Generation of the magnetic field in the compressible plasma streams
Results of the numerical simulations of the compressible plasma flows in the modified quasi-steady plasma accelerator with longitudinal magnetic field are presented. The investigations are carried out within the framework of the one-component two-dimensional MHD-model of the axial-symmetrical plasma...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2008 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2008
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| Цитувати: | Generation of the magnetic field in the compressible plasma streams / A.N. Kozlov // Вопросы атомной науки и техники. — 2008. — № 6. — С. 101-103. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859489367241785344 |
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| author | Kozlov, A.N. |
| author_facet | Kozlov, A.N. |
| citation_txt | Generation of the magnetic field in the compressible plasma streams / A.N. Kozlov // Вопросы атомной науки и техники. — 2008. — № 6. — С. 101-103. — Бібліогр.: 12 назв. — англ. |
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| description | Results of the numerical simulations of the compressible plasma flows in the modified quasi-steady plasma accelerator with longitudinal magnetic field are presented. The investigations are carried out within the framework of the one-component two-dimensional MHD-model of the axial-symmetrical plasma flows taking into account of the finite conductivity of the medium. Features of the compressible streams at an outlet from accelerator are revealed in the presence of a longitudinal field and rotation of plasma. Generation of a magnetic field on a conic shock wave is observed.
Представлено результати чисельного дослідження компресійних потоків плазми у модифікованому квазістаціонарному плазмовому прискорювачі з подовжнім магнітним полем. Дослідження проведено у рамках однорідинної двомірної МГД-моделі вісесиметричних течій плазми з урахуванням кінцевої провідності сердовища. Виявлено особливості компресіонних потоків на виході з прискорювача при наявності подовжнього поля й обертання плазми. Спостережується генерація магнітного поля на канонічній ударній хвилі
Представлены результаты численного исследования компрессионных потоков плазмы в модифицированном квазистационарном плазменном ускорителе с продольным магнитным полем. Исследования проведены в рамках одножидкостной двумерной МГД-модели осесимметричных течений плазмы с учетом конечной проводимости среды. Выявлены особенности компрессионных потоков на выходе из ускорителя при наличии продольного поля и вращения плазмы. Наблюдается генерация магнитного поля на конической ударной волне.
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GENERATION OF THE MAGNETIC FIELD IN THE COMPRESSIBLE
PLASMA STREAMS
A.N. Kozlov
Keldysh Institute for Applied Mathematics, RAS, Moscow, Russia, e-mail: ankoz@keldysh.ru
Results of the numerical simulations of the compressible plasma flows in the modified quasi-steady plasma accelerator
with longitudinal magnetic field are presented. The investigations are carried out within the framework of the one-
component two-dimensional MHD-model of the axial-symmetrical plasma flows taking into account of the finite
conductivity of the medium. Features of the compressible streams at an outlet from accelerator are revealed in the presence
of a longitudinal field and rotation of plasma. Generation of a magnetic field on a conic shock wave is observed.
PACS: 52.30.Cv, 52.59.Dk, 52.65.-y
1. INTRODUCTION
The quasi-steady plasma accelerators (QSPA) with an
azimuthal magnetic field [1,2] are the multipurpose
systems owing to an opportunity of their work both in the
accelerating and in the compressible modes. The possible
use of the accelerators as the electrojet engines assumes
the optimum organization of the accelerating modes in the
channel. Plasma is accelerated along an axis of system
due to the Ampere force [ ]Hj ,1
c
where a plasma current
j between two coaxial electrodes has mainly radial
direction. The accelerating modes answer also to
problems of generation of the high-energy plasma streams
for the various applications including thermonuclear ones.
The compressible flows at an outlet from accelerator with
the truncated central electrode of the special form allow to
receive the high values of the density and temperature in a
vicinity of a system axis due to the plasma compression.
The compressible flow modes are of interest for the
various plasma technologies.
The powerful plasma streams with a high level of
stability and azimuthal symmetry in the moving plasma
have been received in experiments on the QSPA [3-6].
The theoretical and numerical researches of processes in
the accelerators for the relatively dense plasma were
executed within the framework of the MHD-models (see,
for example, [1,2,7,8]). The history of researches of the
compressible flows in the plasma accelerators with an
azimuthal magnetic field begins with work [9].
