The influence of incident and reflected potential wave on the point vortex evolution near a solid surface
The influence of oscillation potential modes upon the point vortex movement near a solid wall is studied. The equations describing the point vortex movement in the given field of a potential wave in the presence of a solid wall are obtained. The character of the vortex movement is shown to change qu...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | The influence of incident and reflected potential wave on the point vortex evolution near a solid surface / K.N. Kulik, A.V. Tur, V.V. Yanovsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 280-284. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1109722017-01-08T03:02:59Z The influence of incident and reflected potential wave on the point vortex evolution near a solid surface Kulik, K.N. Tur, A.V. Yanovsky, V.V. Nonlinear dynamics The influence of oscillation potential modes upon the point vortex movement near a solid wall is studied. The equations describing the point vortex movement in the given field of a potential wave in the presence of a solid wall are obtained. The character of the vortex movement is shown to change qualitatively under the influence of a potential wave. All the possible modes of vortex movement under the influence of a potential wave are analyzed. Вивчена дія потенційних мод коливань на рух точкового вихору поблизу твердої стінки. Отримані рівняння, які описують рух точкового вихора у заданому полі потенційної хвилі у наявності твердої стінки. Показано, що характер руху вихору під дією потенційної хвилі змінюється. Проаналізовані всі можливі режими руху вихору під дією потенційної хвилі. Изучено влияние потенциальных мод колебаний на движение точечного вихря вблизи твердой стенки. Получены уравнения, описывающие движение точечного вихря в заданном поле потенциальной волны при наличии твердой стенки. Показано, что характер движения вихря под влиянием потенциальной волны качественно меняется. Проанализированы все возможные режимы движения вихря под воздействием потенциальной волны. 2007 Article The influence of incident and reflected potential wave on the point vortex evolution near a solid surface / K.N. Kulik, A.V. Tur, V.V. Yanovsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 280-284. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 05.45.-a http://dspace.nbuv.gov.ua/handle/123456789/110972 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Nonlinear dynamics Nonlinear dynamics |
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Nonlinear dynamics Nonlinear dynamics Kulik, K.N. Tur, A.V. Yanovsky, V.V. The influence of incident and reflected potential wave on the point vortex evolution near a solid surface Вопросы атомной науки и техники |
| description |
The influence of oscillation potential modes upon the point vortex movement near a solid wall is studied. The equations describing the point vortex movement in the given field of a potential wave in the presence of a solid wall are obtained. The character of the vortex movement is shown to change qualitatively under the influence of a potential wave. All the possible modes of vortex movement under the influence of a potential wave are analyzed. |
| format |
Article |
| author |
Kulik, K.N. Tur, A.V. Yanovsky, V.V. |
| author_facet |
Kulik, K.N. Tur, A.V. Yanovsky, V.V. |
| author_sort |
Kulik, K.N. |
| title |
The influence of incident and reflected potential wave on the point vortex evolution near a solid surface |
| title_short |
The influence of incident and reflected potential wave on the point vortex evolution near a solid surface |
| title_full |
The influence of incident and reflected potential wave on the point vortex evolution near a solid surface |
| title_fullStr |
The influence of incident and reflected potential wave on the point vortex evolution near a solid surface |
| title_full_unstemmed |
The influence of incident and reflected potential wave on the point vortex evolution near a solid surface |
| title_sort |
influence of incident and reflected potential wave on the point vortex evolution near a solid surface |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Nonlinear dynamics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110972 |
| citation_txt |
The influence of incident and reflected potential wave on the point vortex evolution near a solid surface / K.N. Kulik, A.V. Tur, V.V. Yanovsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 280-284. — Бібліогр.: 15 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T01:26:16Z |
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2025-07-08T01:26:16Z |
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| fulltext |
THE INFLUENCE OF INSIDENT AND REFLECTED POTENTIAL WAVE
ON THE POINT VORTEX EVOLUTION NEAR A SOLID SURFACE
K.N. Kulik, A.V. Tur1, and V.V. Yanovsky
Institute for Single Crystals, National Academy of Sciences of Ukraine,
60, Lenin Av. 61001, Kharkiv, Ukraine,
e-mail: koskul@isc.kharkov.ua;
e-mail: yanovsky@isc.kharkov.ua;
1Centre d'Étude Spatiale des Rayonnements, C.N.R.S.-U.P.S., 9, avenue Colonel-Roche
31028 Toulouse CEDEX 4
The influence of oscillation potential modes upon the point vortex movement near a solid wall is studied. The
equations describing the point vortex movement in the given field of a potential wave in the presence of a solid wall
are obtained. The character of the vortex movement is shown to change qualitatively under the influence of a poten-
tial wave. All the possible modes of vortex movement under the influence of a potential wave are analyzed.
