to calculate the square .
In this way, by selecting the branch of (square root of ) such that ,
one obtains with being the positive root of .
The definition of center and radius (3.9) of the invariant circle, together with
the relation (D.2) are inserted in (D.1), so that the following equation in the
parameter is obtained: where the upper sign holds in
the cases and and the lower one when is external and not including (). The above equation has the following two
real solutions: which are called and (with ): as a consequence must be equal to or to .
Moreover, the sign of the quantity (D.3) specifies if the circle is in the
conditions , (it is positive) or (negative) and a calculation of (if needed) discriminates between the cases () or ().
D.2. Invariant Circle Tangent to
Consider now
another invariant circle which is also tangent to .
The same relations (D.1) hold, with in place of in the left-hand side and in place of in the right one. As before, the first row
holds for inside (sign ,
condition ) or outside (sign ,
condition ), while the second row holds for the
invariant circle external and not including .
By using the forms (3.9), (3.6) of the position of the center and of the radius of
the invariant circle ( and ) and of ( and ), the following equation in : is obtained. As before, the
upper sign holds in the conditions and ,
while the lower one must be selected in the condition .
Notice that (D.5) is identical to the one (D.3) for vortices of kind or with respect to the first property (), while its right-hand side has opposite sign
for vortices of kind ().
D.3. Position of with Respect to and
At this stage,
the following key remark is needed: the circle is the image through the map of the unit one ,
so that if an invariant circle is tangent to ,
it must be also tangent to .
For this reason, some of the previous possibilities (labeled with , ,
and ) must be ruled out, once the relative
position of with respect to is known. The invariant circle lies inside and includes for vortices of kind with respect to the first property: the right-hand
side of (D.3) is positive. The same holds for vortices of kind : lies inside and includes .
For vortices of kind ,
the signs in the right-hand sides of (D.3), (D.5) are necessarily different,
being .
In this way, if the invariant circle includes ,
it is also external to ,
or, on the contrary, if it includes ,
it lies also outside : the tangent cannot include both circles and or lie outside them, at the same time.
Another way to understand this behaviour is based on
the discussion of (D.3), rewritten for the sake of shortness as with , and .
It defines the intersection points between the straight line and the two curves in a plane ,
as shown in Figure 10. Two conditions must be considered, as discussed below.
If solutions exist only for : they stay both above (below) the axis ,
so that only the positive (negative) sign holds in the right-hand side of (D.3).
On the contrary, if the intersection points always exist: one lies
on the curve and the other one on (for and the values are obtained). Vortices of kinds and with respect to the first property satisfy
this condition, having and .
Moreover, from the latter condition, it follows that the right-hand side of
(D.3) must be taken positive, as shown also above. Vortices of kind can verify the first () or the second () of the above conditions, depending on the
positions of the poles and and on the ratio in the Schwarz function (3.1).
Figure 10: Functions
(red lines), their asymptotes
(yellow), and the straight line
(red) versus
,
for (a)
,
and (b)
.
In (a) also the tangent lines at
are drawn with green lines. Intersection
points are marked with blue symbols.
D.4. Choice of the Parameter
The last issue
to be discussed concerns the choice of an invariant circle which neither
intersects
nor .
This choice is based on the values and of the parameter given in (D.4), which correspond to invariant
circles that are tangent to and ,
and on the position of the straight line relative to the curves .
As discussed before, a vortex of kind with respect to the first property has and ,
while for a vortex of kind the relations and are verified. In both the above cases, the
inequality follows (see Figure 10(a)): it is verified for
any and, in particular, for the mean ,
which is the value of the parameter adopted in the paper. As a consequence, the
branch of the inverse map used for vortices of kind satisfies for any the inequality ,
while the branch for vortices of kind verifies the inequality .
Consider now the vortices which are of kind with respect to the first property. If and ,
the roots (D.4) verify (D.3) with the upper sign: both circles , include .