The new direction of researches in the QSPA is connected
with an introduction in the system of an additional longitudinal
magnetic field. It leads to the rotation of plasma. The
theoretical investigations [10] have revealed that owing to a
longitudinal field it is possible to reduce essentially the
influence of the Hall effect in the accelerator channel. The
theoretical analysis of processes in the channel of the QSPA
with a longitudinal field was added by a series of the
numerical researches [11]. The main peculiarities of the
plasma flows and the estimations of the efficiency of the
acceleration process in the channel of the QSPA with a
longitudinal field are presented in [12].
The experimental research of a compression zone [6]
specifies what even the weak longitudinal magnetic field
influences essentially on a compressible plasma streams.
2. MHD-EQUATIONS IN TERMS OF VECTOR
POTENTIAL OF MAGNETIC FIELD
We consider a plasma flow in the channel between
two coaxial profiled electrodes (see Figure) and at an
outlet from the accelerator in case of when the central
electrode is shorter external one. Whole three components
of the magnetic field ( )ϕHHH rz ,,=H and velocity
( )ϕVVV rz ,,=V participate in model in the presence of a
longitudinal field and the arising rotation. To be specific,
we investigate a plasma formed from atomic hydrogen
when the inertia of electrons ( mmm ie =< < ) is
neglected. The medium is assumed to be quasi-neutral
nnn ei == . The construction of the model within
framework of the one-component approximation (
VVV == ie ) is based on the traditional MHD-equations
taking into account of the conductivity. We have the
following equations in the dimensionless form
( ) 0=•∇+ Vρ
∂
ρ∂
t ; HjV ×=∇+ P
td
dρ
(1)
2jV νερ =•∇+ P
td
d
; ( ) ( )jHVH ν
∂
∂ ×∇−××∇=
t
Here, TPPP ei ρβ=+= is the total pressure;
( )1/ −= γβε T is the intrinsic energy per unit mass;
Hj ×∇= is the electric current. We will restrict our
attention to the case of a single-temperature mixture
TTT ei =≈ . In accordance with Ohm’s law, the
electrical field is given by the relation
HVjE ×−= ν (2)
The initial quantities are related to the dimensionless
parameters of the problem as follows: 2/8 oo HPπβ =
is the ratio between the gas-kinetic and magnetic pressure
at inlet, ooo TnkP 2= ; σπν om VLc 4/Re/1 2== is
the magnetic viscosity which is inversely proportional to
the Reynolds number with the Spitzer conductivity
2/3Re Tom σ= . opo cRJH /2= , oR - the radius of
the external electrode, pJ - the discharge current,
ooo HV ρπ4/= - the characteristic velocity.
It is necessary to satisfy a condition 0=•∇ H for a
magnetic field at the numerical integration of a
multivariate MHD problem. In the case of the axial flow
symmetry ( 0/ =∂∂ ϕ ) it is possible to introduce the
vector potential A ( AH ×∇= ) so that 0=•∇ H (see,
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 6. 101
Series: Plasma Physics (14), p. .
e.g., [10-12]). The azimuthal component ϕA of a vector
A defines the components of a magnetic field
z
A
H r ∂
∂
−= ϕ
,
( )
r
Ar
r
Hz ∂
∂
= ϕ1
(3)
With allowance for above remarks, the equations (1) can
be written in the explicit form in terms of azimuthal
components ϕA and ϕH . In accordance with (3), we
obtain the transformed equations in which the right parts
contain the azimuthal component of the plasma current
rHzHj zr ∂∂−∂∂= //ϕ . The equivalent equation
for entropy ( )1/ln −= γρTS is used instead of the
equation for the internal energy (1). As a result, we have
the system of seven equations for seven variables
ϕϕρ HVVVS rz ,,,,, and ϕA .