PACS: 05.45.-a
1. INTRODUCTION
In all the hydrodynamic spheres there are two types
of extremely important objects. They are wave and vor-
tex. Both objects are claimed to explain a lot of phe-
nomena in hydrodynamic media. The problem is that
vortexes, unlike waves, are the solutions of nonlinear
hydrodynamic equations. Linear approximation for vor-
texes has little content. This causes a number of obsta-
cles for their introduction and studying. The latter gets
even more complicated because the fruitful idealization
of nonlinear waves – „one-dimensionalization” – does-
n't work for vortexes. Vortexes are inevitably many-
dimensional. The closest to quasiparticles and well-
studied vortexes appear in two-dimensional, ideal hy-
drodynamics. They are point vortexes. The point vor-
texes movement equations in the Hamiltonian form
were obtained by Kirchhoff. The evolution of interact-
ing point vortexes is studied rather fully. Two vortexes
movement was studied as early as in the works (see, for
example, [1]). Three point vortexes evolution was ana-
lyzed in detail in the papers [2,3] (see also [4,5]). Non-
integrability of four or more vortexes problem, in case
of general position and chaos initiation, is proved in the
papers [6-9]. The studying of liquid boundaries was
started rather long ago by Helmholtz. Today the proper-
ties of waves, on the one hand, and vortexes, on the
other hand, have been studied rather deeply. But in hy-
drodynamic media waves and vortexes are usually pre-
sent simultaneously. So the research on the mutual in-
fluence of the main hydrodynamic objects is of great
importance. The base of such influence studies is made
in Lighthill works [11,12]. They explore the potential
waves generation by vortex movements. The research of
the reverse influence of potential waves on the vortexes
evolution has been started comparatively recently
[14,13]. It has turned out that the potential waves cause
the character of point vortexes evolution to change
qualitatively. The example of it can be the phenomenon
of vortexes (with the same vorticity sign) collapse under
the influence of potential oscillations, even with small
amplitude [14,13]. Such a collapse of point vortexes
with shared vorticity sign is impossible without any
potential oscillations.
The paper studies the influence of oscillation poten-
tial modes upon the point vortex movement near a solid
wall. It is shown that the character of the vortex move-
ments changes qualitatively. All the possible modes of
vortex movement under the influence of a potential
wave are analyzed.
2. VORTEX MOVEMENT EQUATIONS
IN THE FIELD OF A POTENTIAL WAVE
NEAR A WALL
Let us first discuss the velocity field of a potential
wave in the presence of a solid wall in a compressed
liquid. Let us consider the potential waves amplitudes
small. In this case, there is a potential mode in the com-
pressed liquid – sound waves (see, for example, [15]).
In the presence of a wall two the most interesting sta-
tionary situations of sound wave spreading are realized.
They are its spreading parallel to the wall and falling to
the media boundary at some angle. In this paper we
shall consider the field of falling and reflected waves
from the wall as the most general case. Let the liquid
occupy half-space , and the impermeable boundary
. Without loss of generality, let us consider poten-
tial waves spreading in the plane .
>0y
= 0y
( , )x y
Then the velocity field potential is determined by
the falling and reflected wave from the solid boundary
and has the following form. = 0y
( )0= cos cos(s y xa yk k xϕ − Ω )t (1)
The angle α of the wave falling on the medium
boundary and . tan( = /y xk k)α = ckΩ
Now let us consider the movement equations for a
point vortex, which is situated near the wall under the
influence of the given potential wave. To derive these
equations, we use the approach, developed in [13,14]. It
is generally known that the vortex is trapped in the liq-
uid, and so, the vortex velocity coincides with the liquid
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 280-284. 280
velocity in the point of its location, which
means that
( , )V Vx y
1
1 1
= = =
dx
Vx x x y ydt
,
1
1 1
=
= =
dy
Vy x x y ydt
.