Moreover, for any belonging to the interval ,
as shown in Figure 10(a). In the same interval, the inequality is verified: any invariant circle ,
with ,
includes and does not
intersect
.
As before, the mean value of the parameter is selected and the corresponding branch of
the inverse satisfies the inequality ,
for any .
If and ,
the roots (D.4) verify (D.3) with the lower sign: both circles and include .
The inequality is also verified for any ,
it follows that ,
so that each invariant circle does not intersect
.
The mean value of the parameters and is still adopted and the corresponding branch
of the inverse verifies the inequality ,
for any .
Finally, consider the vortices such that ,
having and .
The invariant circle includes ,
while the other one includes .
Moreover, for any and for any (see Figure 10(b)): as a consequence, the
inequalities for and for hold. In the first case, () includes and does not
intersect
,
while in the other range of the parameter () includes and does not
intersect
.
In the paper, the value of the parameter is adopted, leading to a branch of the inverse
map which satisfies the inequality ,
for any .
E. Some Properties of the Map
In the present appendix, the conditions under which
the function (3.2) maps the points of a given circle for ( is the center and the radius) in points lying on another circle
(with center on and radius ) are investigated. The map has the form with , , , , being complex constants. Conditions among the
coefficients , ,
the rescaled ones , and the radius (which is assumed different from and ) have to be specified. As it
results from the following analysis, the center and the radius of the image circle will be also defined.
In order to map points of the circle in points on another circle, the quantity must be (positive and)
independent of the position .
In (E.2), the rescaled coefficient is also used, with being defined as .
The functions and are the following polynomials of second degree
in : The right-hand side of (E.2)
results to be independent of if the zeros of the polynomial are in , and the ones of the polynomial in , .
In terms of the quantity ,
these conditions on and lead to the following two relations: with the condition giving the same coefficient as the one ,
as it occurs for .
The last two sides in the above formula give a constraint on the center (through the quantities and ) and the radius of the circle in the -plane. By conjugating both sides of the
following equation in : a linear system in and is obtained, the solution of which implies
that the following complex quantity: must be real and positive, with and .
Moreover, the squared radius must also be equal to the above quantity. The quantity (E.6) must be real and positive, or is not parallel to and ,
or is parallel to .
The first case is not possible: it implies and the corresponding quantity (E.4) becomes singular. It follows that must be parallel to .
Hereafter, and will indicate the components of and in direction of their sum .
Apices are needed to distinguish (, ) by : actually may be or ,
if the difference between the arguments of and of vanishes or holds ,
respectively. Real quantities and depend on the parameters in (E.1) by the
relations (with the above convection on the apex) and .
Then the radius must satisfy the relation: which leads through the forms of (E.4) to the position of the center of the
image circle: as well as to its radius:
In the present case, we have to consider the images
through the map of the circles of the first family. The center
position assumes the value (3.9) and and become by using the quantities (6.1). As
discussed before, the vectors and must be parallel. Indeed, a direct computation
by using the quantities (E.10) gives which shows that and have just the same argument (, ). The use of (E.10) and (E.11) into the
constraint for (E.6) leads to the radius (3.9) of a circle of the first family
(a similar behaviour is found also for each
invariant circle of the second family, having center on the point (): equation (E.7) still gives the corresponding
radius (3.10)) with
parameter .
It follows that points on an invariant circle of the first family goes with the
map on another circle in the physical plane. By
using the real quantities , ,
it is found that the center of such a circle lies on the point
(E.8): In order to obtain a more useful
form of the position of the center, the conjugates and are written in terms of the corresponding
quantities and from the definitions of and ,
so that the above form of the position of the center becomes where the function has the following form: Finally, the radius (E.9) of the
image circle is written as
Acknowledgments
The second author
acknowledges the support of the Ph.D. in “Scienze e tecnologie
aerospaziali” of the Second University of Naples. The authors acknowledge
an anonymous reviewer for his helpful suggestions, which have deeply enhanced
this work.