3. BOUNDARY CONDITIONS
The conditions at the channel inlet ( )0=z correspond
to subsonic plasma inflow with the known distributions of
the density ( ) ( )rfr 1=ρ and the temperature
( ) ( )rfrT 2= . We assume that the total electric current
flowing in the system is maintained constant. This
assumption brings about the boundary condition at the
inlet for the azimuthal magnetic field 0=zj or
constrHr o ==ϕ , where LRr oo /= . The inflow is
carried out along the coordinate lines, for example. We
suppose also that the plasma at the inlet cross section
0=z is non-rotating and 0=ϕV . We specify
longitudinal field at entrance 0≠zH . In accordance with
[10], for a cold plasma 1< <β in case of the radial-
equilibrium case we have constHH o
zz == at 0=z .
Integrating (3), we obtain 2/2rHAr o
z== ϕψ .
At the outlet 1=z the boundary conditions
correspond to a free supersonic plasma flow.
We suppose that the electrodes with the given profiles
( )zrr k= and ( )zrr a= are equipotential 0=τE and
non-penetrable 0=nV surfaces. The Hall effect does not
take into account in the one-fluid model. Therefore, this
model is suitable for the qualitative description of the
both regimes of the ion and electron current transport.
It is necessary an additional relation in the presence of
longitudinal magnetic field. The equality 0=nH is the typical
and natural condition in plasmadynamics. This condition leads to
the conservation law of the magnetic flux along the channel.
On an axis of system ( 0=r ) we have the obvious
conditions: 0=rV ; 0=ϕV ; 0=ϕH ; 0=rH .
Methods of the numerical solution were described in
detail in [11].
4. RESULTS OF CALCULATIONS
A series of the numerical experiments is carried out on
the basis of the presented model. A choice of values of
the concentration on , temperature oT , discharge current
pJ and lengths of channel L correspond to the
experiments under the QSPA program [3-6]. If
314101.8 −⋅= сmno , eVTo 2= , kAJ p 300= ,
cmL 60= , cmRo 20= , the dimensionless parameters
of a problem are equal 007.0=β and .678=oσ Besides
it is necessary to set a value of a longitudinal magnetic
field at an inlet in the channel, for example, 1.0=o
zH .
Such value of 0
zH is low enough. It gives the possibility
to produce the transonic flow in the channel in accordance
with the analytical model [10,12]. The channel geometry
(see Figure) also corresponds to the analytical research of
the plasma flows. In this case the density at the inlet
varies according to the law ( ) 22 /,0 rrrz o==ρ .
Assuming also that the plasma injected into the channel is
isentropic ( constPcs v == γρ/ln ) we have 1−= γρT
at 0=z . It is possible to consider the quasi steady-state
solutions obtained by means of the relaxation method.
The compressible plasma flow in the presence of
the longitudinal magnetic field
The compressible plasma flow in the presence of a
longitudinal field 1.0=o
zH is presented in Figure under the
condition of the non-uniform inflow of plasma at the inlet
according to the analytical model. Here we have plotted: (a) the
distribution of the magnetic field H in the ( )zr, plane and
the plasma current ( )zr jj , (solid contour lines of function
ϕHr ); (b) the level lines of function ϕψ Ar= or magnetic
flux (dashed curves) and the contour lines of the azimuthal
velocity ( )zrV ,ϕ (solid curves); (c) the distribution of the
density ( )zr,ρ ; (d) the projections of the velocity vector V
onto the ( )zr, plane; and (e) the contour lines of temperature
( )rzT , . The dashed curve in Figure (a) shows that in the
central part of the accelerator channel the flow velocity increases
102
from below to above the local speed of a fast magnetosonic
wave. The level lines of function constHr =ϕ in Figure (a)
define a direction of an electric current depending on a polarity
of the electrodes. To be specific, we consider that an external
electrode is the anode. The length of vectors in Figure d is equal
to the dimensionless value of velocity at the given point. The
scale of vectors is defined by the characteristic velocity oV
indicated in the Figure.
Also as well as in the absence of a longitudinal field,
the conic shock wave is distinctly observed. There are a
break of lines of a plasma stream (Figure (d)) and of a
magnetic flux (Figure (b)) and the corresponding
spasmodic change of a tangential component of a
longitudinal magnetic field (Figure (a)). To similarly
previous case on a conic shock wave there is a spasmodic
change of the density, temperatures and pressure.