Here is the vortex location in the positive half-
space . According to Helmholtz theorem, the ve-
locity field can be split to a sum of vortex and potential
components . The potential component
consists of a given external flow and waves induced by
the vortex movements. But the contributions connected
with the induced potential waves are proportional to
square Mach number (see. [11]) and can be neglected in
the general approximation [13,14].
1 1( , )x y
> 0y
p
y
= vV v v+
The vortex velocity field for a single vortex near a
solid wall is well known [10] v , and the
potential component of the velocity (the ve-
locity potential is shown above). Then the equations of
the vortex movement in the field of the given falling
and reflected wave (1) can be written in the following
form
= ( /4 , 0)v πΓ
=pv ϕ∇
( )= 1 cos sin(
dX
Y X
d Y
δ
ε
τ
− − ) (2)
( )= sin cos( )
dY
Y X
dτ
∆ (3)
where dimensionless variables τ , are
introduced and the transition to moving base is made
. In this case, dimensionless parameters
= tΩ = yY yk
=X k xx τ−
0
2
=
a kxε
Ω
, 0
2
=
a ky
∆
Ω
and =
4
x yk k
ckπ
Γ
δ , which charac-
terize the wave velocity components amplitudes and the
vortex intensity respectively, appear.
In the absence of potential oscillations, this equa-
tions system is reduced to the well-known vortex
movement equations near a solid wall (see, for example,
[10]). The character of vortex movement in this case is
very simple. The vortex moves along the wall preserv-
ing the distance to it and with constant speed
, depending on the vorticity value and
the initial distance to the wall. Using the equations ob-
tained, let us pass to the analysis of the possible vortex
movement modes under the influence of a potential
wave near a solid wall.
0= /4Vv πΓ y Γ
3. VORTEX MOVEMENT MODES
IN THE FIELD OF FALLING
AND REFLECTED SOUND WAVES
Let us qualitatively analyze the equations system
(2), (3). The coordinates of fixed points are defined by
the zeros of the right parts in these equations
( )1 cos sin( ) =Y X
Y
δ
ε− − 0 , (4)
( )sin cos( ) = 0Y X∆ . (5)
The equation (5) has two types of solutions and, so,
there appear two sets of fixed points. We shall mark
their coordinates by indices and . The fixed points
form a point system, which is periodical along axis
with the period . This means the periodicity of the
phase portraits for the system (2), (3) along axis with
the period .
A B
x
2π
x
2π
In case of , coordinates of the fixes points are A x
=
2
X kA
π∗ ± π (6)
here . = 0,1, 2,k …
From the equation (5), we can also find coordinates
of the fixed points B
=Y nB π∗ ± , (7)
where . It is natural that only fixed points
with the coordinates , situated in the part filled
with liquid, have physical meaning. So, in the case of
one should limit oneself with only Y n .
= 1, 2,n …
> 0y
B =B π∗
0 5 10 15 20 δ25
0
5
10
15
Y
A
BBBB
Fig. 1. The position of fixed points and along
axis at different values of δ and fixed value of
. The solid lines on the diagram correspond to
type fixed points, and dotted line – to type points.
The dotted line corresponds to two fixed points, be-
cause they have the same coordinates, but different
coordinates. It is easy to see that the change of the
fixed points number with δ increase is connected with
the birth and annihilation of the fixed points
A B
y
ε
A
x
= 0.5
B
B
y
The location of the fixed points along axis is de-
termined by the equation
A y
= 1 cos( )YAkYA
δ
α ε ∗−∗ , (8)
where
2
= sin( ) = ( 1)k
kkπ
α π± −
A
ε
A
δ ε
. The number of the
solutions of this equation and, so, the number of fixed
points per period, essentially depends on the pa-
rameters and δ . In Fig. 1 it is shown how the num-
ber of fixed type points changes with the change of
parameter and fixed . In fact, the figure can be
281
regarded as a bifurcational diagram at changing. The
bifurcations leading to the change of fixed points num-
ber are the birth or annihilation of fixed point and
pairs. Analyzing the equation (8), one can find areas
with different numbers of type fixed points in the
parameter plane.
δ
A B
B
A
a022
b022
c110
c202
d202
( ,ε δ
A
,AX Y∗
sin(∗ ∗
2Y
A
δ
∆ −
∗
Y
) <Aε
in(α
∆
B
)
∗
2 2os− ) =
Analogously, the equation defining the fixed
points location and number, has the form
sin( ) =
n
X nB n
δ π
α
π ε
−∗ , (9)
where . This equation has solutions at
fixed if the following inequality is true:
= cos( )nnα
n
π
1 /− |nδ π ε≤
B
| . These conditions at different val-
ues, define the areas on the parameter plane with differ-
ent number of fixed points on a period of the phase
portrait. The locations of the fixed points on Y axis
at different values of δ and fixed , are also shown in
Fig. 1.
n
B
ε
π 2π 3π δ
ε
0.8
0.6
0.4
0.2
0
a110
a044
a132
a220
a154
a`132
b220
b
1
3
2
a066
b
1
5
4
b110
b044
d110
d220
d514
d312
d`312
d404
d606
c220
c312
c514
c404
-0.2
-0.4
-0.6
-0.8
-1
π 2π 3π δ
ε
0.8
0.6
0.4
0.2
0
c110
c202
c404
c312
c220
c514
c`321
d220
d
3
1
2
c606
d
5
1
4
d110
d202
d404
a110
a220
a154
a132
a`132
a044
a606
b110
b220
b1322
b154
a022
b022
b044
-0.2
-0.4
-0.6
-0.8
-1
Fig. 2. The spaces of parameters , defining the number and type of fixed points in the phase portraits, are
shown. On the left there are the parameter spaces at , on the right — at . The first digit in the space
number denotes the number of fixed points with hyperbolic type, the second one shows the number of elliptic
fixed points , and the third digit means the number of hyperbolic fixed points for numbers starting with and
or the number of nodes for numbers starting from and . Phase portraits with the same numbers, but differ-
ent letters differ one from another with their shift along axis on a half-period equal to π , for example,
, c at π shift
)
> 0∆
c
< 0∆
BA
d
a
b d
x
a b↔ ↔
Now let us discuss the types of fixed points. The
characteristic equation for fixed points has
the form
( )A
∗
.)A
2 = sin( )A k A k
Y
λ α α ε
A
It is easy to
see that fixed points can be only elliptic (in case
sin( ) sin( 02Y Yk A k
YA
δ
α α∗ ∗∆ −
∗ is true) or hyperbolic
(at sin( ) s ) > 02Y Yk A k A
YA
δ
α ε∗∆ −
∗
). Parameter is
included in these condition trivially, and only its sign
matters. When the sign of changes, hyperbolic fixed
points turn into elliptic ones, and elliptic — into hyper-
bolic without changing their position in the phase space.
∆
The type of fixed points is defined analogously
by the following characteristic equation
cos( )( c ( 0X Xn B Bλ α λ ε ε∗ ∗− ∆ − ∆
the solutions of which have the form:
cos( )
= [ ( ) (
2
X B
nBλ α ε± ∆ − ± ∆ + ,
∗
)]ε
whence and points have hy-
perbolic type, and in case of or
the fixed points are stable and unstable
nodes. In this case the influence of is more essential.
Summing up the information about the number of fixed
points on a period and about their types, one can plot
the parameter spaces for any phase portraits types real-
ized in this mode (see Fig. 2). All the parameter space is
divided into an infinite number of areas with different
phase portraits types. A part of these areas, for com-
paratively simple phase portraits, is shown in Fig. 2.
The choice of spaces is defined by physically sensible
restrictions and not too large δ (for example,
). Plotting all the rest of the spaces and
their geometric position can be easily continued on all
the parameter plane. The areas numeration is chosen
according to the type and number of the fixed points
existing in the phase portrait at these values of parame-
ters.