Accordingly, the temperature, density and pressure increase
on a shock wave. As a whole the compression zone
represents an region of the compressed and heat plasmas.
However, under the influence of a longitudinal
magnetic field the density and temperature noticeably
decrease on an axis of the system. According to Figure (b)
the rotation of plasma in a vicinity of a conic shock wave
occurs to rather high speed. Besides under the influence
even of a weak longitudinal field we observe the increase
in a corner between the forming of the conic surface of a
shock wave and an axis of system. This corner increases
with the increase in a longitudinal field. That
circumstance is essential also that behind a shock wave
the region with rather high values of a longitudinal field is
formed. There is a spasmodic change of the tangential
component of a magnetic field on a conic shock wave.
The presented numerical model takes into account the
finite conductivity of plasma. In spite of the fact that the
known relations on the MHD-breaks including the shock
waves answer the ideally conducting plasma, these
relations can be used for an investigated shock wave. The
analysis of processes on the shock transition has shown in
particular that according to the elementary theoretical
constructions on a shock wave we have
{ }τττ VHH |||| LR where the jump is { } RL
τττ VVV −= . In
this case we assume that the condition to the right of
break (R) belongs to the flow before a shock wave.
5. CONCLUSIONS
The peculiarities of the compressible plasma streams
at an outlet from the channel of the quasi-steady plasma
accelerators (QSPA) with a longitudinal magnetic field
are revealed. As a result of the numerical experiments, the
essential influence of a weak longitudinal field on the
compressible plasma streams is discovered. The zone of a
compression contains a conic shock wave on which there
is a sharp change of the density and temperature. The
characteristic break of lines of a plasma stream and lines
of a magnetic flux is observed on a shock wave. Under
the influence of a weak longitudinal field the plasma
starts to rotate with the rather high speed in a vicinity of
the conic shock wave. We observe a generation of a
longitudinal magnetic field with rather high values behind
a shock wave. Such parameters of plasma as the density
and temperature decrease noticeably on an axis of system
under the influence of a longitudinal field.
This work is supported by RFBR (N 06-02-16707).
REFERENCES
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2. A.I. Morozov. Introduction in Plasmadynamics.
Moscow: “Fizmatlit”, 2nd issue, 2008 (in Russian).
3. V.G. Belan, S.P. Zolotarev, V.F. Levashov,
V.S. Mainashev, A.I. Morozov, V.L. Podkoviirov,
Iu.V. Skvortsov // Fiz.Plasmy. 1990, v.16, N 2, p.96.
4. V.I. Tereshin, A.N. Bandura, O.V. Byrka,
V.V. Chebotarev, I.E. Garkusha, I. Landman,
V.A. Makhlaj, I.M. Neklyudov, D.G. Solyakov,
A.V. Tsarenko//Plasma Phys. Contr. Fusion. 2007,
v. 49, p. А231.
5. S.I. Ananin, V.M. Astashinskii, E.A. Kostyukevich,
A.A. Man’kovskii, L.Ya. Min’ko // Fiz. Plasmy.
1998, v. 24, N 11, p. 1003 (in Russian).
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v. 20, N 6, p. 533 (in Russian).
7. K.V. Brushlinsky, A.M. Zaborov, A.N. Kozlov,
A.I. Morozov, V.V. Savelyev // Fiz. Plasmy. 1990,
v. 16, N 2, p. 147 (in Russian).
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9. A.I. Morozov// Tech. Fiz. 1967, v. 37, N 12, p. 2147.
10. A.N. Kozlov // Fluid Dynamics. 2003, v. 38, p. 653.
11. A.N.Kozlov//Plasma Phys.Reports. 2006,v.32,p.378.
12. A.N. Kozlov // Plasma Physics. 2008, v. 74, p. 261.
Article received 30.09.08.
ГЕНЕРАЦИЯ МАГНИТНОГО ПОЛЯ В КОМПРЕССИОННЫХ ПОТОКАХ ПЛАЗМЫ
А.Н. Козлов
Представлены результаты численного исследования компрессионных потоков плазмы в модифицированном
квазистационарном плазменном ускорителе с продольным магнитным полем. Исследования проведены в
рамках одножидкостной двумерной МГД-модели осесимметричных течений плазмы с учетом конечной
проводимости среды. Выявлены особенности компрессионных потоков на выходе из ускорителя при наличии
продольного поля и вращения плазмы. Наблюдается генерация магнитного поля на конической ударной волне.