> 0, > 0ε∆
< 0
| | 1ε ≤
| ε≤
< 0, < 0ε∆
∆
< 0, > 0ε∆
> 0,ε∆
|1 /4δ π−
282
Let us now describe the modes of vortex movement
in the moving base. Of course, the complexity of the
phase portraits and, so, that of the vortex movement
modes, increases with the increase of fixed points num-
ber on a period. The phase space division with sepa-
ratrices into separate cells is common. The number of
such cells increases with the increase of fixed points
number on a period. In this case, only three types of
behavior can be realized in the cells. These types are
non-linear oscillations of trapped vortexes with zero
velocity of movement along axis, non-linear oscilla-
tions of passing vortexes with nonzero average velocity
of motion along axis and nonlinear relaxation into a
stable node inside the cell. The last behavior type is
unusual. During the relaxation process, the memory
about the initial vortex state is entirely lost. Such a be-
havior is typical for dissipative systems. In (Fig.3) sim-
ple examples of phase portraits, characteristic for differ-
ent parameter spaces, are shown.
x
x
Phase portraits in Fig. 3 on the left, are plotted at
and , and on the right – at
and ∆ ; they have a certain
symmetry.
> 0, > 0ε∆
< 0, > 0ε∆
< 0, < 0ε∆
> 0, < 0ε
0
420-2-4-6
Y
X
1
0.5
0
420-2-4-6
Y
X
1
0.5
4
3
2
1
0
420-2-4-6
Y
X
4
3
2
1
0
420-2-4-6
Y
X
4
3
2
1
0
420-2-4-6
Y
X
4
3
2
1
0
420-2-4-6
Y
X
8
6
4
2
0
420-2-4-6
Y
X
8
6
4
2
0
420-2-4-6
Y
X
Fig. 3. In the left figure (top-down) one can see the
typical phase portraits for values of parameters belong-
ing to the spaces , ε , ),
δ , , ), δ ,
, ), δ , , ).
On the right c (δ , , ),
δ , , ), δ ,
, ), = 3δ , ,
)
110a ( = 1δ = 0.4 = 0.5∆
220a ( = 1.5 = 0.ε
.
5 ∆
2a (
= 3
5
20
= 0.5
= 3 ε
ε
= 0.5∆ −
2c (
132
= 0.4
= 0.4
312c (
a ( =1.2
= 0.5∆
= 0.5∆ −
= 1.2
= 0.4ε
= 0.ε
220
= 0.ε
= 0∆ −
7 = 0∆
c ( = 4
7
.5
5 02
110
= 0.ε
= 0.5∆ −
It is easy to notice, that they turn into each other in
case of replacement of hyperbolic points with elliptic
ones and vice versa. Let us start from the description of
the phase portraits shown in the left figure. We can see
that the phase portraits are periodical along axis with
period and consist of cells, limited with separatrices
of hyperbolic fixed points. Inside these cells there are
elliptic fixed points. The vortex, whose initial coordi-
nates are in such a cell, is trapped and starts nonlinear
oscillations lengthwise and crosswise near the elliptic
point. If the initial location of the vortex is higher of
lower than the cell, then the vortex moves along the
boundary, making nonlinear oscillations crosswise.
x
2π
The phase portraits on the right (see Fig. 3) also
consist of cells. On the two top phase portraits on the
right such cells are analogous to those on the left, so the
vortex movement modes are analogous. On the two
bottom phase portraits on the right, there appear cells of
a different type. In the vertexes of these cells there are
hyperbolic fixed points, and on two separatrices, joining
the vertexes, there are a stable node and an unstable
one. Such areas can be distinctly seen on the bottom
phase portrait. The appearance of the nodes means that
a vortex with the initial conditions, which belong to the
corresponding cell, will inevitable be drawn into the
stable node. This is the manifestation of one more vor-
tex movement mode — its trapping by a stable node,
followed by changing the vortex velocity to zero (in a
moving base).