ГЕНЕРАЦІЯ МАГНІТНОГО ПОЛЯ У КОМПРЕСІЙНИХ ПОТОКАХ ПЛАЗМИ
А.М. Козлов
Представлено результати чисельного дослідження компресійних потоків плазми у модифікованому
квазістаціонарному плазмовому прискорювачі з подовжнім магнітним полем. Дослідження проведено у рамках
однорідинної двомірної МГД-моделі вісесиметричних течій плазми з урахуванням кінцевої провідності
сердовища. Виявлено особливості компресіонних потоків на виході з прискорювача при наявності подовжнього
поля й обертання плазми. Спостережується генерація магнітного поля на канонічній ударній хвилі
103
GENERATION OF THE MAGNETIC FIELD IN THE COMPRESSIBLE PLASMA STREAMS
A.N. Kozlov
1. INTRODUCTION
3. BOUNDARY CONDITIONS
5. CONCLUSIONS
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-110796 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T16:25:15Z |
| publishDate | 2008 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kozlov, A.N. 2017-01-06T12:59:36Z 2017-01-06T12:59:36Z 2008 Generation of the magnetic field in the compressible plasma streams / A.N. Kozlov // Вопросы атомной науки и техники. — 2008. — № 6. — С. 101-103. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 52.30.Cv, 52.59.Dk, 52.65.-y https://nasplib.isofts.kiev.ua/handle/123456789/110796 Results of the numerical simulations of the compressible plasma flows in the modified quasi-steady plasma accelerator with longitudinal magnetic field are presented. The investigations are carried out within the framework of the one-component two-dimensional MHD-model of the axial-symmetrical plasma flows taking into account of the finite conductivity of the medium. Features of the compressible streams at an outlet from accelerator are revealed in the presence of a longitudinal field and rotation of plasma. Generation of a magnetic field on a conic shock wave is observed. Представлено результати чисельного дослідження компресійних потоків плазми у модифікованому квазістаціонарному плазмовому прискорювачі з подовжнім магнітним полем. Дослідження проведено у рамках однорідинної двомірної МГД-моделі вісесиметричних течій плазми з урахуванням кінцевої провідності сердовища. Виявлено особливості компресіонних потоків на виході з прискорювача при наявності подовжнього поля й обертання плазми. Спостережується генерація магнітного поля на канонічній ударній хвилі Представлены результаты численного исследования компрессионных потоков плазмы в модифицированном квазистационарном плазменном ускорителе с продольным магнитным полем. Исследования проведены в рамках одножидкостной двумерной МГД-модели осесимметричных течений плазмы с учетом конечной проводимости среды. Выявлены особенности компрессионных потоков на выходе из ускорителя при наличии продольного поля и вращения плазмы. Наблюдается генерация магнитного поля на конической ударной волне. This work is supported by RFBR (N 06-02-16707). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma dynamics and plasma wall interaction Generation of the magnetic field in the compressible plasma streams Генерація магнітного поля у компресійних потоках плазми Генерация магнитного поля в компрессионных потоках плазмы Article published earlier |
| spellingShingle | Generation of the magnetic field in the compressible plasma streams Kozlov, A.N. Plasma dynamics and plasma wall interaction |
| title | Generation of the magnetic field in the compressible plasma streams |
| title_alt | Генерація магнітного поля у компресійних потоках плазми Генерация магнитного поля в компрессионных потоках плазмы |
| title_full | Generation of the magnetic field in the compressible plasma streams |
| title_fullStr | Generation of the magnetic field in the compressible plasma streams |
| title_full_unstemmed | Generation of the magnetic field in the compressible plasma streams |
| title_short | Generation of the magnetic field in the compressible plasma streams |
| title_sort | generation of the magnetic field in the compressible plasma streams |
| topic | Plasma dynamics and plasma wall interaction |
| topic_facet | Plasma dynamics and plasma wall interaction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110796 |
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