At the end of this part, we shall consider the
parameter space, in which it is easy to prove the inte-
grability of the vortex movement equations. Equations
system (2), (3) can be reduced to a quasi-Hamiltonian
form
=
dX H
dt Y
∂
∆
∂
, (10)
=
dY H
dt X
ε
∂
−
∂
. (11)
The part of Hamiltonian is played by the function
. In case of pa-
rameters , the equations system (10),(11) be-
comes Hamiltonian, and its Hamiltonian does not de-
pend on time. So, according to Liouville theorem about
the integrability of Hamiltonian systems, this system is
integrated in quadratures. Let us note that under given
condition ε the integrable systems belong to the
parameter spaces ( and
. In the general case, the initial
equations system cannot be reduced to Hamiltonian
form. It is easy to understand, taking into account the
presence of modes with fixed points such as nodes,
which cannot appear in Hamiltonian systems.
H
si∆= ln( ) n( ) sin(H Y Y Yδ ε∆ − ∆ −
= ε∆
= ∆
> 0, > 0,ε δ
( < 0, > 0, < 0)ε δ ∆
)X
> 0)∆
4. CONCLUSION
In summary, we shall discuss the main qualitative
changes in vortex evolution influenced by a potential
wave. First of all, we shall note that the state of vortex
283
uniform motion with preserved distance to the wall is
easily broken under the influence of the wave. In this
case the distance changes as well as lengthwise and
crosswise (to the wall) velocities. In the moving base,
even the direction of the vortex movement can change
to the opposite. Taking into account that a point vortex
in a real liquid corresponds to a linear extended vortex,
one can expect horseshoe-shaped and more complicated
structures to appear due to the interaction of the linear
vortex with heterogeneous wave packets of sound
waves.
Now let us consider the influence of small correc-
tions, appearing due to the potential waves influence
induced by the vortex movement, on the vortex move-
ment. Basing upon the general properties of dynamic
systems, one can expect that taking the corrections into
account should make separatrices to disappear and thin
stochastic layers to appear about them. This, in its turn,
means the possibility of vortex wandering about the
overlapping stochastic layers. In the rest of phase space
the qualitative picture of vortex behavior should not
have any qualitative changes.
It is interesting to note that the modes causing the
memory loss about the initial state of the vortex, lead to
vortex component energy changes. This implies that in
wave–vortex system there can an effect appear, which is
analogous to collisionless wave attenuation in plasma.
In other words, the interaction of a potential wave with
point vortexes in the ideal liquid can lead to the change
of its amplitude. But the analysis of such effects re-
quires self-consistent description of waves and vortexes
interaction in the quasilinear theory manner, which is
beyond the subject of this work.
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ВЛИЯНИЕ ПАДАЮЩЕЙ И ОТРАЖЕННОЙ ПОТЕНЦИАЛЬНОЙ ВОЛНЫ НА ЭВОЛЮЦИЮ
ТОЧЕЧНОГО ВИХРЯ У СТЕНКИ
К.Н. Кулик, А.В. Тур, В.В. Яновский
Изучено влияние потенциальных мод колебаний на движение точечного вихря вблизи твердой стенки.
Получены уравнения, описывающие движение точечного вихря в заданном поле потенциальной волны при
наличии твердой стенки. Показано, что характер движения вихря под влиянием потенциальной волны каче-
ственно меняется. Проанализированы все возможные режимы движения вихря под воздействием потенци-
альной волны.
ВПЛИВ ПАДАЮЧОЇ ТА ВІДБИТОЇ ПОТЕНЦІЙНОЇ ХВИЛІ НА EВОЛЮЦІЮ ТОЧКОВОГО
ВИХОРА БІЛЯ СТІНКИ
К.М.Кулик, А.В.Тур, В.В.Яновський
Вивчена дія потенційних мод коливань на рух точкового вихору поблизу твердої стінки. Отримані рів-
няння, які описують рух точкового вихора у заданому полі потенційної хвилі у наявності твердої стінки.
Показано, що характер руху вихору під дією потенційної хвилі змінюється. Проаналізовані всі можливі ре-
жими руху вихору під дією потенційної хвилі.
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